Optimal. Leaf size=150 \[ -\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.225859, 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, 2611, 3768, 3770, 2607, 14} \[ -\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{a}-\frac{\int \sec ^4(c+d x) \tan ^5(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,\tan (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,\tan (c+d x)\right )}{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{\tan ^6(c+d x)}{6 a d}-\frac{\tan ^8(c+d x)}{8 a d}+\frac{3 \int \sec ^3(c+d x) \, dx}{64 a}\\ &=\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{\tan ^6(c+d x)}{6 a d}-\frac{\tan ^8(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{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{\tan ^6(c+d x)}{6 a d}-\frac{\tan ^8(c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 0.632404, size = 101, normalized size = 0.67 \[ \frac{\frac{-9 \sin ^6(c+d x)-9 \sin ^5(c+d x)+24 \sin ^4(c+d x)-72 \sin ^3(c+d x)-39 \sin ^2(c+d x)+25 \sin (c+d x)+16}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+9 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 162, 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{1}{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}{24\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{64\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{32\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\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 = 1.11721, 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} + 72 \, \sin \left (d x + c\right )^{3} + 39 \, \sin \left (d x + c\right )^{2} - 25 \, \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 = 1.46116, size = 452, normalized size = 3.01 \begin{align*} -\frac{18 \, \cos \left (d x + c\right )^{6} - 6 \, \cos \left (d x + c\right )^{4} + 36 \, \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} - 90 \, \cos \left (d x + c\right )^{2} + 56\right )} \sin \left (d x + c\right ) - 16}{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 = 1.33109, 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} - 87 \, \sin \left (d x + c\right )^{2} + 39 \, \sin \left (d x + c\right ) - 1\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac{75 \, \sin \left (d x + c\right )^{4} + 396 \, \sin \left (d x + c\right )^{3} + 786 \, \sin \left (d x + c\right )^{2} + 556 \, \sin \left (d x + c\right ) + 139}{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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