Optimal. Leaf size=162 \[ \frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{13 \tan ^5(c+d x)}{5 a^3 d}+\frac{5 \tan ^3(c+d x)}{a^3 d}+\frac{7 \tan (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.347003, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2875, 2873, 3767, 2622, 302, 207, 2620, 270, 2606, 30} \[ \frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{13 \tan ^5(c+d x)}{5 a^3 d}+\frac{5 \tan ^3(c+d x)}{a^3 d}+\frac{7 \tan (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 3767
Rule 2622
Rule 302
Rule 207
Rule 2620
Rule 270
Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \csc ^2(c+d x) \sec ^8(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (3 a^3 \sec ^8(c+d x)-3 a^3 \csc (c+d x) \sec ^8(c+d x)+a^3 \csc ^2(c+d x) \sec ^8(c+d x)-a^3 \sec ^7(c+d x) \tan (c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \csc ^2(c+d x) \sec ^8(c+d x) \, dx}{a^3}-\frac{\int \sec ^7(c+d x) \tan (c+d x) \, dx}{a^3}+\frac{3 \int \sec ^8(c+d x) \, dx}{a^3}-\frac{3 \int \csc (c+d x) \sec ^8(c+d x) \, dx}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^2} \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \frac{x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{a^3 d}\\ &=-\frac{\sec ^7(c+d x)}{7 a^3 d}+\frac{3 \tan (c+d x)}{a^3 d}+\frac{3 \tan ^3(c+d x)}{a^3 d}+\frac{9 \tan ^5(c+d x)}{5 a^3 d}+\frac{3 \tan ^7(c+d x)}{7 a^3 d}+\frac{\operatorname{Subst}\left (\int \left (4+\frac{1}{x^2}+6 x^2+4 x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (1+x^2+x^4+x^6+\frac{1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac{\cot (c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{7 \tan (c+d x)}{a^3 d}+\frac{5 \tan ^3(c+d x)}{a^3 d}+\frac{13 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{7 \tan (c+d x)}{a^3 d}+\frac{5 \tan ^3(c+d x)}{a^3 d}+\frac{13 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}\\ \end{align*}
Mathematica [B] time = 0.848432, size = 351, normalized size = 2.17 \[ \frac{\csc ^3(c+d x) \left (-1316 \sin (c+d x)+3520 \sin (2 (c+d x))-1380 \sin (3 (c+d x))-1056 \sin (4 (c+d x))+176 \sin (5 (c+d x))-440 \cos (2 (c+d x))-2640 \cos (3 (c+d x))+846 \cos (4 (c+d x))+176 \cos (5 (c+d x))-2100 \sin (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+630 \sin (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-1575 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+105 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+14 \cos (c+d x) \left (-105 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+176\right )+1575 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-105 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2100 \sin (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-630 \sin (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-966\right )}{140 a^3 d (\sin (c+d x)+1)^3 \left (\csc ^2\left (\frac{1}{2} (c+d x)\right )-\sec ^2\left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.15, size = 224, normalized size = 1.4 \begin{align*}{\frac{1}{2\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{8}{7\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-7}}+4\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{46}{5\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}+13\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-{\frac{31}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{49}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{111}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-3\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06049, size = 533, normalized size = 3.29 \begin{align*} -\frac{\frac{\frac{934 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3854 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6566 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3556 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{3710 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{7070 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{4270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{1015 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 35}{\frac{a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{14 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{14 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{6 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac{210 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{35 \, \sin \left (d x + c\right )}{a^{3}{\left (\cos \left (d x + c\right ) + 1\right )}}}{70 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5394, size = 713, normalized size = 4.4 \begin{align*} \frac{846 \, \cos \left (d x + c\right )^{4} - 956 \, \cos \left (d x + c\right )^{2} + 105 \,{\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} -{\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 105 \,{\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} -{\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (176 \, \cos \left (d x + c\right )^{4} - 477 \, \cos \left (d x + c\right )^{2} + 15\right )} \sin \left (d x + c\right ) + 40}{70 \,{\left (a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + 4 \, a^{3} d \cos \left (d x + c\right ) -{\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\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 = 1.30927, size = 252, normalized size = 1.56 \begin{align*} -\frac{\frac{840 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{3}} - \frac{35 \,{\left (12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 17 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} a^{3}} + \frac{3885 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 19880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 45465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 57120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 41671 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 16632 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2931}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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