Optimal. Leaf size=112 \[ -\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d}+\frac{2 a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+3 C)+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
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Rubi [A] time = 0.39064, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3044, 2976, 2968, 3023, 2735, 3770} \[ -\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d}+\frac{2 a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+3 C)+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2976
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{\int (a+a \cos (c+d x))^2 (2 a A-a (2 A-C) \cos (c+d x)) \sec (c+d x) \, dx}{a}\\ &=-\frac{(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{\int (a+a \cos (c+d x)) \left (4 a^2 A-a^2 (2 A-3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac{(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{\int \left (4 a^3 A+\left (4 a^3 A-a^3 (2 A-3 C)\right ) \cos (c+d x)-a^3 (2 A-3 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{\int \left (4 a^3 A+a^3 (2 A+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=\frac{1}{2} a^2 (2 A+3 C) x-\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\left (2 a^2 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^2 (2 A+3 C) x+\frac{2 a^2 A \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac{A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.394473, size = 109, normalized size = 0.97 \[ \frac{a^2 \left (4 A \tan (c+d x)-8 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+8 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 A c+4 A d x+8 C \sin (c+d x)+C \sin (2 (c+d x))+6 c C+6 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 107, normalized size = 1. \begin{align*} A{a}^{2}x+{\frac{A{a}^{2}c}{d}}+{\frac{{a}^{2}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{2}Cx}{2}}+{\frac{3\,{a}^{2}Cc}{2\,d}}+2\,{\frac{A{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{{a}^{2}C\sin \left ( dx+c \right ) }{d}}+{\frac{A{a}^{2}\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 = 1.05289, size = 136, normalized size = 1.21 \begin{align*} \frac{4 \,{\left (d x + c\right )} A a^{2} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 4 \,{\left (d x + c\right )} C a^{2} + 4 \, A a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, C a^{2} \sin \left (d x + c\right ) + 4 \, A a^{2} \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 = 1.4864, size = 296, normalized size = 2.64 \begin{align*} \frac{{\left (2 \, A + 3 \, C\right )} a^{2} d x \cos \left (d x + c\right ) + 2 \, A a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, A a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (C a^{2} \cos \left (d x + c\right )^{2} + 4 \, C a^{2} \cos \left (d x + c\right ) + 2 \, A a^{2}\right )} \sin \left (d x + c\right )}{2 \, 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 = 1.21522, size = 193, normalized size = 1.72 \begin{align*} \frac{4 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 4 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{4 \, A a^{2} \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} +{\left (2 \, A a^{2} + 3 \, C a^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, C a^{2} \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}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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