Integrand size = 13, antiderivative size = 110 \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=-\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac {2 b^4 \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \sqrt {a^2-b^2}}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a} \]
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Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3938, 4189, 4004, 3916, 2739, 632, 212} \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\frac {b \sin (x) \cos (x)}{2 a^2}-\frac {2 b^4 \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \sqrt {a^2-b^2}}-\frac {b x \left (a^2+2 b^2\right )}{2 a^4}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}-\frac {\sin ^2(x) \cos (x)}{3 a} \]
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Rule 212
Rule 632
Rule 2739
Rule 3916
Rule 3938
Rule 4004
Rule 4189
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (x) \sin ^2(x)}{3 a}+\frac {\int \frac {\left (-3 b+2 a \csc (x)+2 b \csc ^2(x)\right ) \sin ^2(x)}{a+b \csc (x)} \, dx}{3 a} \\ & = \frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a}-\frac {\int \frac {\left (-2 \left (2 a^2+3 b^2\right )-a b \csc (x)+3 b^2 \csc ^2(x)\right ) \sin (x)}{a+b \csc (x)} \, dx}{6 a^2} \\ & = -\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a}+\frac {\int \frac {-3 b \left (a^2+2 b^2\right )-3 a b^2 \csc (x)}{a+b \csc (x)} \, dx}{6 a^3} \\ & = -\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a}+\frac {b^4 \int \frac {\csc (x)}{a+b \csc (x)} \, dx}{a^4} \\ & = -\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a}+\frac {b^3 \int \frac {1}{1+\frac {a \sin (x)}{b}} \, dx}{a^4} \\ & = -\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a}+\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^4} \\ & = -\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {x}{2}\right )\right )}{a^4} \\ & = -\frac {b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac {2 b^4 \text {arctanh}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \sqrt {a^2-b^2}}-\frac {\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac {b \cos (x) \sin (x)}{2 a^2}-\frac {\cos (x) \sin ^2(x)}{3 a} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\frac {-6 b \left (a^2+2 b^2\right ) x+\frac {24 b^4 \arctan \left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-3 a \left (3 a^2+4 b^2\right ) \cos (x)+a^3 \cos (3 x)+3 a^2 b \sin (2 x)}{12 a^4} \]
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Time = 0.72 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.32
method | result | size |
default | \(\frac {2 b^{4} \arctan \left (\frac {2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a^{4} \sqrt {-a^{2}+b^{2}}}+\frac {\frac {2 \left (-\frac {a^{2} b \tan \left (\frac {x}{2}\right )^{5}}{2}-a \,b^{2} \tan \left (\frac {x}{2}\right )^{4}+\left (-2 a^{3}-2 a \,b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}+\frac {a^{2} b \tan \left (\frac {x}{2}\right )}{2}-\frac {2 a^{3}}{3}-a \,b^{2}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}-b \left (a^{2}+2 b^{2}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{4}}\) | \(145\) |
risch | \(-\frac {x b}{2 a^{2}}-\frac {x \,b^{3}}{a^{4}}-\frac {3 \,{\mathrm e}^{i x}}{8 a}-\frac {{\mathrm e}^{i x} b^{2}}{2 a^{3}}-\frac {3 \,{\mathrm e}^{-i x}}{8 a}-\frac {{\mathrm e}^{-i x} b^{2}}{2 a^{3}}-\frac {i b^{4} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {-a^{2}+b^{2}}\, b +a^{2}-b^{2}\right )}{a \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{4}}+\frac {i b^{4} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {-a^{2}+b^{2}}\, b -a^{2}+b^{2}\right )}{a \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{4}}+\frac {\cos \left (3 x \right )}{12 a}+\frac {b \sin \left (2 x \right )}{4 a^{2}}\) | \(215\) |
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Time = 0.26 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.99 \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\left [\frac {3 \, \sqrt {a^{2} - b^{2}} b^{4} \log \left (-\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2} - 2 \, {\left (b \cos \left (x\right ) \sin \left (x\right ) + a \cos \left (x\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{5} - a^{3} b^{2}\right )} \cos \left (x\right )^{3} + 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - 3 \, {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} x - 6 \, {\left (a^{5} - a b^{4}\right )} \cos \left (x\right )}{6 \, {\left (a^{6} - a^{4} b^{2}\right )}}, -\frac {6 \, \sqrt {-a^{2} + b^{2}} b^{4} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (x\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (x\right )}\right ) - 2 \, {\left (a^{5} - a^{3} b^{2}\right )} \cos \left (x\right )^{3} - 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \, {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} x + 6 \, {\left (a^{5} - a b^{4}\right )} \cos \left (x\right )}{6 \, {\left (a^{6} - a^{4} b^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.35 \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} b^{4}}{\sqrt {-a^{2} + b^{2}} a^{4}} - \frac {{\left (a^{2} b + 2 \, b^{3}\right )} x}{2 \, a^{4}} - \frac {3 \, a b \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 12 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, x\right ) + 4 \, a^{2} + 6 \, b^{2}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a^{3}} \]
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Time = 19.02 (sec) , antiderivative size = 1218, normalized size of antiderivative = 11.07 \[ \int \frac {\sin ^3(x)}{a+b \csc (x)} \, dx=\text {Too large to display} \]
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