Optimal. Leaf size=103 \[ \frac{\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac{a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.278564, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3050, 3033, 3023, 2735, 3770} \[ \frac{\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac{a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 3050
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 (c+d x) \, dx &=\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{3} \int (a+b \cos (c+d x)) \left (3 a A+b (3 A+2 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{6} \int \left (6 a^2 A+6 a b (2 A+C) \cos (c+d x)+2 \left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac{a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{6} \int \left (6 a^2 A+6 a b (2 A+C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=a b (2 A+C) x+\frac{\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac{a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx\\ &=a b (2 A+C) x+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac{a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.248575, size = 145, normalized size = 1.41 \[ \frac{3 \left (4 a^2 C+4 A b^2+3 b^2 C\right ) \sin (c+d x)-12 a^2 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 a^2 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+24 a A b c+24 a A b d x+6 a b C \sin (2 (c+d x))+12 a b c C+12 a b C d x+b^2 C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 137, normalized size = 1.3 \begin{align*}{\frac{A{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{2}}{3\,d}}+{\frac{2\,{b}^{2}C\sin \left ( dx+c \right ) }{3\,d}}+2\,aAbx+2\,{\frac{Aabc}{d}}+{\frac{abC\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+abCx+{\frac{abCc}{d}}+{\frac{A{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}C\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987378, size = 142, normalized size = 1.38 \begin{align*} \frac{12 \,{\left (d x + c\right )} A a b + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b - 2 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{2} + 6 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 6 \, C a^{2} \sin \left (d x + c\right ) + 6 \, A b^{2} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5569, size = 250, normalized size = 2.43 \begin{align*} \frac{6 \,{\left (2 \, A + C\right )} a b d x + 3 \, A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (C b^{2} \cos \left (d x + c\right )^{2} + 3 \, C a b \cos \left (d x + c\right ) + 3 \, C a^{2} +{\left (3 \, A + 2 \, C\right )} b^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \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 [B] time = 1.37585, size = 346, normalized size = 3.36 \begin{align*} \frac{3 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3 \,{\left (2 \, A a b + C a b\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 )^{5} - 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A b^{2} \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 )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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