Evaluate the integral.

Integral from nothing to nothing StartFraction $4$ sine x plus $5$ secant x tangent x Over cosine x EndFraction dx$4$sinx$+5$secxtanxcosx$$dx

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Part $1$

A$\mathrm{.}5$ secant squared x plus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C or $5$ tangent squared x plus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C

$5$ secant squared x plus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C or $5$ tangent squared x plus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C$5$sec$2$x$+4$lncosx$+$C$$or$5$tan$2$x$+4$lncosx$+$C

B$.$one half secant squared x minus ln StartAbsoluteValue secant x EndAbsoluteValue plus Upper C or one half tangent squared x minus ln StartAbsoluteValue secant x EndAbsoluteValue plus Upper C

one half secant squared x minus ln StartAbsoluteValue secant x EndAbsoluteValue plus Upper C or one half tangent squared x minus ln StartAbsoluteValue secant x EndAbsoluteValue plus Upper C$12$sec$2$x$$lnsecx$+$C$$or$12$tan$2$x$$lnsecx$+$C

C$.$five halves tangent squared x minus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C or five halves tangent squared x plus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C

five halves tangent squared x minus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C or five halves tangent squared x plus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C$52$tan$2$x$4$lncosx$+$C$$or$52$tan$2$x$+4$lncosx$+$C

D$.$five halves secant squared x minus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C or five halves tangent squared x minus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C

five halves secant squared x minus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C or five halves tangent squared x minus $4$ ln StartAbsoluteValue cosine x EndAbsoluteValue plus Upper C$52$sec$2$x$4$lncosx$+$C$$or$52$tan$2$x$4$lncosx$+$C