Optimal. Leaf size=91 \[ -\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{3 \sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.288258, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2875, 2873, 2606, 14, 2607, 270} \[ -\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{3 \sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2606
Rule 14
Rule 2607
Rule 270
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \sec ^5(c+d x) (a-a \sin (c+d x))^2 \tan ^3(c+d x) \, dx}{a^4}\\ &=\frac{\int \left (a^2 \sec ^5(c+d x) \tan ^3(c+d x)-2 a^2 \sec ^4(c+d x) \tan ^4(c+d x)+a^2 \sec ^3(c+d x) \tan ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sec ^5(c+d x) \tan ^3(c+d x) \, dx}{a^2}+\frac{\int \sec ^3(c+d x) \tan ^5(c+d x) \, dx}{a^2}-\frac{2 \int \sec ^4(c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{3 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^7(c+d x)}{7 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.324578, size = 126, normalized size = 1.38 \[ \frac{\sec ^3(c+d x) (448 \sin (c+d x)-104 \sin (2 (c+d x))-144 \sin (3 (c+d x))-52 \sin (4 (c+d x))+48 \sin (5 (c+d x))-182 \cos (c+d x)-736 \cos (2 (c+d x))-39 \cos (3 (c+d x))+192 \cos (4 (c+d x))+13 \cos (5 (c+d x))+672)}{6720 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 130, normalized size = 1.4 \begin{align*} 16\,{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{192\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{3}}}-{\frac{1}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+1/28\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-7}-1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-6}+{\frac{7}{40\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}-1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}+{\frac{7}{192\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}+{\frac{1}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23808, size = 454, normalized size = 4.99 \begin{align*} \frac{4 \,{\left (\frac{4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{91 \, \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{105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\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 = 1.61421, size = 267, normalized size = 2.93 \begin{align*} -\frac{24 \, \cos \left (d x + c\right )^{4} - 47 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (6 \, \cos \left (d x + c\right )^{4} - 9 \, \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 = 1.28072, size = 162, normalized size = 1.78 \begin{align*} -\frac{\frac{35 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1015 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1330 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1302 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 469 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 67}{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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