Optimal. Leaf size=152 \[ \frac{\tan ^8(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.244631, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2835, 2607, 14, 2611, 3768, 3770} \[ \frac{\tan ^8(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2607
Rule 14
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^4(c+d x) \tan ^5(c+d x) \, dx}{a}-\frac{\int \sec ^3(c+d x) \tan ^6(c+d x) \, dx}{a}\\ &=-\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac{5 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac{5 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{16 a}+\frac{\operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac{5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{\tan ^8(c+d x)}{8 a d}+\frac{5 \int \sec ^3(c+d x) \, dx}{64 a}\\ &=\frac{5 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{\tan ^8(c+d x)}{8 a d}+\frac{5 \int \sec (c+d x) \, dx}{128 a}\\ &=\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{5 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac{5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac{\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{\tan ^8(c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 1.1749, size = 92, normalized size = 0.61 \[ \frac{-\frac{15}{\sin (c+d x)-1}-\frac{15}{(\sin (c+d x)-1)^2}+\frac{30}{(\sin (c+d x)+1)^2}-\frac{4}{(\sin (c+d x)-1)^3}-\frac{24}{(\sin (c+d x)+1)^3}+\frac{6}{(\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 144, normalized size = 1. \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{5}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{5}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{256\,da}}+{\frac{1}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{16\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02561, size = 234, normalized size = 1.54 \begin{align*} -\frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{6} + 15 \, \sin \left (d x + c\right )^{5} + 88 \, \sin \left (d x + c\right )^{4} - 8 \, \sin \left (d x + c\right )^{3} - 63 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right ) + 16\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac{15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63004, size = 460, normalized size = 3.03 \begin{align*} -\frac{30 \, \cos \left (d x + c\right )^{6} - 266 \, \cos \left (d x + c\right )^{4} + 316 \, \cos \left (d x + c\right )^{2} - 15 \,{\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (15 \, \cos \left (d x + c\right )^{4} - 22 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \,{\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36729, size = 184, normalized size = 1.21 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (55 \, \sin \left (d x + c\right )^{3} - 225 \, \sin \left (d x + c\right )^{2} + 225 \, \sin \left (d x + c\right ) - 71\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{125 \, \sin \left (d x + c\right )^{4} + 500 \, \sin \left (d x + c\right )^{3} + 510 \, \sin \left (d x + c\right )^{2} + 212 \, \sin \left (d x + c\right ) + 29}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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