Optimal. Leaf size=145 \[ \frac {a \left (2 a^2 A+9 a b B+8 A b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 (3 a B+5 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} x \left (a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right )+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {b^3 B \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.35, antiderivative size = 145, normalized size of antiderivative = 1.00, 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 = {4025, 4074, 4047, 8, 4045, 3770} \[ \frac {a \left (2 a^2 A+9 a b B+8 A b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} x \left (3 a^2 A b+a^3 B+6 a b^2 B+2 A b^3\right )+\frac {a^2 (3 a B+5 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {b^3 B \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 4025
Rule 4045
Rule 4047
Rule 4074
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (-a (5 A b+3 a B)-\left (2 a^2 A+3 A b^2+6 a b B\right ) \sec (c+d x)-3 b^2 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (2 a^2 A+8 A b^2+9 a b B\right )+3 \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) \sec (c+d x)+6 b^3 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (2 a^2 A+8 A b^2+9 a b B\right )+6 b^3 B \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) \int 1 \, dx\\ &=\frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) x+\frac {a \left (2 a^2 A+8 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\left (b^3 B\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) x+\frac {b^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (2 a^2 A+8 A b^2+9 a b B\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 A b+3 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 159, normalized size = 1.10 \[ \frac {a^3 A \sin (3 (c+d x))+9 a \left (a^2 A+4 a b B+4 A b^2\right ) \sin (c+d x)+3 a^2 (a B+3 A b) \sin (2 (c+d x))+6 (c+d x) \left (a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right )-12 b^3 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^3 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 131, normalized size = 0.90 \[ \frac {3 \, B b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} d x + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{2} + 4 \, A a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 314, normalized size = 2.17 \[ \frac {6 \, B b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, B b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.00, size = 207, normalized size = 1.43 \[ \frac {A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{3 d}+\frac {2 a^{3} A \sin \left (d x +c \right )}{3 d}+\frac {a^{3} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a^{3} B x}{2}+\frac {a^{3} B c}{2 d}+\frac {3 A \,a^{2} b \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 A x \,a^{2} b}{2}+\frac {3 A \,a^{2} b c}{2 d}+\frac {3 a^{2} b B \sin \left (d x +c \right )}{d}+\frac {3 A a \,b^{2} \sin \left (d x +c \right )}{d}+3 B x a \,b^{2}+\frac {3 B a \,b^{2} c}{d}+A x \,b^{3}+\frac {A \,b^{3} c}{d}+\frac {b^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 152, normalized size = 1.05 \[ -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 36 \, {\left (d x + c\right )} B a b^{2} - 12 \, {\left (d x + c\right )} A b^{3} - 6 \, B b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{2} b \sin \left (d x + c\right ) - 36 \, A a b^{2} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 1924, normalized size = 13.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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