Optimal. Leaf size=182 \[ \frac{b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{2 d}+\frac{a \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x \]
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Rubi [A] time = 0.599905, antiderivative size = 182, 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, 3031, 3021, 2735, 3770} \[ \frac{b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{2 d}+\frac{a \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3047
Rule 3031
Rule 3021
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 ^5(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int (a+b \cos (c+d x))^2 \left (3 A b+a (3 A+4 C) \cos (c+d x)+4 b C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{A b (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{12} \int (a+b \cos (c+d x)) \left (3 \left (2 A b^2+a^2 (3 A+4 C)\right )+3 a b (5 A+8 C) \cos (c+d x)+12 b^2 C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a \left (2 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{A b (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{24} \int \left (-12 b \left (A b^2+a^2 (4 A+6 C)\right )-3 a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \cos (c+d x)-24 b^3 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{b \left (A b^2+a^2 (4 A+6 C)\right ) \tan (c+d x)}{2 d}+\frac{a \left (2 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{A b (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{24} \int \left (-3 a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right )-24 b^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^3 C x+\frac{b \left (A b^2+a^2 (4 A+6 C)\right ) \tan (c+d x)}{2 d}+\frac{a \left (2 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{A b (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} \left (a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right )\right ) \int \sec (c+d x) \, dx\\ &=b^3 C x+\frac{a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left (A b^2+a^2 (4 A+6 C)\right ) \tan (c+d x)}{2 d}+\frac{a \left (2 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{A b (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.942651, size = 127, normalized size = 0.7 \[ \frac{a \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (a \left (a^2 (3 A+4 C)+12 A b^2\right ) \sec (c+d x)+8 b \left (3 a^2 (A+C)+A b^2\right )+2 a^3 A \sec ^3(c+d x)\right )+8 a^2 A b \tan ^3(c+d x)+8 b^3 C d x}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 267, normalized size = 1.5 \begin{align*}{\frac{A{b}^{3}\tan \left ( dx+c \right ) }{d}}+{b}^{3}Cx+{\frac{C{b}^{3}c}{d}}+{\frac{3\,aA{b}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{3\,aA{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{Ca{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{A{a}^{2}b\tan \left ( dx+c \right ) }{d}}+{\frac{A{a}^{2}b\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+3\,{\frac{{a}^{2}bC\tan \left ( dx+c \right ) }{d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02143, size = 352, normalized size = 1.93 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b + 16 \,{\left (d x + c\right )} C b^{3} - A a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, C 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 )} - 12 \, A a b^{2}{\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 )} + 24 \, C 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 )} + 48 \, C a^{2} b \tan \left (d x + c\right ) + 16 \, A b^{3} \tan \left (d x + c\right )}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55796, size = 485, normalized size = 2.66 \begin{align*} \frac{16 \, C b^{3} d x \cos \left (d x + c\right )^{4} +{\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \,{\left (A + 2 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \,{\left (A + 2 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \, A a^{2} b \cos \left (d x + c\right ) + 2 \, A a^{3} + 8 \,{\left ({\left (2 \, A + 3 \, C\right )} a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \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 = 1.37252, size = 710, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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