Optimal. Leaf size=130 \[ \frac{\sec ^8(c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan (c+d x) \sec ^7(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{48 a d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.147518, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2835, 2606, 30, 2611, 3768, 3770} \[ \frac{\sec ^8(c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan (c+d x) \sec ^7(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{48 a d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2606
Rule 30
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^8(c+d x) \tan (c+d x) \, dx}{a}-\frac{\int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=-\frac{\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac{\int \sec ^7(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^7 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{\sec ^8(c+d x)}{8 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac{\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac{5 \int \sec ^5(c+d x) \, dx}{48 a}\\ &=\frac{\sec ^8(c+d x)}{8 a d}+\frac{5 \sec ^3(c+d x) \tan (c+d x)}{192 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac{\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac{5 \int \sec ^3(c+d x) \, dx}{64 a}\\ &=\frac{\sec ^8(c+d x)}{8 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)}{192 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac{\sec ^7(c+d x) \tan (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{\sec ^8(c+d x)}{8 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)}{192 a d}+\frac{\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac{\sec ^7(c+d x) \tan (c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 0.870656, size = 92, normalized size = 0.71 \[ \frac{-\frac{15}{\sin (c+d x)-1}+\frac{9}{(\sin (c+d x)-1)^2}+\frac{6}{(\sin (c+d x)+1)^2}-\frac{4}{(\sin (c+d x)-1)^3}+\frac{8}{(\sin (c+d x)+1)^3}+\frac{6}{(\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 144, normalized size = 1.1 \begin{align*} -{\frac{1}{96\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{3}{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}{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{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 = 1.27393, size = 236, normalized size = 1.82 \begin{align*} -\frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{6} + 15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{4} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )^{2} + 33 \, \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{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 = 1.6565, size = 456, normalized size = 3.51 \begin{align*} -\frac{30 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} - 4 \, \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} + 10 \, \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 = 1.28742, size = 184, normalized size = 1.42 \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} + 321 \, \sin \left (d x + c\right ) - 167\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} + 702 \, \sin \left (d x + c\right )^{2} + 340 \, \sin \left (d x + c\right ) - 35}{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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