Optimal. Leaf size=168 \[ \frac{a \left (a^2 (A+2 C)+6 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left (C \left (6 a^2+b^2\right )+2 A b^2\right )-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{3 A b \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}-\frac{b^3 (4 A-C) \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.579688, antiderivative size = 168, 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 = {3048, 3047, 3033, 3023, 2735, 3770} \[ \frac{a \left (a^2 (A+2 C)+6 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left (C \left (6 a^2+b^2\right )+2 A b^2\right )-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{3 A b \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}-\frac{b^3 (4 A-C) \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3047
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+b \cos (c+d x))^2 \left (3 A b+a (A+2 C) \cos (c+d x)-2 b (A-C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{3 A b (a+b \cos (c+d x))^2 \tan (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+b \cos (c+d x)) \left (6 A b^2+a^2 (A+2 C)-a b (A-4 C) \cos (c+d x)-2 b^2 (4 A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^3 (4 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{3 A b (a+b \cos (c+d x))^2 \tan (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{4} \int \left (2 a \left (6 A b^2+a^2 (A+2 C)\right )+2 b \left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \cos (c+d x)-6 a b^2 (3 A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}-\frac{b^3 (4 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{3 A b (a+b \cos (c+d x))^2 \tan (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{4} \int \left (2 a \left (6 A b^2+a^2 (A+2 C)\right )+2 b \left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) x-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}-\frac{b^3 (4 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{3 A b (a+b \cos (c+d x))^2 \tan (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a \left (6 A b^2+a^2 (A+2 C)\right )\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) x+\frac{a \left (6 A b^2+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}-\frac{b^3 (4 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{3 A b (a+b \cos (c+d x))^2 \tan (c+d x)}{2 d}+\frac{A (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.51221, size = 285, normalized size = 1.7 \[ \frac{2 b (c+d x) \left (C \left (6 a^2+b^2\right )+2 A b^2\right )-2 a \left (a^2 (A+2 C)+6 A b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 a \left (a^2 (A+2 C)+6 A b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{12 a^2 A b \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{12 a^2 A b \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{a^3 A}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{a^3 A}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+12 a b^2 C \sin (c+d x)+b^3 C \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 196, normalized size = 1.2 \begin{align*} A{b}^{3}x+{\frac{A{b}^{3}c}{d}}+{\frac{C{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{3}Cx}{2}}+{\frac{C{b}^{3}c}{2\,d}}+3\,{\frac{aA{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{Ca{b}^{2}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{A{a}^{2}b\tan \left ( dx+c \right ) }{d}}+3\,{a}^{2}bCx+3\,{\frac{C{a}^{2}bc}{d}}+{\frac{A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{a}^{3}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 = 1.01112, size = 242, normalized size = 1.44 \begin{align*} \frac{12 \,{\left (d x + c\right )} C a^{2} b + 4 \,{\left (d x + c\right )} A b^{3} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} - A a^{3}{\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 \, C a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A a b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C a b^{2} \sin \left (d x + c\right ) + 12 \, A a^{2} b \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 = 1.58239, size = 419, normalized size = 2.49 \begin{align*} \frac{2 \,{\left (6 \, C a^{2} b +{\left (2 \, A + C\right )} b^{3}\right )} d x \cos \left (d x + c\right )^{2} +{\left ({\left (A + 2 \, C\right )} a^{3} + 6 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left ({\left (A + 2 \, C\right )} a^{3} + 6 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (C b^{3} \cos \left (d x + c\right )^{3} + 6 \, C a b^{2} \cos \left (d x + c\right )^{2} + 6 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3}\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 [B] time = 1.30672, size = 520, normalized size = 3.1 \begin{align*} \frac{{\left (6 \, C a^{2} b + 2 \, A b^{3} + C b^{3}\right )}{\left (d x + c\right )} +{\left (A a^{3} + 2 \, C a^{3} + 6 \, A a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (A a^{3} + 2 \, C a^{3} + 6 \, A a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 6 \, A a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 6 \, C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - C b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, A a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, C b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, A a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C b^{3} \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 )^{4} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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