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 = 0.067309, 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 2706
Rule 2606
Rule 30
Rule 2611
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*}
Mathematica [A] time = 0.038978, 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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Maple [A] time = 0.058, size = 54, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{4\,da}}+{\frac{1}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{4\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08759, size = 63, normalized size = 1.7 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63615, size = 163, normalized size = 4.41 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34095, size = 78, normalized size = 2.11 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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