Optimal. Leaf size=176 \[ -\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{32 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rubi [A] time = 0.266218, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2835, 2611, 3768, 3770, 2607, 14} \[ -\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{32 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^5(c+d x) \tan ^6(c+d x) \, dx}{a}-\frac{\int \sec ^4(c+d x) \tan ^7(c+d x) \, dx}{a}\\ &=\frac{\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac{\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{2 a}-\frac{\operatorname{Subst}\left (\int x^7 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac{\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}+\frac{3 \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx}{16 a}-\frac{\operatorname{Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac{\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{\int \sec ^5(c+d x) \, dx}{32 a}\\ &=-\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac{\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{3 \int \sec ^3(c+d x) \, dx}{128 a}\\ &=-\frac{3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac{\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{3 \int \sec (c+d x) \, dx}{256 a}\\ &=-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac{\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac{\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^{10}(c+d x)}{10 a d}\\ \end{align*}
Mathematica [A] time = 2.76586, size = 122, normalized size = 0.69 \[ -\frac{30 \tanh ^{-1}(\sin (c+d x))-\frac{2 \left (15 \sin ^8(c+d x)+15 \sin ^7(c+d x)-55 \sin ^6(c+d x)+265 \sin ^5(c+d x)+137 \sin ^4(c+d x)-183 \sin ^3(c+d x)-113 \sin ^2(c+d x)+47 \sin (c+d x)+32\right )}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}}{2560 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 198, normalized size = 1.1 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}+{\frac{1}{64\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{9}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{1}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,da}}-{\frac{1}{160\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{7}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5}{128\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03646, size = 289, normalized size = 1.64 \begin{align*} \frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{8} + 15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{6} + 265 \, \sin \left (d x + c\right )^{5} + 137 \, \sin \left (d x + c\right )^{4} - 183 \, \sin \left (d x + c\right )^{3} - 113 \, \sin \left (d x + c\right )^{2} + 47 \, \sin \left (d x + c\right ) + 32\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, 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}}{2560 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56189, size = 518, normalized size = 2.94 \begin{align*} \frac{30 \, \cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 124 \, \cos \left (d x + c\right )^{4} - 112 \, \cos \left (d x + c\right )^{2} - 15 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (15 \, \cos \left (d x + c\right )^{6} - 310 \, \cos \left (d x + c\right )^{4} + 392 \, \cos \left (d x + c\right )^{2} - 144\right )} \sin \left (d x + c\right ) + 32}{2560 \,{\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\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 = 1.33182, size = 211, normalized size = 1.2 \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{5 \,{\left (25 \, \sin \left (d x + c\right )^{4} - 84 \, \sin \left (d x + c\right )^{3} + 66 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) - 3\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{137 \, \sin \left (d x + c\right )^{5} + 885 \, \sin \left (d x + c\right )^{4} + 2270 \, \sin \left (d x + c\right )^{3} + 2470 \, \sin \left (d x + c\right )^{2} + 1265 \, \sin \left (d x + c\right ) + 253}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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