Optimal. Leaf size=216 \[ \frac{a \left (16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{6 d}+\frac{\left (24 a^2 A b^2+3 a^4 A+16 a^3 b B+32 a b^3 B+8 A b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^4 B x \]
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Rubi [A] time = 0.597489, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2989, 3047, 3031, 3021, 2735, 3770} \[ \frac{a \left (16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{6 d}+\frac{\left (24 a^2 A b^2+3 a^4 A+16 a^3 b B+32 a b^3 B+8 A b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^4 B x \]
Antiderivative was successfully verified.
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Rule 2989
Rule 3047
Rule 3031
Rule 3021
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx &=\frac{a 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 (a (7 A b+4 a B)+\left (3 a^2 A+4 A b^2+8 a b B\right ) \cos (c+d x)+4 b^2 B \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a 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 (a \left (9 a^2 A+26 A b^2+32 a b B\right )+\left (23 a^2 A b+12 A b^3+8 a^3 B+36 a b^2 B\right ) \cos (c+d x)+12 b^3 B \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{24} \int \left (-4 a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right )-3 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \cos (c+d x)-24 b^4 B \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac{a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a 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 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right )-24 b^4 B \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^4 B x+\frac{a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac{a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{8} \left (-3 a^4 A-24 a^2 A b^2-8 A b^4-16 a^3 b B-32 a b^3 B\right ) \int \sec (c+d x) \, dx\\ &=b^4 B x+\frac{\left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac{a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac{a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 1.01386, size = 160, normalized size = 0.74 \[ \frac{3 \left (24 a^2 A b^2+3 a^4 A+16 a^3 b B+32 a b^3 B+8 A b^4\right ) \tanh ^{-1}(\sin (c+d x))+3 a \tan (c+d x) \left (a \left (3 a^2 A+16 a b B+24 A b^2\right ) \sec (c+d x)+8 \left (4 a^2 A b+a^3 B+6 a b^2 B+4 A b^3\right )+2 a^3 A \sec ^3(c+d x)\right )+8 a^3 (a B+4 A b) \tan ^3(c+d x)+24 b^4 B d x}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 338, normalized size = 1.6 \begin{align*}{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{2\,{a}^{4}B\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{8\,A{a}^{3}b\tan \left ( dx+c \right ) }{3\,d}}+{\frac{4\,A{a}^{3}b\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+2\,{\frac{B{a}^{3}b\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{d}}+2\,{\frac{B{a}^{3}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{A{a}^{2}{b}^{2}\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{d}}+3\,{\frac{A{a}^{2}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{B{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}}+4\,{\frac{Aa{b}^{3}\tan \left ( dx+c \right ) }{d}}+4\,{\frac{Ba{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{A{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{b}^{4}Bx+{\frac{B{b}^{4}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13339, size = 428, normalized size = 1.98 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 48 \,{\left (d x + c\right )} B b^{4} - 3 \, A a^{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 )} - 48 \, B a^{3} 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 )} - 72 \, A 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 )} + 96 \, B a b^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 288 \, B a^{2} b^{2} \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 = 1.54221, size = 603, normalized size = 2.79 \begin{align*} \frac{48 \, B b^{4} d x \cos \left (d x + c\right )^{4} + 3 \,{\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (6 \, A a^{4} + 16 \,{\left (B a^{4} + 4 \, A a^{3} b + 9 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\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 = 1.86199, size = 857, normalized size = 3.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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