Optimal. Leaf size=107 \[ -\frac{2 a^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^3 d \sqrt{a^2-b^2}}+\frac{x \left (2 a^2+b^2\right )}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d} \]
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Rubi [A] time = 0.18554, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2793, 3023, 2735, 2660, 618, 204} \[ -\frac{2 a^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^3 d \sqrt{a^2-b^2}}+\frac{x \left (2 a^2+b^2\right )}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\int \frac{a+b \sin (c+d x)-2 a \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{2 b}\\ &=\frac{a \cos (c+d x)}{b^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\int \frac{a b+\left (2 a^2+b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2 b^2}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}-\frac{a^3 \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^3}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a 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 ) x}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\left (4 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^3 d}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}-\frac{2 a^3 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^3 \sqrt{a^2-b^2} d}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}\\ \end{align*}
Mathematica [A] time = 0.24686, size = 97, normalized size = 0.91 \[ \frac{2 \left (2 a^2+b^2\right ) (c+d x)-\frac{8 a^3 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+4 a b \cos (c+d x)-b^2 \sin (2 (c+d x))}{4 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 216, normalized size = 2. \begin{align*}{\frac{1}{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{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}{{b}^{2}d \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{1}{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{a}{{b}^{2}d \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ){a}^{2}}{d{b}^{3}}}+{\frac{1}{bd}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-2\,{\frac{{a}^{3}}{d{b}^{3}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \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 = 2.27219, size = 768, normalized size = 7.18 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2}} a^{3} \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) -{\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d x +{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )}{2 \,{\left (a^{2} b^{3} - b^{5}\right )} d}, \frac{2 \, \sqrt{a^{2} - b^{2}} a^{3} \arctan \left (-\frac{a \sin \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) +{\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d x -{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \,{\left (a^{3} b - a b^{3}\right )} \cos \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 = 1.12006, size = 204, normalized size = 1.91 \begin{align*} -\frac{\frac{4 \,{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )} a^{3}}{\sqrt{a^{2} - b^{2}} b^{3}} - \frac{{\left (2 \, a^{2} + b^{2}\right )}{\left (d x + c\right )}}{b^{3}} - \frac{2 \,{\left (b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a\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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