Optimal. Leaf size=144 \[ -\frac{2 \left (3 a^2 b^2 C-2 a^4 C+A b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.346346, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3032, 3023, 2735, 2659, 205} \[ -\frac{2 \left (3 a^2 b^2 C-2 a^4 C+A b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 3032
Rule 3023
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx &=\frac{a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{b \left (A b^2+a^2 C\right )+a \left (a^2-b^2\right ) C \cos (c+d x)-b \left (a^2-b^2\right ) C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=\frac{C \sin (c+d x)}{b^2 d}+\frac{a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{b^2 \left (A b^2+a^2 C\right )+2 a b \left (a^2-b^2\right ) C \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )}\\ &=-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d}+\frac{a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\left (A b^4-2 a^4 C+3 a^2 b^2 C\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )}\\ &=-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d}+\frac{a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\left (2 \left (A b^4-2 a^4 C+3 a^2 b^2 C\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 )}{b^3 \left (a^2-b^2\right ) d}\\ &=-\frac{2 a C x}{b^3}-\frac{2 \left (A b^4-2 a^4 C+3 a^2 b^2 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac{C \sin (c+d x)}{b^2 d}+\frac{a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.03547, size = 136, normalized size = 0.94 \[ \frac{\frac{a b \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{2 \left (3 a^2 b^2 C-2 a^4 C+A b^4\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}-2 a C (c+d x)+b C \sin (c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 359, normalized size = 2.5 \begin{align*} 2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{d{b}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}-4\,{\frac{C\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) a}{d{b}^{3}}}+2\,{\frac{a\tan \left ( 1/2\,dx+c/2 \right ) A}{d \left ({a}^{2}-{b}^{2} \right ) \left ( a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }}+2\,{\frac{{a}^{3}\tan \left ( 1/2\,dx+c/2 \right ) C}{d{b}^{2} \left ({a}^{2}-{b}^{2} \right ) \left ( a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }}-2\,{\frac{Ab}{d \left ( a+b \right ) \left ( a-b \right ) \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 ) }+4\,{\frac{{a}^{4}C}{d{b}^{3} \left ( a+b \right ) \left ( a-b \right ) \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 ) }-6\,{\frac{{a}^{2}C}{db \left ( a+b \right ) \left ( a-b \right ) \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 [B] time = 1.73507, size = 1388, normalized size = 9.64 \begin{align*} \left [-\frac{4 \,{\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} d x \cos \left (d x + c\right ) + 4 \,{\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} d x +{\left (2 \, C a^{5} - 3 \, C a^{3} b^{2} - A a b^{4} +{\left (2 \, C a^{4} b - 3 \, C a^{2} b^{3} - A b^{5}\right )} \cos \left (d x + c\right )\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 (2 \, C a^{5} b +{\left (A - 3 \, C\right )} a^{3} b^{3} -{\left (A - C\right )} a b^{5} +{\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \,{\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right ) +{\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d\right )}}, -\frac{2 \,{\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} d x \cos \left (d x + c\right ) + 2 \,{\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} d x -{\left (2 \, C a^{5} - 3 \, C a^{3} b^{2} - A a b^{4} +{\left (2 \, C a^{4} b - 3 \, C a^{2} b^{3} - A b^{5}\right )} \cos \left (d x + c\right )\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 (2 \, C a^{5} b +{\left (A - 3 \, C\right )} a^{3} b^{3} -{\left (A - C\right )} a b^{5} +{\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right ) +{\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\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 [B] time = 1.43825, size = 478, normalized size = 3.32 \begin{align*} -\frac{2 \,{\left (\frac{{\left (2 \, C a^{4} - 3 \, C a^{2} b^{2} - A b^{4}\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 )}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt{a^{2} - b^{2}}} + \frac{{\left (d x + c\right )} C a}{b^{3}} - \frac{2 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + A a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a + b\right )}{\left (a^{2} b^{2} - b^{4}\right )}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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