Optimal. Leaf size=75 \[ \frac {1}{2} a^3 x (5 A+7 B)+\frac {5}{2} a^3 (A+B) \sin (x)+\frac {1}{6} (5 A+3 B) \sin (x) \left (a^3 \cos (x)+a^3\right )+a^3 B \tanh ^{-1}(\sin (x))+\frac {1}{3} a A \sin (x) (a \cos (x)+a)^2 \]
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Rubi [A] time = 0.30, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2828, 2976, 2968, 3023, 2735, 3770} \[ \frac {1}{2} a^3 x (5 A+7 B)+\frac {5}{2} a^3 (A+B) \sin (x)+\frac {1}{6} (5 A+3 B) \sin (x) \left (a^3 \cos (x)+a^3\right )+a^3 B \tanh ^{-1}(\sin (x))+\frac {1}{3} a A \sin (x) (a \cos (x)+a)^2 \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2828
Rule 2968
Rule 2976
Rule 3023
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (x))^3 (A+B \sec (x)) \, dx &=\int (a+a \cos (x))^3 (B+A \cos (x)) \sec (x) \, dx\\ &=\frac {1}{3} a A (a+a \cos (x))^2 \sin (x)+\frac {1}{3} \int (a+a \cos (x))^2 (3 a B+a (5 A+3 B) \cos (x)) \sec (x) \, dx\\ &=\frac {1}{3} a A (a+a \cos (x))^2 \sin (x)+\frac {1}{6} (5 A+3 B) \left (a^3+a^3 \cos (x)\right ) \sin (x)+\frac {1}{6} \int (a+a \cos (x)) \left (6 a^2 B+15 a^2 (A+B) \cos (x)\right ) \sec (x) \, dx\\ &=\frac {1}{3} a A (a+a \cos (x))^2 \sin (x)+\frac {1}{6} (5 A+3 B) \left (a^3+a^3 \cos (x)\right ) \sin (x)+\frac {1}{6} \int \left (6 a^3 B+\left (6 a^3 B+15 a^3 (A+B)\right ) \cos (x)+15 a^3 (A+B) \cos ^2(x)\right ) \sec (x) \, dx\\ &=\frac {5}{2} a^3 (A+B) \sin (x)+\frac {1}{3} a A (a+a \cos (x))^2 \sin (x)+\frac {1}{6} (5 A+3 B) \left (a^3+a^3 \cos (x)\right ) \sin (x)+\frac {1}{6} \int \left (6 a^3 B+3 a^3 (5 A+7 B) \cos (x)\right ) \sec (x) \, dx\\ &=\frac {1}{2} a^3 (5 A+7 B) x+\frac {5}{2} a^3 (A+B) \sin (x)+\frac {1}{3} a A (a+a \cos (x))^2 \sin (x)+\frac {1}{6} (5 A+3 B) \left (a^3+a^3 \cos (x)\right ) \sin (x)+\left (a^3 B\right ) \int \sec (x) \, dx\\ &=\frac {1}{2} a^3 (5 A+7 B) x+a^3 B \tanh ^{-1}(\sin (x))+\frac {5}{2} a^3 (A+B) \sin (x)+\frac {1}{3} a A (a+a \cos (x))^2 \sin (x)+\frac {1}{6} (5 A+3 B) \left (a^3+a^3 \cos (x)\right ) \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 80, normalized size = 1.07 \[ \frac {1}{12} a^3 \left (9 (5 A+4 B) \sin (x)+3 (3 A+B) \sin (2 x)+30 A x+A \sin (3 x)+42 B x-12 B \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+12 B \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 77, normalized size = 1.03 \[ \frac {1}{2} \, {\left (5 \, A + 7 \, B\right )} a^{3} x + \frac {1}{2} \, B a^{3} \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, B a^{3} \log \left (-\sin \relax (x) + 1\right ) + \frac {1}{6} \, {\left (2 \, A a^{3} \cos \relax (x)^{2} + 3 \, {\left (3 \, A + B\right )} a^{3} \cos \relax (x) + 2 \, {\left (11 \, A + 9 \, B\right )} a^{3}\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 125, normalized size = 1.67 \[ B a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - B a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) + \frac {1}{2} \, {\left (5 \, A a^{3} + 7 \, B a^{3}\right )} x + \frac {15 \, A a^{3} \tan \left (\frac {1}{2} \, x\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, x\right )^{5} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 36 \, B a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 33 \, A a^{3} \tan \left (\frac {1}{2} \, x\right ) + 21 \, B a^{3} \tan \left (\frac {1}{2} \, x\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 77, normalized size = 1.03 \[ \frac {A \,a^{3} \left (2+\cos ^{2}\relax (x )\right ) \sin \relax (x )}{3}+\frac {B \,a^{3} \sin \relax (x ) \cos \relax (x )}{2}+\frac {7 B \,a^{3} x}{2}+\frac {3 A \,a^{3} \sin \relax (x ) \cos \relax (x )}{2}+\frac {5 A \,a^{3} x}{2}+3 B \,a^{3} \sin \relax (x )+3 A \,a^{3} \sin \relax (x )+B \,a^{3} \ln \left (\sec \relax (x )+\tan \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 84, normalized size = 1.12 \[ -\frac {1}{3} \, {\left (\sin \relax (x)^{3} - 3 \, \sin \relax (x)\right )} A a^{3} + \frac {3}{4} \, A a^{3} {\left (2 \, x + \sin \left (2 \, x\right )\right )} + \frac {1}{4} \, B a^{3} {\left (2 \, x + \sin \left (2 \, x\right )\right )} + A a^{3} x + 3 \, B a^{3} x + B a^{3} \log \left (\sec \relax (x) + \tan \relax (x)\right ) + 3 \, A a^{3} \sin \relax (x) + 3 \, B a^{3} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 431, normalized size = 5.75 \[ \frac {\left (5\,A\,a^3+5\,B\,a^3\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (\frac {40\,A\,a^3}{3}+12\,B\,a^3\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (11\,A\,a^3+7\,B\,a^3\right )\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}+a^3\,\mathrm {atan}\left (\frac {1000\,A^3\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{1000\,A^3\,a^9+4200\,A^2\,B\,a^9+6040\,A\,B^2\,a^9+2968\,B^3\,a^9}+\frac {2968\,B^3\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{1000\,A^3\,a^9+4200\,A^2\,B\,a^9+6040\,A\,B^2\,a^9+2968\,B^3\,a^9}+\frac {6040\,A\,B^2\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{1000\,A^3\,a^9+4200\,A^2\,B\,a^9+6040\,A\,B^2\,a^9+2968\,B^3\,a^9}+\frac {4200\,A^2\,B\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{1000\,A^3\,a^9+4200\,A^2\,B\,a^9+6040\,A\,B^2\,a^9+2968\,B^3\,a^9}\right )\,\left (5\,A+7\,B\right )+2\,B\,a^3\,\mathrm {atanh}\left (\frac {848\,B^3\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{400\,A^2\,B\,a^9+1120\,A\,B^2\,a^9+848\,B^3\,a^9}+\frac {1120\,A\,B^2\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{400\,A^2\,B\,a^9+1120\,A\,B^2\,a^9+848\,B^3\,a^9}+\frac {400\,A^2\,B\,a^9\,\mathrm {tan}\left (\frac {x}{2}\right )}{400\,A^2\,B\,a^9+1120\,A\,B^2\,a^9+848\,B^3\,a^9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.63, size = 92, normalized size = 1.23 \[ \frac {5 A a^{3} x}{2} - \frac {A a^{3} \sin ^{3}{\relax (x )}}{3} + 4 A a^{3} \sin {\relax (x )} + \frac {3 A a^{3} \sin {\left (2 x \right )}}{4} + \frac {7 B a^{3} x}{2} + B a^{3} \log {\left (\tan {\relax (x )} + \sec {\relax (x )} \right )} + \frac {B a^{3} \sin {\relax (x )} \cos {\relax (x )}}{2} + 3 B a^{3} \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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