Optimal. Leaf size=37 \[ \frac {1}{2 d (a \sin (c+d x)+a)}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d} \]
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Rubi [A] time = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.57, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2706, 2606, 30, 2611, 3770} \[ \frac {\sec ^2(c+d x)}{2 a d}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {\tan (c+d x) \sec (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2706
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^2(c+d x) \tan (c+d x) \, dx}{a}-\frac {\int \sec (c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{2 a d}+\frac {\int \sec (c+d x) \, dx}{2 a}+\frac {\operatorname {Subst}(\int x \, dx,x,\sec (c+d x))}{a d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {\sec ^2(c+d x)}{2 a d}-\frac {\sec (c+d x) \tan (c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 28, normalized size = 0.76 \[ \frac {\frac {1}{\sin (c+d x)+1}+\tanh ^{-1}(\sin (c+d x))}{2 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 58, normalized size = 1.57 \[ \frac {{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2}{4 \, {\left (a d \sin \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 58, normalized size = 1.57 \[ \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {\sin \left (d x + c\right ) - 1}{a {\left (\sin \left (d x + c\right ) + 1\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 54, normalized size = 1.46 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{4 a d}+\frac {1}{2 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{4 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 47, normalized size = 1.27 \[ \frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {2}{a \sin \left (d x + c\right ) + a}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.66, size = 61, normalized size = 1.65 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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