Optimal. Leaf size=175 \[ \frac{a^3 (7 A+6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A+2 B-C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}+\frac{(3 A+2 B) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 a d}+\frac{1}{2} a^3 x (2 A+6 B+7 C)-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d} \]
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Rubi [A] time = 0.533527, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3043, 2975, 2976, 2968, 3023, 2735, 3770} \[ \frac{a^3 (7 A+6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A+2 B-C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}+\frac{(3 A+2 B) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 a d}+\frac{1}{2} a^3 x (2 A+6 B+7 C)-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d} \]
Antiderivative was successfully verified.
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Rule 3043
Rule 2975
Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x))^3 (a (3 A+2 B)-2 a (A-C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 a}\\ &=\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x))^2 \left (a^2 (7 A+6 B+2 C)-2 a^2 (4 A+2 B-C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac{(4 A+2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x)) \left (2 a^3 (7 A+6 B+2 C)-10 a^3 (A-C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{4 a}\\ &=-\frac{(4 A+2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int \left (2 a^4 (7 A+6 B+2 C)+\left (-10 a^4 (A-C)+2 a^4 (7 A+6 B+2 C)\right ) \cos (c+d x)-10 a^4 (A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{4 a}\\ &=-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}-\frac{(4 A+2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int \left (2 a^4 (7 A+6 B+2 C)+2 a^4 (2 A+6 B+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{4 a}\\ &=\frac{1}{2} a^3 (2 A+6 B+7 C) x-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}-\frac{(4 A+2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a^3 (7 A+6 B+2 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^3 (2 A+6 B+7 C) x+\frac{a^3 (7 A+6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}-\frac{(4 A+2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 a d}+\frac{A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.47749, size = 256, normalized size = 1.46 \[ \frac{a^3 \left (2 (2 A+6 B+7 C) (c+d x)-2 (7 A+6 B+2 C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 (7 A+6 B+2 C) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 (3 A+B) \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{4 (3 A+B) \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{A}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{A}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+4 (B+3 C) \sin (c+d x)+C \sin (2 (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 219, normalized size = 1.3 \begin{align*}{\frac{A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{7\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{a}^{3}B\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{A{a}^{3}\tan \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{7\,{a}^{3}Cx}{2}}+{\frac{7\,{a}^{3}Cc}{2\,d}}+3\,{a}^{3}Bx+3\,{\frac{B{a}^{3}c}{d}}+3\,{\frac{{a}^{3}C\sin \left ( dx+c \right ) }{d}}+A{a}^{3}x+{\frac{A{a}^{3}c}{d}}+{\frac{{a}^{3}B\sin \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02162, size = 320, normalized size = 1.83 \begin{align*} \frac{4 \,{\left (d x + c\right )} A a^{3} + 12 \,{\left (d x + c\right )} B a^{3} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 12 \,{\left (d x + c\right )} C a^{3} - 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 )} + 6 \, B a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, 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 )} + 4 \, B a^{3} \sin \left (d x + c\right ) + 12 \, C a^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} \tan \left (d x + c\right ) + 4 \, B a^{3} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15055, size = 410, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (2 \, A + 6 \, B + 7 \, C\right )} a^{3} d x \cos \left (d x + c\right )^{2} +{\left (7 \, A + 6 \, B + 2 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (7 \, A + 6 \, B + 2 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (C a^{3} \cos \left (d x + c\right )^{3} + 2 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sin \left (d x + c\right )}{4 \, 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 = 1.24958, size = 378, normalized size = 2.16 \begin{align*} \frac{{\left (2 \, A a^{3} + 6 \, B a^{3} + 7 \, C a^{3}\right )}{\left (d x + c\right )} +{\left (7 \, A a^{3} + 6 \, B a^{3} + 2 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (7 \, A a^{3} + 6 \, B a^{3} + 2 \, 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 (5 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 7 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 7 \, 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 )^{4} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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