Optimal. Leaf size=198 \[ \frac{a^2 \left (2 a^2 A+9 a b B+9 A b^2\right ) \sin (c+d x)}{3 d}-\frac{b^2 \left (3 a^2 B+8 a A b-6 b^2 B\right ) \tan (c+d x)}{6 d}+\frac{1}{2} a x \left (4 a^2 A b+a^3 B+12 a b^2 B+8 A b^3\right )+\frac{b^3 (4 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (a B+2 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.590815, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4025, 4094, 4076, 4047, 8, 4045, 3770} \[ \frac{a^2 \left (2 a^2 A+9 a b B+9 A b^2\right ) \sin (c+d x)}{3 d}-\frac{b^2 \left (3 a^2 B+8 a A b-6 b^2 B\right ) \tan (c+d x)}{6 d}+\frac{1}{2} a x \left (4 a^2 A b+a^3 B+12 a b^2 B+8 A b^3\right )+\frac{b^3 (4 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (a B+2 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 4025
Rule 4094
Rule 4076
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (-3 a (2 A b+a B)-\left (2 a^2 A+3 A b^2+6 a b B\right ) \sec (c+d x)+b (a A-3 b B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac{1}{6} \int \cos (c+d x) (a+b \sec (c+d x)) \left (-2 a \left (2 a^2 A+9 A b^2+9 a b B\right )-\left (8 a^2 A b+6 A b^3+3 a^3 B+18 a b^2 B\right ) \sec (c+d x)+b \left (8 a A b+3 a^2 B-6 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac{b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}-\frac{1}{6} \int \cos (c+d x) \left (-2 a^2 \left (2 a^2 A+9 A b^2+9 a b B\right )-3 a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) \sec (c+d x)-6 b^3 (A b+4 a B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac{b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}-\frac{1}{6} \int \cos (c+d x) \left (-2 a^2 \left (2 a^2 A+9 A b^2+9 a b B\right )-6 b^3 (A b+4 a B) \sec ^2(c+d x)\right ) \, dx+\frac{1}{2} \left (a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right )\right ) \int 1 \, dx\\ &=\frac{1}{2} a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) x+\frac{a^2 \left (2 a^2 A+9 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac{a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac{b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}+\left (b^3 (A b+4 a B)\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a \left (4 a^2 A b+8 A b^3+a^3 B+12 a b^2 B\right ) x+\frac{b^3 (A b+4 a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \left (2 a^2 A+9 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac{a (2 A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac{b^2 \left (8 a A b+3 a^2 B-6 b^2 B\right ) \tan (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 1.07503, size = 257, normalized size = 1.3 \[ \frac{6 a (c+d x) \left (4 a^2 A b+a^3 B+12 a b^2 B+8 A b^3\right )+3 a^2 \left (3 a^2 A+16 a b B+24 A b^2\right ) \sin (c+d x)+3 a^3 (a B+4 A b) \sin (2 (c+d x))+a^4 A \sin (3 (c+d x))-12 b^3 (4 a B+A b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 b^3 (4 a B+A b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{12 b^4 B \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{12 b^4 B \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 255, normalized size = 1.3 \begin{align*}{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{3\,d}}+{\frac{2\,A{a}^{4}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{B{a}^{4}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+{\frac{B{a}^{4}x}{2}}+{\frac{B{a}^{4}c}{2\,d}}+2\,{\frac{A{a}^{3}b\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{d}}+2\,A{a}^{3}bx+2\,{\frac{A{a}^{3}bc}{d}}+4\,{\frac{B{a}^{3}b\sin \left ( dx+c \right ) }{d}}+6\,{\frac{A{a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{d}}+6\,B{a}^{2}{b}^{2}x+6\,{\frac{B{a}^{2}{b}^{2}c}{d}}+4\,Aa{b}^{3}x+4\,{\frac{Aa{b}^{3}c}{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}}+{\frac{B{b}^{4}\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 = 0.989478, size = 266, normalized size = 1.34 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 12 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 72 \,{\left (d x + c\right )} B a^{2} b^{2} - 48 \,{\left (d x + c\right )} A a b^{3} - 24 \, 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 )} - 6 \, 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 )} - 48 \, B a^{3} b \sin \left (d x + c\right ) - 72 \, A a^{2} b^{2} \sin \left (d x + c\right ) - 12 \, B b^{4} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.593142, size = 471, normalized size = 2.38 \begin{align*} \frac{3 \,{\left (B a^{4} + 4 \, A a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, A a^{4} \cos \left (d x + c\right )^{3} + 6 \, B b^{4} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (A a^{4} + 6 \, B a^{3} b + 9 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \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 [A] time = 1.28818, size = 501, normalized size = 2.53 \begin{align*} -\frac{\frac{12 \, B b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )}{\left (d x + c\right )} - 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, A a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 24 \, B a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 36 \, A a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 48 \, B a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 72 \, A a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, A a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, B a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 36 \, A a^{2} 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}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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