Optimal. Leaf size=229 \[ -\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac{\left (24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+4 a^3 A b x-\frac{b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]
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Rubi [A] time = 0.492718, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4095, 4056, 4048, 3770, 3767, 8} \[ -\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac{\left (24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+4 a^3 A b x-\frac{b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]
Antiderivative was successfully verified.
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Rule 4095
Rule 4056
Rule 4048
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\int (a+b \sec (c+d x))^3 \left (4 A b+a C \sec (c+d x)-b (4 A-C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{4} \int (a+b \sec (c+d x))^2 \left (16 a A b+\left (4 A b^2+4 a^2 C+3 b^2 C\right ) \sec (c+d x)-a b (12 A-7 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{12} \int (a+b \sec (c+d x)) \left (48 a^2 A b+a \left (36 A b^2+12 a^2 C+23 b^2 C\right ) \sec (c+d x)-b \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{24} \int \left (96 a^3 A b+3 \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \sec (c+d x)-4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=4 a^3 A b x+\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{1}{6} \left (a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{8} \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=4 a^3 A b x+\frac{\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{\left (a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=4 a^3 A b x+\frac{\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [B] time = 6.5741, size = 1357, normalized size = 5.93 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 316, normalized size = 1.4 \begin{align*}{\frac{A{a}^{4}\sin \left ( dx+c \right ) }{d}}+{\frac{{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{a}^{3}Abx+4\,{\frac{A{a}^{3}bc}{d}}+4\,{\frac{{a}^{3}bC\tan \left ( dx+c \right ) }{d}}+6\,{\frac{A{a}^{2}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{C{a}^{2}{b}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+3\,{\frac{C{a}^{2}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{Aa{b}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{8\,Ca{b}^{3}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{4\,Ca{b}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{A{b}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{C{b}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,C{b}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,C{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992264, size = 413, normalized size = 1.8 \begin{align*} \frac{192 \,{\left (d x + c\right )} A a^{3} b + 64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{3} - 3 \, C b^{4}{\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 )} - 72 \, C a^{2} 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 )} - 12 \, A b^{4}{\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^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, A a^{2} 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 \, A a^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.59003, size = 575, normalized size = 2.51 \begin{align*} \frac{192 \, A a^{3} b d x \cos \left (d x + c\right )^{4} + 3 \,{\left (8 \, C a^{4} + 24 \,{\left (2 \, A + C\right )} a^{2} b^{2} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (8 \, C a^{4} + 24 \,{\left (2 \, A + C\right )} a^{2} b^{2} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, C a b^{3} \cos \left (d x + c\right ) + 6 \, C b^{4} + 32 \,{\left (3 \, C a^{3} b +{\left (3 \, A + 2 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (24 \, C a^{2} b^{2} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, 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 = 1.26404, size = 797, normalized size = 3.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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