Optimal. Leaf size=112 \[ -\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d}+2 a^2 C x+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.358719, 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, 2975, 2968, 3023, 2735, 3770} \[ -\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d}+2 a^2 C x+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
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))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x))^2 (2 a A-a (A-2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 a}\\ &=\frac{A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int (a+a \cos (c+d x)) \left (a^2 (3 A+2 C)-a^2 (3 A-2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=\frac{A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int \left (a^3 (3 A+2 C)+\left (-a^3 (3 A-2 C)+a^3 (3 A+2 C)\right ) \cos (c+d x)-a^3 (3 A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{\int \left (a^3 (3 A+2 C)+4 a^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=2 a^2 C x-\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a^2 (3 A+2 C)\right ) \int \sec (c+d x) \, dx\\ &=2 a^2 C x+\frac{a^2 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac{A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 2.1381, size = 293, normalized size = 2.62 \[ \frac{1}{16} a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{2 (3 A+2 C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{2 (3 A+2 C) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{8 A \sin \left (\frac{d x}{2}\right )}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{8 A \sin \left (\frac{d x}{2}\right )}{d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{A}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{A}{d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{4 C \sin (c) \cos (d x)}{d}+\frac{4 C \cos (c) \sin (d x)}{d}+8 C x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 114, normalized size = 1. \begin{align*}{\frac{3\,A{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{a}^{2}C\sin \left ( dx+c \right ) }{d}}+2\,{\frac{A{a}^{2}\tan \left ( dx+c \right ) }{d}}+2\,{a}^{2}Cx+2\,{\frac{C{a}^{2}c}{d}}+{\frac{A{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{2}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 = 1.08465, size = 192, normalized size = 1.71 \begin{align*} \frac{8 \,{\left (d x + c\right )} C a^{2} - A a^{2}{\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 )} + 2 \, 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 )} + 2 \, C a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C a^{2} \sin \left (d x + c\right ) + 8 \, 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 = 1.50356, size = 320, normalized size = 2.86 \begin{align*} \frac{8 \, C a^{2} d x \cos \left (d x + c\right )^{2} +{\left (3 \, A + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (3 \, A + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, C a^{2} \cos \left (d x + c\right )^{2} + 4 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}\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 = 1.23067, size = 205, normalized size = 1.83 \begin{align*} \frac{4 \,{\left (d x + c\right )} C a^{2} + \frac{4 \, C 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 (3 \, A a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (3 \, A a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5 \, A 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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