Optimal. Leaf size=150 \[ \frac{\sec ^8(c+d x)}{8 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.224721, antiderivative size = 150, 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, 2606, 14, 2611, 3768, 3770} \[ \frac{\sec ^8(c+d x)}{8 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2606
Rule 14
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^6(c+d x) \tan ^3(c+d x) \, dx}{a}-\frac{\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{a}\\ &=-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}+\frac{3 \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^5 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac{\int \sec ^5(c+d x) \, dx}{16 a}+\frac{\operatorname{Subst}\left (\int \left (-x^5+x^7\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{8 a d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac{3 \int \sec ^3(c+d x) \, dx}{64 a}\\ &=-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{8 a d}-\frac{3 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac{3 \int \sec (c+d x) \, dx}{128 a}\\ &=-\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{\sec ^8(c+d x)}{8 a d}-\frac{3 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 0.878819, size = 92, normalized size = 0.61 \[ -\frac{-\frac{9}{\sin (c+d x)-1}+\frac{3}{(\sin (c+d x)-1)^2}+\frac{6}{(\sin (c+d x)+1)^2}+\frac{4}{(\sin (c+d x)-1)^3}+\frac{8}{(\sin (c+d x)+1)^3}-\frac{6}{(\sin (c+d x)+1)^4}+9 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 144, normalized size = 1. \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}-{\frac{1}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}+{\frac{3}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{3\,\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}{48\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,\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 = 1.26131, size = 236, normalized size = 1.57 \begin{align*} \frac{\frac{2 \,{\left (9 \, \sin \left (d x + c\right )^{6} + 9 \, \sin \left (d x + c\right )^{5} - 24 \, \sin \left (d x + c\right )^{4} - 24 \, \sin \left (d x + c\right )^{3} - 57 \, \sin \left (d x + c\right )^{2} + 7 \, \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{9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{9 \, \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 = 1.5533, size = 451, normalized size = 3.01 \begin{align*} \frac{18 \, \cos \left (d x + c\right )^{6} - 6 \, \cos \left (d x + c\right )^{4} - 156 \, \cos \left (d x + c\right )^{2} - 9 \,{\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 ) + 9 \,{\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 (9 \, \cos \left (d x + c\right )^{4} + 6 \, \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 = 1.40873, size = 184, normalized size = 1.23 \begin{align*} -\frac{\frac{36 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{36 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (33 \, \sin \left (d x + c\right )^{3} - 135 \, \sin \left (d x + c\right )^{2} + 183 \, \sin \left (d x + c\right ) - 65\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{75 \, \sin \left (d x + c\right )^{4} + 300 \, \sin \left (d x + c\right )^{3} + 402 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) + 11}{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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