Optimal. Leaf size=112 \[ \frac{\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{3 d}+\frac{a b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x \]
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Rubi [A] time = 0.314508, antiderivative size = 112, 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, 3031, 3021, 2735, 3770} \[ \frac{\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{3 d}+\frac{a b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3031
Rule 3021
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 ^4(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int (a+b \cos (c+d x)) \left (2 A b+a (2 A+3 C) \cos (c+d x)+3 b C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a A b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{1}{6} \int \left (-2 \left (2 A b^2+a^2 (2 A+3 C)\right )-6 a b (A+2 C) \cos (c+d x)-6 b^2 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{\left (2 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac{a A b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{1}{6} \int \left (-6 a b (A+2 C)-6 b^2 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^2 C x+\frac{\left (2 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac{a A b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+(a b (A+2 C)) \int \sec (c+d x) \, dx\\ &=b^2 C x+\frac{a b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (2 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac{a A b \sec (c+d x) \tan (c+d x)}{3 d}+\frac{A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.439717, size = 76, normalized size = 0.68 \[ \frac{3 \tan (c+d x) \left (a^2 (A+C)+a A b \sec (c+d x)+A b^2\right )+a^2 A \tan ^3(c+d x)+3 a b (A+2 C) \tanh ^{-1}(\sin (c+d x))+3 b^2 C d x}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 145, normalized size = 1.3 \begin{align*}{\frac{A{b}^{2}\tan \left ( dx+c \right ) }{d}}+{b}^{2}Cx+{\frac{C{b}^{2}c}{d}}+{\frac{aAb\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{aAb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{abC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{2\,A{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{A{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{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 = 1.01538, size = 184, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} + 6 \,{\left (d x + c\right )} C b^{2} - 3 \, A a b{\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 )} + 6 \, 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 )} + 6 \, C a^{2} \tan \left (d x + c\right ) + 6 \, A b^{2} \tan \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 = 1.45657, size = 347, normalized size = 3.1 \begin{align*} \frac{6 \, C b^{2} d x \cos \left (d x + c\right )^{3} + 3 \,{\left (A + 2 \, C\right )} a b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (A + 2 \, C\right )} a b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (3 \, A a b \cos \left (d x + c\right ) + A a^{2} +{\left ({\left (2 \, A + 3 \, C\right )} a^{2} + 3 \, A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \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 = 1.40049, size = 354, normalized size = 3.16 \begin{align*} \frac{3 \,{\left (d x + c\right )} C b^{2} + 3 \,{\left (A a b + 2 \, C a b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (A a b + 2 \, C a b\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 )^{5} + 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, A 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} - 2 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 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} + 3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A 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 )\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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