Optimal. Leaf size=156 \[ \frac{a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{3 d}-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 a d}+3 a^3 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.499718, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3044, 2975, 2968, 3023, 2735, 3770} \[ \frac{a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{3 d}-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 a d}+3 a^3 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2975
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x))^3 (3 a A-a (A-3 C) \cos (c+d x)) \sec ^3(c+d x) \, dx}{3 a}\\ &=\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x))^2 \left (2 a^2 (5 A+3 C)-a^2 (5 A-6 C) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{6 a}\\ &=\frac{(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int (a+a \cos (c+d x)) \left (3 a^3 (5 A+6 C)-15 a^3 A \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac{(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int \left (3 a^4 (5 A+6 C)+\left (-15 a^4 A+3 a^4 (5 A+6 C)\right ) \cos (c+d x)-15 a^4 A \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{\int \left (3 a^4 (5 A+6 C)+18 a^4 C \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=3 a^3 C x-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} \left (a^3 (5 A+6 C)\right ) \int \sec (c+d x) \, dx\\ &=3 a^3 C x+\frac{a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 6.34054, size = 832, normalized size = 5.33 \[ \frac{3}{8} C x (\cos (c+d x) a+a)^3 \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )+\frac{(-5 A-6 C) (\cos (c+d x) a+a)^3 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d}+\frac{(5 A+6 C) (\cos (c+d x) a+a)^3 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{16 d}+\frac{C \cos (d x) (\cos (c+d x) a+a)^3 \sin (c) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 d}+\frac{C \cos (c) (\cos (c+d x) a+a)^3 \sin (d x) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{8 d}+\frac{(\cos (c+d x) a+a)^3 \left (11 A \sin \left (\frac{d x}{2}\right )+3 C \sin \left (\frac{d x}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 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{(\cos (c+d x) a+a)^3 \left (11 A \sin \left (\frac{d x}{2}\right )+3 C \sin \left (\frac{d x}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 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{(\cos (c+d x) a+a)^3 \left (5 A \cos \left (\frac{c}{2}\right )-4 A \sin \left (\frac{c}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{48 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 )^2}+\frac{(\cos (c+d x) a+a)^3 \left (-5 A \cos \left (\frac{c}{2}\right )-4 A \sin \left (\frac{c}{2}\right )\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{48 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 )^2}+\frac{A (\cos (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{48 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 )^3}+\frac{A (\cos (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{48 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 )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 152, normalized size = 1. \begin{align*}{\frac{5\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{a}^{3}C\sin \left ( dx+c \right ) }{d}}+{\frac{11\,A{a}^{3}\tan \left ( dx+c \right ) }{3\,d}}+3\,{a}^{3}Cx+3\,{\frac{C{a}^{3}c}{d}}+{\frac{3\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+3\,{\frac{{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,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 = 1.04456, size = 239, normalized size = 1.53 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 36 \,{\left (d x + c\right )} C a^{3} - 9 \, A a^{3}{\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 )} + 6 \, A a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 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 )} + 12 \, C a^{3} \sin \left (d x + c\right ) + 36 \, A a^{3} \tan \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 = 1.49857, size = 379, normalized size = 2.43 \begin{align*} \frac{36 \, C a^{3} d x \cos \left (d x + c\right )^{3} + 3 \,{\left (5 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (5 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (6 \, C a^{3} \cos \left (d x + c\right )^{3} + 2 \,{\left (11 \, A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 9 \, A a^{3} \cos \left (d x + c\right ) + 2 \, A a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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 = 1.2188, size = 296, normalized size = 1.9 \begin{align*} \frac{18 \,{\left (d x + c\right )} C a^{3} + \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 )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (5 \, A a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\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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