Optimal. Leaf size=179 \[ -\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{3 a^2 \left (-5 a^2 b^2+2 a^4+4 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2}}+\frac{a^2 \sin ^2(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{3 a x}{b^4} \]
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Rubi [A] time = 0.411073, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {2792, 3031, 3023, 2735, 2660, 618, 204} \[ -\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{3 a^2 \left (-5 a^2 b^2+2 a^4+4 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2}}+\frac{a^2 \sin ^2(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{3 a x}{b^4} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3031
Rule 3023
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\sin ^4(x)}{(a+b \sin (x))^3} \, dx &=\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{\int \frac{\sin (x) \left (2 a^2-2 a b \sin (x)-\left (3 a^2-2 b^2\right ) \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{\int \frac{3 a^2 b \left (a^2-2 b^2\right )+a \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin (x)-b \left (3 a^2-2 b^2\right ) \left (a^2-b^2\right ) \sin ^2(x)}{a+b \sin (x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{\int \frac{3 a^2 b^2 \left (a^2-2 b^2\right )+6 a b \left (a^2-b^2\right )^2 \sin (x)}{a+b \sin (x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{3 a x}{b^4}-\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{\left (3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right )\right ) \int \frac{1}{a+b \sin (x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{3 a x}{b^4}-\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac{\left (3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{3 a x}{b^4}-\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac{\left (6 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right )\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{x}{2}\right )\right )}{b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{3 a x}{b^4}+\frac{3 a^2 \left (2 a^4-5 a^2 b^2+4 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{5/2}}-\frac{\left (3 a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )}+\frac{a^2 \cos (x) \sin ^2(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac{3 a^3 \left (a^2-2 b^2\right ) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end{align*}
Mathematica [A] time = 0.756829, size = 144, normalized size = 0.8 \[ \frac{\frac{6 a^2 \left (-5 a^2 b^2+2 a^4+4 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{a^3 b \left (8 b^2-5 a^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}+\frac{a^4 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}-6 a x-2 b \cos (x)}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 634, normalized size = 3.5 \begin{align*} \text{result too large to display} \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 = 2.28055, size = 2013, normalized size = 11.25 \begin{align*} \text{result too large to display} \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 = 1.98173, size = 346, normalized size = 1.93 \begin{align*} \frac{3 \,{\left (2 \, a^{6} - 5 \, a^{4} b^{2} + 4 \, a^{2} b^{4}\right )}{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt{a^{2} - b^{2}}} - \frac{3 \, a^{5} b \tan \left (\frac{1}{2} \, x\right )^{3} - 6 \, a^{3} b^{3} \tan \left (\frac{1}{2} \, x\right )^{3} + 4 \, a^{6} \tan \left (\frac{1}{2} \, x\right )^{2} + a^{4} b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} - 14 \, a^{2} b^{4} \tan \left (\frac{1}{2} \, x\right )^{2} + 13 \, a^{5} b \tan \left (\frac{1}{2} \, x\right ) - 22 \, a^{3} b^{3} \tan \left (\frac{1}{2} \, x\right ) + 4 \, a^{6} - 7 \, a^{4} b^{2}}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )}{\left (a \tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, x\right ) + a\right )}^{2}} - \frac{3 \, a x}{b^{4}} - \frac{2}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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