Optimal. Leaf size=97 \[ \frac{2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d} \]
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Rubi [A] time = 0.147348, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {3023, 2735, 2659, 205} \[ \frac{2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{A+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{A b+(b B-a C) \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}-\left (-A+\frac{a (b B-a C)}{b^2}\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx\\ &=\frac{(b B-a C) x}{b^2}+\frac{C \sin (c+d x)}{b d}+\frac{\left (2 \left (A-\frac{a (b B-a C)}{b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{(b B-a C) x}{b^2}+\frac{2 \left (A-\frac{a (b B-a C)}{b^2}\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b} d}+\frac{C \sin (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.224974, size = 92, normalized size = 0.95 \[ \frac{-\frac{2 \left (a (a C-b B)+A b^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+(c+d x) (b B-a C)+b C \sin (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 216, normalized size = 2.2 \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{A}{d\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{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 = 1.86926, size = 710, normalized size = 7.32 \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 + A b^{2}\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 + A b^{2}\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 = 1.18304, size = 198, normalized size = 2.04 \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 + A b^{2}\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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