Optimal. Leaf size=114 \[ -\frac{2 a^3 B \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{B x \left (2 a^2+b^2\right )}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d} \]
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Rubi [A] time = 0.21655, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {21, 2793, 3023, 2735, 2659, 205} \[ -\frac{2 a^3 B \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{B x \left (2 a^2+b^2\right )}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2793
Rule 3023
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx &=B \int \frac{\cos ^3(c+d x)}{a+b \cos (c+d x)} \, dx\\ &=\frac{B \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{B \int \frac{a+b \cos (c+d x)-2 a \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b}\\ &=-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{B \int \frac{a b+\left (2 a^2+b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2}\\ &=\frac{\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (a^3 B\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^3}\\ &=\frac{\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (2 a^3 B\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 d}\\ &=\frac{\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac{2 a^3 B \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^3 \sqrt{a+b} d}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \cos (c+d x) \sin (c+d x)}{2 b d}\\ \end{align*}
Mathematica [A] time = 0.228605, size = 98, normalized size = 0.86 \[ \frac{B \left (2 \left (2 a^2+b^2\right ) (c+d x)+\frac{8 a^3 \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}-4 a b \sin (c+d x)+b^2 \sin (2 (c+d x))\right )}{4 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.122, size = 229, normalized size = 2. \begin{align*} -2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}Ba}{{b}^{2}d \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{B}{bd} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) a}{{b}^{2}d \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{B}{bd}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B{a}^{2}}{d{b}^{3}}}+{\frac{B}{bd}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-2\,{\frac{{a}^{3}B}{d{b}^{3}\sqrt{ \left ( a-b \right ) \left ( a+b \right ) }}\arctan \left ({\frac{\tan \left ( 1/2\,dx+c/2 \right ) \left ( a-b \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 = 1.60562, size = 759, normalized size = 6.66 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2}} B a^{3} \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 ) -{\left (2 \, B a^{4} - B a^{2} b^{2} - B b^{4}\right )} d x +{\left (2 \, B a^{3} b - 2 \, B a b^{3} -{\left (B a^{2} b^{2} - B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{2} b^{3} - b^{5}\right )} d}, -\frac{2 \, \sqrt{a^{2} - b^{2}} B a^{3} \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 \, B a^{4} - B a^{2} b^{2} - B b^{4}\right )} d x +{\left (2 \, B a^{3} b - 2 \, B a b^{3} -{\left (B a^{2} b^{2} - B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{2} b^{3} - b^{5}\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 = 1.46652, size = 250, normalized size = 2.19 \begin{align*} -\frac{\frac{4 \,{\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 )} B a^{3}}{\sqrt{a^{2} - b^{2}} b^{3}} - \frac{{\left (2 \, B a^{2} + B b^{2}\right )}{\left (d x + c\right )}}{b^{3}} + \frac{2 \,{\left (2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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