Optimal. Leaf size=130 \[ \frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.161036, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2611, 3770} \[ \frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^2(c+d x) \tan ^7(c+d x) \, dx}{a}-\frac{\int \sec (c+d x) \tan ^8(c+d x) \, dx}{a}\\ &=-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{7 \int \sec (c+d x) \tan ^6(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^7 \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{48 a}\\ &=-\frac{35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}+\frac{35 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{64 a}\\ &=\frac{35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \int \sec (c+d x) \, dx}{128 a}\\ &=-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac{7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac{\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac{\tan ^8(c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 1.00892, size = 101, normalized size = 0.78 \[ -\frac{\frac{279 \sin ^6(c+d x)+87 \sin ^5(c+d x)-424 \sin ^4(c+d x)-136 \sin ^3(c+d x)+249 \sin ^2(c+d x)+57 \sin (c+d x)-48}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+105 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 162, normalized size = 1.3 \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{9}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{29}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{35\,\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{5}{48\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{19}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{35\,\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 = 1.02832, size = 236, normalized size = 1.82 \begin{align*} -\frac{\frac{2 \,{\left (279 \, \sin \left (d x + c\right )^{6} + 87 \, \sin \left (d x + c\right )^{5} - 424 \, \sin \left (d x + c\right )^{4} - 136 \, \sin \left (d x + c\right )^{3} + 249 \, \sin \left (d x + c\right )^{2} + 57 \, \sin \left (d x + c\right ) - 48\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{105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{105 \, \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 = 2.05396, size = 464, normalized size = 3.57 \begin{align*} -\frac{558 \, \cos \left (d x + c\right )^{6} - 826 \, \cos \left (d x + c\right )^{4} + 476 \, \cos \left (d x + c\right )^{2} + 105 \,{\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 ) - 105 \,{\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 (87 \, \cos \left (d x + c\right )^{4} - 38 \, \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 = 8.06961, size = 184, normalized size = 1.42 \begin{align*} -\frac{\frac{420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (385 \, \sin \left (d x + c\right )^{3} - 807 \, \sin \left (d x + c\right )^{2} + 567 \, \sin \left (d x + c\right ) - 129\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{875 \, \sin \left (d x + c\right )^{4} + 1964 \, \sin \left (d x + c\right )^{3} + 1554 \, \sin \left (d x + c\right )^{2} + 396 \, \sin \left (d x + c\right ) - 21}{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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