Optimal. Leaf size=93 \[ \frac{8 \tan ^3(c+d x)}{105 a^2 d}+\frac{8 \tan (c+d x)}{35 a^2 d}-\frac{2 \sec ^3(c+d x)}{35 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.11545, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2859, 2672, 3767} \[ \frac{8 \tan ^3(c+d x)}{105 a^2 d}+\frac{8 \tan (c+d x)}{35 a^2 d}-\frac{2 \sec ^3(c+d x)}{35 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2859
Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}+\frac{2 \int \frac{\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{7 a}\\ &=\frac{\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac{2 \sec ^3(c+d x)}{35 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac{8 \int \sec ^4(c+d x) \, dx}{35 a^2}\\ &=\frac{\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac{2 \sec ^3(c+d x)}{35 d \left (a^2+a^2 \sin (c+d x)\right )}-\frac{8 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 a^2 d}\\ &=\frac{\sec ^3(c+d x)}{7 d (a+a \sin (c+d x))^2}-\frac{2 \sec ^3(c+d x)}{35 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac{8 \tan (c+d x)}{35 a^2 d}+\frac{8 \tan ^3(c+d x)}{105 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.301531, size = 134, normalized size = 1.44 \[ -\frac{\sec ^3(c+d x) \left (-56 \sin (c+d x)+3 \sin (2 (c+d x))-12 \sin (3 (c+d x))+\frac{3}{2} \sin (4 (c+d x))+4 \sin (5 (c+d x))+\frac{21}{4} \cos (c+d x)+32 \cos (2 (c+d x))+\frac{9}{8} \cos (3 (c+d x))+16 \cos (4 (c+d x))-\frac{3}{8} \cos (5 (c+d x))-84\right )}{420 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 160, normalized size = 1.7 \begin{align*} 4\,{\frac{1}{d{a}^{2}} \left ( -1/48\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-3}-1/32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-2}-1/16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}+1/7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-7}-1/2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-6}+{\frac{9}{10\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}+{\frac{35}{48\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{11}{32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+1/16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16389, size = 508, normalized size = 5.46 \begin{align*} \frac{2 \,{\left (\frac{36 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{132 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{68 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{14 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{140 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{140 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{105 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 9\right )}}{105 \,{\left (a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61269, size = 267, normalized size = 2.87 \begin{align*} \frac{32 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 25}{105 \,{\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\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 = 1.19218, size = 197, normalized size = 2.12 \begin{align*} -\frac{\frac{35 \,{\left (6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 175 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 910 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 756 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 427 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 31}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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