Integrand size = 13, antiderivative size = 118 \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\frac {\left (a^2+2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
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Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2869, 2833, 12, 2739, 632, 210} \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\frac {\left (a^2+2 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2869
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {\int \frac {2 a b+\left (a^2-2 b^2\right ) \sin (x)}{(a+b \sin (x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = \frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {b \left (a^2+2 b^2\right )}{a+b \sin (x)} \, dx}{2 b \left (a^2-b^2\right )^2} \\ & = \frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (a^2+2 b^2\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = \frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (a^2+2 b^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2} \\ & = \frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (2 \left (a^2+2 b^2\right )\right ) \text {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 )}{\left (a^2-b^2\right )^2} \\ & = \frac {\left (a^2+2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a^2 \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {a \left (a^2-4 b^2\right ) \cos (x)}{2 b \left (a^2-b^2\right )^2 (a+b \sin (x))} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\frac {\left (a^2+2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a \cos (x) \left (3 a b-\left (a^2-4 b^2\right ) \sin (x)\right )}{2 (a-b)^2 (a+b)^2 (a+b \sin (x))^2} \]
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Time = 0.66 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {\frac {8 \left (a^{2}+2 b^{2}\right ) a \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {3 b \left (a^{2}+2 b^{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {a \left (a^{2}-10 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}+\frac {3 a^{2} b}{a^{4}-2 a^{2} b^{2}+b^{4}}}{{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}^{2}}+\frac {\left (a^{2}+2 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\) | \(213\) |
risch | \(\frac {i a \left (-2 i a^{3} b \,{\mathrm e}^{3 i x}+5 i a \,b^{3} {\mathrm e}^{3 i x}+2 i a^{3} b \,{\mathrm e}^{i x}-11 i a \,b^{3} {\mathrm e}^{i x}+2 a^{4} {\mathrm e}^{2 i x}-7 a^{2} b^{2} {\mathrm e}^{2 i x}-4 b^{4} {\mathrm e}^{2 i x}-a^{2} b^{2}+4 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} \left (a^{2}-b^{2}\right )^{2} b^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(434\) |
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (108) = 216\).
Time = 0.34 (sec) , antiderivative size = 516, normalized size of antiderivative = 4.37 \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\left [-\frac {2 \, {\left (a^{5} - 5 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{3} b + 2 \, a b^{3}\right )} \sin \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - 6 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right )}{4 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (x\right )\right )}}, -\frac {{\left (a^{5} - 5 \, a^{3} b^{2} + 4 \, a b^{4}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{3} b + 2 \, a b^{3}\right )} \sin \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) - 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right )}{2 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (x\right )\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.54 \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\frac {{\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} {\left (a^{2} + 2 \, b^{2}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 3 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} + 6 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, x\right ) + 10 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) + 3 \, a^{2} b}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} \]
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Time = 6.94 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.69 \[ \int \frac {\sin ^2(x)}{(a+b \sin (x))^3} \, dx=\frac {\frac {3\,a^2\,b}{a^4-2\,a^2\,b^2+b^4}-\frac {a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2-10\,b^2\right )}{a^4-2\,a^2\,b^2+b^4}+\frac {a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2+2\,b^2\right )}{a^4-2\,a^2\,b^2+b^4}+\frac {3\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^2+2\,b^2\right )}{a^4-2\,a^2\,b^2+b^4}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2+4\,b^2\right )+a^2+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}+\frac {\mathrm {atan}\left (\frac {\left (\frac {\left (a^2+2\,b^2\right )\,\left (2\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{2\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+2\,b^2\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{a^2+2\,b^2}\right )\,\left (a^2+2\,b^2\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \]
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