Optimal. Leaf size=87 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a+b}}\right )}{b e \sqrt{a-b} \sqrt{a+b}}-\frac{C \log (a+b \cos (d+e x))}{b e}+\frac{B x}{b} \]
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Rubi [A] time = 0.143702, antiderivative size = 87, normalized size of antiderivative = 1., 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 = {4377, 2735, 2659, 205, 2668, 31} \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a+b}}\right )}{b e \sqrt{a-b} \sqrt{a+b}}-\frac{C \log (a+b \cos (d+e x))}{b e}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Rule 4377
Rule 2735
Rule 2659
Rule 205
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{a+b \cos (d+e x)} \, dx &=C \int \frac{\sin (d+e x)}{a+b \cos (d+e x)} \, dx+\int \frac{A+B \cos (d+e x)}{a+b \cos (d+e x)} \, dx\\ &=\frac{B x}{b}-\frac{(-A b+a B) \int \frac{1}{a+b \cos (d+e x)} \, dx}{b}-\frac{C \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \cos (d+e x)\right )}{b e}\\ &=\frac{B x}{b}-\frac{C \log (a+b \cos (d+e x))}{b e}+\frac{(2 (A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{b e}\\ &=\frac{B x}{b}+\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b \sqrt{a+b} e}-\frac{C \log (a+b \cos (d+e x))}{b e}\\ \end{align*}
Mathematica [A] time = 0.224411, size = 82, normalized size = 0.94 \[ \frac{\frac{2 (a B-A b) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}-C \log (a+b \cos (d+e x))+B (d+e x)}{b e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 226, normalized size = 2.6 \begin{align*}{\frac{C}{eb}\ln \left ( \left ( \tan \left ({\frac{ex}{2}}+{\frac{d}{2}} \right ) \right ) ^{2}+1 \right ) }+2\,{\frac{B\arctan \left ( \tan \left ( 1/2\,ex+d/2 \right ) \right ) }{eb}}-{\frac{Ca}{eb \left ( a-b \right ) }\ln \left ( \left ( \tan \left ({\frac{ex}{2}}+{\frac{d}{2}} \right ) \right ) ^{2}a- \left ( \tan \left ({\frac{ex}{2}}+{\frac{d}{2}} \right ) \right ) ^{2}b+a+b \right ) }+{\frac{C}{e \left ( a-b \right ) }\ln \left ( \left ( \tan \left ({\frac{ex}{2}}+{\frac{d}{2}} \right ) \right ) ^{2}a- \left ( \tan \left ({\frac{ex}{2}}+{\frac{d}{2}} \right ) \right ) ^{2}b+a+b \right ) }+2\,{\frac{A}{e\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,ex+d/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-2\,{\frac{Ba}{eb\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,ex+d/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9228, size = 721, normalized size = 8.29 \begin{align*} \left [\frac{2 \,{\left (B a^{2} - B b^{2}\right )} e x +{\left (B a - A b\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (e x + d\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (e x + d\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (e x + d\right ) + b\right )} \sin \left (e x + d\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}}\right ) -{\left (C a^{2} - C b^{2}\right )} \log \left (b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}\right )}{2 \,{\left (a^{2} b - b^{3}\right )} e}, \frac{2 \,{\left (B a^{2} - B b^{2}\right )} e x - 2 \,{\left (B a - A b\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (e x + d\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (e x + d\right )}\right ) -{\left (C a^{2} - C b^{2}\right )} \log \left (b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}\right )}{2 \,{\left (a^{2} b - b^{3}\right )} e}\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 [B] time = 1.188, size = 225, normalized size = 2.59 \begin{align*}{\left (\frac{{\left (x e + d\right )} B}{b} - \frac{C \log \left (-a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} - a - b\right )}{b} + \frac{C \log \left (\tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 1\right )}{b} + \frac{2 \,{\left (\pi \left \lfloor \frac{x e + d}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )}{\left (B a - A b\right )}}{\sqrt{a^{2} - b^{2}} b}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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