Optimal. Leaf size=186 \[ -\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(15 A-2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}-\frac{(9 A-4 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3 d}+2 a^4 x (2 A+3 C)+\frac{2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d} \]
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Rubi [A] time = 0.607624, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3044, 2975, 2976, 2968, 3023, 2735, 3770} \[ -\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(15 A-2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}-\frac{(9 A-4 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3 d}+2 a^4 x (2 A+3 C)+\frac{2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2975
Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x))^4 (4 a A-a (3 A-2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 a}\\ &=\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x))^3 \left (a^2 (13 A+2 C)-a^2 (15 A-2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac{(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}+\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x))^2 \left (3 a^3 (13 A+2 C)-4 a^3 (9 A-4 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=-\frac{(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac{(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x)) \left (6 a^4 (13 A+2 C)-30 a^4 (A-2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac{(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac{(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int \left (6 a^5 (13 A+2 C)+\left (-30 a^5 (A-2 C)+6 a^5 (13 A+2 C)\right ) \cos (c+d x)-30 a^5 (A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}-\frac{(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac{(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int \left (6 a^5 (13 A+2 C)+24 a^5 (2 A+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=2 a^4 (2 A+3 C) x-\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}-\frac{(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac{(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a^4 (13 A+2 C)\right ) \int \sec (c+d x) \, dx\\ &=2 a^4 (2 A+3 C) x+\frac{a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}-\frac{(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac{(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 6.22573, size = 756, normalized size = 4.06 \[ \frac{1}{8} x (2 A+3 C) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4+\frac{(4 A+27 C) \sin (c) \cos (d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d}+\frac{(4 A+27 C) \cos (c) \sin (d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d}+\frac{(-13 A-2 C) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{32 d}+\frac{(13 A+2 C) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4 \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{32 d}+\frac{A \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{4 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{A \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{4 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{A \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}-\frac{A \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{C \sin (2 c) \cos (2 d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{16 d}+\frac{C \sin (3 c) \cos (3 d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{192 d}+\frac{C \cos (2 c) \sin (2 d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{16 d}+\frac{C \cos (3 c) \sin (3 d x) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^4}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 190, normalized size = 1. \begin{align*}{\frac{A{a}^{4}\sin \left ( dx+c \right ) }{d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{20\,{a}^{4}C\sin \left ( dx+c \right ) }{3\,d}}+4\,A{a}^{4}x+4\,{\frac{A{a}^{4}c}{d}}+2\,{\frac{{a}^{4}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+6\,{a}^{4}Cx+6\,{\frac{{a}^{4}Cc}{d}}+{\frac{13\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+4\,{\frac{A{a}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01986, size = 285, normalized size = 1.53 \begin{align*} \frac{48 \,{\left (d x + c\right )} A a^{4} - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 12 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 48 \,{\left (d x + c\right )} C a^{4} - 3 \, A a^{4}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{4} \sin \left (d x + c\right ) + 72 \, C a^{4} \sin \left (d x + c\right ) + 48 \, A a^{4} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57148, size = 432, normalized size = 2.32 \begin{align*} \frac{24 \,{\left (2 \, A + 3 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{2} + 3 \,{\left (13 \, A + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (13 \, A + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, C a^{4} \cos \left (d x + c\right )^{4} + 12 \, C a^{4} \cos \left (d x + c\right )^{3} + 2 \,{\left (3 \, A + 20 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 24 \, A a^{4} \cos \left (d x + c\right ) + 3 \, A a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \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 = 1.29635, size = 335, normalized size = 1.8 \begin{align*} \frac{12 \,{\left (2 \, A a^{4} + 3 \, C a^{4}\right )}{\left (d x + c\right )} + 3 \,{\left (13 \, A a^{4} + 2 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (13 \, A a^{4} + 2 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{6 \,{\left (7 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, A a^{4} \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}} + \frac{4 \,{\left (3 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 38 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 27 \, C a^{4} \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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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