Optimal. Leaf size=57 \[ \frac {1}{2} a^2 x (3 A+4 B)+\frac {1}{2} a^2 (3 A+2 B) \sin (x)+\frac {1}{2} A \sin (x) \left (a^2 \cos (x)+a^2\right )+a^2 B \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.20, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {1}{2} a^2 x (3 A+4 B)+\frac {1}{2} a^2 (3 A+2 B) \sin (x)+\frac {1}{2} A \sin (x) \left (a^2 \cos (x)+a^2\right )+a^2 B \tanh ^{-1}(\sin (x)) \]
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))^2 (A+B \sec (x)) \, dx &=\int (a+a \cos (x))^2 (B+A \cos (x)) \sec (x) \, dx\\ &=\frac {1}{2} A \left (a^2+a^2 \cos (x)\right ) \sin (x)+\frac {1}{2} \int (a+a \cos (x)) (2 a B+a (3 A+2 B) \cos (x)) \sec (x) \, dx\\ &=\frac {1}{2} A \left (a^2+a^2 \cos (x)\right ) \sin (x)+\frac {1}{2} \int \left (2 a^2 B+\left (2 a^2 B+a^2 (3 A+2 B)\right ) \cos (x)+a^2 (3 A+2 B) \cos ^2(x)\right ) \sec (x) \, dx\\ &=\frac {1}{2} a^2 (3 A+2 B) \sin (x)+\frac {1}{2} A \left (a^2+a^2 \cos (x)\right ) \sin (x)+\frac {1}{2} \int \left (2 a^2 B+a^2 (3 A+4 B) \cos (x)\right ) \sec (x) \, dx\\ &=\frac {1}{2} a^2 (3 A+4 B) x+\frac {1}{2} a^2 (3 A+2 B) \sin (x)+\frac {1}{2} A \left (a^2+a^2 \cos (x)\right ) \sin (x)+\left (a^2 B\right ) \int \sec (x) \, dx\\ &=\frac {1}{2} a^2 (3 A+4 B) x+a^2 B \tanh ^{-1}(\sin (x))+\frac {1}{2} a^2 (3 A+2 B) \sin (x)+\frac {1}{2} A \left (a^2+a^2 \cos (x)\right ) \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 67, normalized size = 1.18 \[ \frac {1}{4} a^2 \left (4 (2 A+B) \sin (x)+6 A x+A \sin (2 x)+8 B x-4 B \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+4 B \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 60, normalized size = 1.05 \[ \frac {1}{2} \, {\left (3 \, A + 4 \, B\right )} a^{2} x + \frac {1}{2} \, B a^{2} \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, B a^{2} \log \left (-\sin \relax (x) + 1\right ) + \frac {1}{2} \, {\left (A a^{2} \cos \relax (x) + 2 \, {\left (2 \, A + B\right )} a^{2}\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 100, normalized size = 1.75 \[ B a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - B a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) + \frac {1}{2} \, {\left (3 \, A a^{2} + 4 \, B a^{2}\right )} x + \frac {3 \, A a^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, B a^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 5 \, A a^{2} \tan \left (\frac {1}{2} \, x\right ) + 2 \, B a^{2} \tan \left (\frac {1}{2} \, x\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 52, normalized size = 0.91 \[ \frac {a^{2} A \sin \relax (x ) \cos \relax (x )}{2}+\frac {3 a^{2} A x}{2}+a^{2} B \sin \relax (x )+2 a^{2} A \sin \relax (x )+2 a^{2} B x +a^{2} B \ln \left (\sec \relax (x )+\tan \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 54, normalized size = 0.95 \[ \frac {1}{4} \, A a^{2} {\left (2 \, x + \sin \left (2 \, x\right )\right )} + A a^{2} x + 2 \, B a^{2} x + B a^{2} \log \left (\sec \relax (x) + \tan \relax (x)\right ) + 2 \, A a^{2} \sin \relax (x) + B a^{2} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.46, size = 403, normalized size = 7.07 \[ \frac {\left (3\,A\,a^2+2\,B\,a^2\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (5\,A\,a^2+2\,B\,a^2\right )\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}+a^2\,\mathrm {atan}\left (\frac {216\,A^3\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{216\,A^3\,a^6+864\,A^2\,B\,a^6+1248\,A\,B^2\,a^6+640\,B^3\,a^6}+\frac {640\,B^3\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{216\,A^3\,a^6+864\,A^2\,B\,a^6+1248\,A\,B^2\,a^6+640\,B^3\,a^6}+\frac {1248\,A\,B^2\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{216\,A^3\,a^6+864\,A^2\,B\,a^6+1248\,A\,B^2\,a^6+640\,B^3\,a^6}+\frac {864\,A^2\,B\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{216\,A^3\,a^6+864\,A^2\,B\,a^6+1248\,A\,B^2\,a^6+640\,B^3\,a^6}\right )\,\left (3\,A+4\,B\right )+2\,B\,a^2\,\mathrm {atanh}\left (\frac {320\,B^3\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{144\,A^2\,B\,a^6+384\,A\,B^2\,a^6+320\,B^3\,a^6}+\frac {384\,A\,B^2\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{144\,A^2\,B\,a^6+384\,A\,B^2\,a^6+320\,B^3\,a^6}+\frac {144\,A^2\,B\,a^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{144\,A^2\,B\,a^6+384\,A\,B^2\,a^6+320\,B^3\,a^6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.54, size = 61, normalized size = 1.07 \[ \frac {3 A a^{2} x}{2} + 2 A a^{2} \sin {\relax (x )} + \frac {A a^{2} \sin {\left (2 x \right )}}{4} + 2 B a^{2} x + B a^{2} \log {\left (\tan {\relax (x )} + \sec {\relax (x )} \right )} + B a^{2} \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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