Optimal. Leaf size=103 \[ \frac{1}{2} x \left (a^2 (A+2 C)+2 A b^2\right )+\frac{a A b \sin (c+d x)}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{2 a b C \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d} \]
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Rubi [A] time = 0.289607, antiderivative size = 103, 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 = {4095, 4076, 4047, 8, 4045, 3770} \[ \frac{1}{2} x \left (a^2 (A+2 C)+2 A b^2\right )+\frac{a A b \sin (c+d x)}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{2 a b C \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 4095
Rule 4076
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{1}{2} \int \cos (c+d x) (a+b \sec (c+d x)) \left (2 A b+a (A+2 C) \sec (c+d x)-b (A-2 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d}+\frac{1}{2} \int \cos (c+d x) \left (2 a A b+\left (2 A b^2+a^2 (A+2 C)\right ) \sec (c+d x)+4 a b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d}+\frac{1}{2} \int \cos (c+d x) \left (2 a A b+4 a b C \sec ^2(c+d x)\right ) \, dx+\frac{1}{2} \left (2 A b^2+a^2 (A+2 C)\right ) \int 1 \, dx\\ &=\frac{1}{2} \left (2 A b^2+a^2 (A+2 C)\right ) x+\frac{a A b \sin (c+d x)}{d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d}+(2 a b C) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (2 A b^2+a^2 (A+2 C)\right ) x+\frac{2 a b C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A b \sin (c+d x)}{d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.748399, size = 130, normalized size = 1.26 \[ \frac{2 (c+d x) \left (a^2 (A+2 C)+2 A b^2\right )+\tan (c+d x) \left (a^2 A \cos (2 (c+d x))+a^2 A+4 b^2 C\right )+8 a A b \sin (c+d x)-8 a b C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+8 a b C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 120, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}A\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{2}Ax}{2}}+{\frac{{a}^{2}Ac}{2\,d}}+{a}^{2}Cx+{\frac{C{a}^{2}c}{d}}+2\,{\frac{Aab\sin \left ( dx+c \right ) }{d}}+2\,{\frac{abC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+A{b}^{2}x+{\frac{A{b}^{2}c}{d}}+{\frac{{b}^{2}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 = 1.02662, size = 134, normalized size = 1.3 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 4 \,{\left (d x + c\right )} C a^{2} + 4 \,{\left (d x + c\right )} A b^{2} + 4 \, C 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 \, A a b \sin \left (d x + c\right ) + 4 \, C b^{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 = 0.5304, size = 309, normalized size = 3. \begin{align*} \frac{2 \, C a b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, C a b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left ({\left (A + 2 \, C\right )} a^{2} + 2 \, A b^{2}\right )} d x \cos \left (d x + c\right ) +{\left (A a^{2} \cos \left (d x + c\right )^{2} + 4 \, A a b \cos \left (d x + c\right ) + 2 \, C b^{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 = 1.19432, size = 236, normalized size = 2.29 \begin{align*} \frac{4 \, C a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 4 \, C a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{4 \, C b^{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 (A a^{2} + 2 \, C a^{2} + 2 \, A b^{2}\right )}{\left (d x + c\right )} - \frac{2 \,{\left (A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, A a b \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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