Optimal. Leaf size=109 \[ \frac{1}{2} x \left (C \left (2 a^2+b^2\right )+2 A b^2\right )-\frac{2 a b (A-C) \sin (c+d x)}{d}+\frac{2 a A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.314727, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3048, 3033, 3023, 2735, 3770} \[ \frac{1}{2} x \left (C \left (2 a^2+b^2\right )+2 A b^2\right )-\frac{2 a b (A-C) \sin (c+d x)}{d}+\frac{2 a A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\int (a+b \cos (c+d x)) \left (2 A b+a C \cos (c+d x)-b (2 A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{1}{2} \int \left (4 a A b+\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \cos (c+d x)-4 a b (A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{2 a b (A-C) \sin (c+d x)}{d}-\frac{b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac{1}{2} \int \left (4 a A b+\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) x-\frac{2 a b (A-C) \sin (c+d x)}{d}-\frac{b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+(2 a A b) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) x+\frac{2 a A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a b (A-C) \sin (c+d x)}{d}-\frac{b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.713461, size = 132, normalized size = 1.21 \[ \frac{2 (c+d x) \left (C \left (2 a^2+b^2\right )+2 A b^2\right )+\tan (c+d x) \left (4 a^2 A+b^2 C \cos (2 (c+d x))+b^2 C\right )-8 a A b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+8 a A b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 a b C \sin (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 120, normalized size = 1.1 \begin{align*} A{b}^{2}x+{\frac{A{b}^{2}c}{d}}+{\frac{{b}^{2}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{2}Cx}{2}}+{\frac{{b}^{2}Cc}{2\,d}}+2\,{\frac{aAb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{abC\sin \left ( dx+c \right ) }{d}}+{\frac{A{a}^{2}\tan \left ( dx+c \right ) }{d}}+{a}^{2}Cx+{\frac{C{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01696, size = 134, normalized size = 1.23 \begin{align*} \frac{4 \,{\left (d x + c\right )} C a^{2} + 4 \,{\left (d x + c\right )} A b^{2} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 4 \, A a b{\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 b \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 = 1.46055, size = 309, normalized size = 2.83 \begin{align*} \frac{2 \, A a b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, A a b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, C a^{2} +{\left (2 \, A + C\right )} b^{2}\right )} d x \cos \left (d x + c\right ) +{\left (C b^{2} \cos \left (d x + c\right )^{2} + 4 \, C a b \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 = 1.67502, size = 236, normalized size = 2.17 \begin{align*} \frac{4 \, A a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 4 \, A a b \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 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (4 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C b^{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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