Optimal. Leaf size=156 \[ -\frac{(5 A-6 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 a d}+\frac{1}{2} a^3 x (5 A+6 C)+\frac{3 a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.396585, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {4087, 4017, 4018, 3996, 3770} \[ -\frac{(5 A-6 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 a d}+\frac{1}{2} a^3 x (5 A+6 C)+\frac{3 a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 4087
Rule 4017
Rule 4018
Rule 3996
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (3 a A-a (A-3 C) \sec (c+d x)) \, dx}{3 a}\\ &=\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x))^2 \left (2 a^2 (5 A+3 C)-a^2 (5 A-6 C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac{(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^3 A+18 a^3 C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac{(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac{\int \left (-3 a^4 (5 A+6 C)-18 a^4 C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{1}{2} a^3 (5 A+6 C) x+\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac{(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\left (3 a^3 C\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^3 (5 A+6 C) x+\frac{3 a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac{(5 A-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [B] time = 6.17475, size = 1014, normalized size = 6.5 \[ a^3 \left (-\frac{3 C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(5 A+6 C) x \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 (\cos (2 c+2 d x) A+A+2 C)}+\frac{(15 A+4 C) \cos (d x) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin (c) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 A \cos (2 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin (2 c) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (3 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin (3 c) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{48 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(15 A+4 C) \cos (c) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin (d x) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 A \cos (2 c) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin (2 d x) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (3 c) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin (3 d x) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{48 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (\cos (2 c+2 d x) A+A+2 C) \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{C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left (C \sec ^2(c+d x)+A\right ) \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (\cos (2 c+2 d x) A+A+2 C) \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 )}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 146, normalized size = 0.9 \begin{align*}{\frac{A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ){a}^{3}}{3\,d}}+{\frac{11\,A{a}^{3}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{3}C\sin \left ( dx+c \right ) }{d}}+{\frac{3\,A{a}^{3}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+{\frac{5\,{a}^{3}Ax}{2}}+{\frac{5\,A{a}^{3}c}{2\,d}}+3\,{a}^{3}Cx+3\,{\frac{C{a}^{3}c}{d}}+3\,{\frac{{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944634, size = 185, normalized size = 1.19 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 9 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 12 \,{\left (d x + c\right )} A a^{3} - 36 \,{\left (d x + c\right )} C a^{3} - 18 \, C a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a^{3} \sin \left (d x + c\right ) - 12 \, C a^{3} \sin \left (d x + c\right ) - 12 \, C a^{3} \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 = 0.530032, size = 350, normalized size = 2.24 \begin{align*} \frac{3 \,{\left (5 \, A + 6 \, C\right )} a^{3} d x \cos \left (d x + c\right ) + 9 \, C a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, C a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 9 \, A a^{3} \cos \left (d x + c\right )^{2} + 2 \,{\left (11 \, A + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 6 \, C a^{3}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\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 = 1.25528, size = 284, normalized size = 1.82 \begin{align*} \frac{18 \, C a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, C a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{12 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} + 3 \,{\left (5 \, A a^{3} + 6 \, C a^{3}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (15 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 33 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C a^{3} \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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