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 = 0.201573, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 2828
Rule 2976
Rule 2968
Rule 3023
Rule 2735
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*}
Mathematica [A] time = 0.0823791, 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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Maple [A] time = 0.046, size = 52, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}A\sin \left ( x \right ) \cos \left ( x \right ) }{2}}+{\frac{3\,{a}^{2}Ax}{2}}+{a}^{2}B\sin \left ( x \right ) +2\,{a}^{2}A\sin \left ( x \right ) +2\,{a}^{2}Bx+{a}^{2}B\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01363, size = 73, normalized size = 1.28 \begin{align*} \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 \left (x\right ) + \tan \left (x\right )\right ) + 2 \, A a^{2} \sin \left (x\right ) + B a^{2} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.57262, size = 170, normalized size = 2.98 \begin{align*} \frac{1}{2} \,{\left (3 \, A + 4 \, B\right )} a^{2} x + \frac{1}{2} \, B a^{2} \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, B a^{2} \log \left (-\sin \left (x\right ) + 1\right ) + \frac{1}{2} \,{\left (A a^{2} \cos \left (x\right ) + 2 \,{\left (2 \, A + B\right )} a^{2}\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.7605, size = 61, normalized size = 1.07 \begin{align*} \frac{3 A a^{2} x}{2} + 2 A a^{2} \sin{\left (x \right )} + \frac{A a^{2} \sin{\left (2 x \right )}}{4} + 2 B a^{2} x + B a^{2} \log{\left (\tan{\left (x \right )} + \sec{\left (x \right )} \right )} + B a^{2} \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18303, size = 135, normalized size = 2.37 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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