Optimal. Leaf size=224 \[ -\frac{7664 \tan (c+d x)}{315 a^5 d}+\frac{31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac{3832 \tan (c+d x) \sec (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]
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Rubi [A] time = 0.539683, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2766, 2978, 2748, 3768, 3770, 3767, 8} \[ -\frac{7664 \tan (c+d x)}{315 a^5 d}+\frac{31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac{3832 \tan (c+d x) \sec (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{\int \frac{(11 a-6 a \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{\int \frac{\left (111 a^2-85 a^2 \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (947 a^3-784 a^3 \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\left (6303 a^4-5193 a^4 \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac{\int \left (29295 a^5-22992 a^5 \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{945 a^{10}}\\ &=-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}-\frac{7664 \int \sec ^2(c+d x) \, dx}{315 a^5}+\frac{31 \int \sec ^3(c+d x) \, dx}{a^5}\\ &=\frac{31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac{31 \int \sec (c+d x) \, dx}{2 a^5}+\frac{7664 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{315 a^5 d}\\ &=\frac{31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{7664 \tan (c+d x)}{315 a^5 d}+\frac{31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac{\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac{577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 6.32031, size = 507, normalized size = 2.26 \[ -\frac{496 \cos ^{10}\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)^5}+\frac{496 \cos ^{10}\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (a \cos (c+d x)+a)^5}+\frac{\sec \left (\frac{c}{2}\right ) \sec (c) \left (3057654 \sin \left (c-\frac{d x}{2}\right )-1885854 \sin \left (c+\frac{d x}{2}\right )+2644362 \sin \left (2 c+\frac{d x}{2}\right )+867048 \sin \left (c+\frac{3 d x}{2}\right )-1868436 \sin \left (2 c+\frac{3 d x}{2}\right )+1821498 \sin \left (3 c+\frac{3 d x}{2}\right )-2083537 \sin \left (c+\frac{5 d x}{2}\right )+339885 \sin \left (2 c+\frac{5 d x}{2}\right )-1456687 \sin \left (3 c+\frac{5 d x}{2}\right )+966735 \sin \left (4 c+\frac{5 d x}{2}\right )-1195641 \sin \left (2 c+\frac{7 d x}{2}\right )+46515 \sin \left (3 c+\frac{7 d x}{2}\right )-874341 \sin \left (4 c+\frac{7 d x}{2}\right )+367815 \sin \left (5 c+\frac{7 d x}{2}\right )-494579 \sin \left (3 c+\frac{9 d x}{2}\right )-31815 \sin \left (4 c+\frac{9 d x}{2}\right )-374879 \sin \left (5 c+\frac{9 d x}{2}\right )+87885 \sin \left (6 c+\frac{9 d x}{2}\right )-128187 \sin \left (4 c+\frac{11 d x}{2}\right )-18585 \sin \left (5 c+\frac{11 d x}{2}\right )-99837 \sin \left (6 c+\frac{11 d x}{2}\right )+9765 \sin \left (7 c+\frac{11 d x}{2}\right )-15328 \sin \left (5 c+\frac{13 d x}{2}\right )-3150 \sin \left (6 c+\frac{13 d x}{2}\right )-12178 \sin \left (7 c+\frac{13 d x}{2}\right )+1472562 \sin \left (\frac{d x}{2}\right )-2822886 \sin \left (\frac{3 d x}{2}\right )\right ) \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2(c+d x)}{40320 d (a \cos (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 219, normalized size = 1. \begin{align*} -{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{5}{56\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{3}{5\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{25}{8\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{351}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{11}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{31}{2\,d{a}^{5}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{1}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{11}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{31}{2\,d{a}^{5}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13478, size = 339, normalized size = 1.51 \begin{align*} -\frac{\frac{5040 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} - \frac{2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac{78120 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac{78120 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{5040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74552, size = 807, normalized size = 3.6 \begin{align*} \frac{9765 \,{\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \,{\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (15328 \, \cos \left (d x + c\right )^{6} + 66875 \, \cos \left (d x + c\right )^{5} + 112119 \, \cos \left (d x + c\right )^{4} + 87440 \, \cos \left (d x + c\right )^{3} + 28828 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right ) - 315\right )} \sin \left (d x + c\right )}{1260 \,{\left (a^{5} d \cos \left (d x + c\right )^{7} + 5 \, a^{5} d \cos \left (d x + c\right )^{6} + 10 \, a^{5} d \cos \left (d x + c\right )^{5} + 10 \, a^{5} d \cos \left (d x + c\right )^{4} + 5 \, a^{5} d \cos \left (d x + c\right )^{3} + a^{5} d \cos \left (d x + c\right )^{2}\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 = 1.43188, size = 231, normalized size = 1.03 \begin{align*} \frac{\frac{78120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac{78120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac{5040 \,{\left (11 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2} a^{5}} - \frac{35 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 450 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15750 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{5040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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