Optimal. Leaf size=89 \[ -\frac{2 a (b B-a C) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d} \]
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Rubi [A] time = 0.131223, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3023, 12, 2735, 2659, 205} \[ -\frac{2 a (b B-a C) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 12
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx &=\frac{C \sin (c+d x)}{b d}+\frac{\int \frac{(b B-a C) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b}\\ &=\frac{C \sin (c+d x)}{b d}+\frac{(b B-a C) \int \frac{\cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b}\\ &=\frac{(b B-a C) x}{b^2}+\frac{C \sin (c+d x)}{b d}-\frac{(a (b B-a C)) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^2}\\ &=\frac{(b B-a C) x}{b^2}+\frac{C \sin (c+d x)}{b d}-\frac{(2 a (b B-a C)) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^2 d}\\ &=\frac{(b B-a C) x}{b^2}-\frac{2 a (b B-a C) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^2 \sqrt{a+b} d}+\frac{C \sin (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.213006, size = 85, normalized size = 0.96 \[ \frac{-\frac{2 a (a C-b B) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+(c+d x) (b B-a C)+b C \sin (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 172, normalized size = 1.9 \begin{align*} 2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{db \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{db}}-2\,{\frac{C\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) a}{d{b}^{2}}}-2\,{\frac{Ba}{db\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{{a}^{2}C}{d{b}^{2}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/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 = 1.55324, size = 689, normalized size = 7.74 \begin{align*} \left [-\frac{2 \,{\left (C a^{3} - B a^{2} b - C a b^{2} + B b^{3}\right )} d x -{\left (C a^{2} - B a b\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \,{\left (C a^{2} b - C b^{3}\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{2} b^{2} - b^{4}\right )} d}, -\frac{{\left (C a^{3} - B a^{2} b - C a b^{2} + B b^{3}\right )} d x -{\left (C a^{2} - B a b\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) -{\left (C a^{2} b - C b^{3}\right )} \sin \left (d x + c\right )}{{\left (a^{2} b^{2} - b^{4}\right )} d}\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 [A] time = 1.66405, size = 192, normalized size = 2.16 \begin{align*} -\frac{\frac{{\left (C a - B b\right )}{\left (d x + c\right )}}{b^{2}} - \frac{2 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} b} + \frac{2 \,{\left (C a^{2} - B a b\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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