Integrand size = 13, antiderivative size = 144 \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {2 b^5 \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2}}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a} \]
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Time = 0.64 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3938, 4189, 4004, 3916, 2739, 632, 212} \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\frac {b \sin ^2(x) \cos (x)}{3 a^2}+\frac {2 b^5 \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2}}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \sin (x) \cos (x)}{8 a^3}+\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{8 a^5}-\frac {\sin ^3(x) \cos (x)}{4 a} \]
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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 ^3(x)}{4 a}+\frac {\int \frac {\left (-4 b+3 a \csc (x)+3 b \csc ^2(x)\right ) \sin ^3(x)}{a+b \csc (x)} \, dx}{4 a} \\ & = \frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {\int \frac {\left (-3 \left (3 a^2+4 b^2\right )-a b \csc (x)+8 b^2 \csc ^2(x)\right ) \sin ^2(x)}{a+b \csc (x)} \, dx}{12 a^2} \\ & = -\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {\int \frac {\left (-8 b \left (2 a^2+3 b^2\right )+a \left (9 a^2-4 b^2\right ) \csc (x)+3 b \left (3 a^2+4 b^2\right ) \csc ^2(x)\right ) \sin (x)}{a+b \csc (x)} \, dx}{24 a^3} \\ & = \frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {\int \frac {-3 \left (3 a^4+4 a^2 b^2+8 b^4\right )-3 a b \left (3 a^2+4 b^2\right ) \csc (x)}{a+b \csc (x)} \, dx}{24 a^4} \\ & = \frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {b^5 \int \frac {\csc (x)}{a+b \csc (x)} \, dx}{a^5} \\ & = \frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {b^4 \int \frac {1}{1+\frac {a \sin (x)}{b}} \, dx}{a^5} \\ & = \frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {\left (2 b^4\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^5} \\ & = \frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {\left (4 b^4\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^5} \\ & = \frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {2 b^5 \text {arctanh}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2}}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a} \\ \end{align*}
Time = 1.45 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\frac {36 a^4 x+48 a^2 b^2 x+96 b^4 x-\frac {192 b^5 \arctan \left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+24 a b \left (3 a^2+4 b^2\right ) \cos (x)-8 a^3 b \cos (3 x)-24 a^4 \sin (2 x)-24 a^2 b^2 \sin (2 x)+3 a^4 \sin (4 x)}{96 a^5} \]
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Time = 0.98 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.62
method | result | size |
default | \(-\frac {2 b^{5} \arctan \left (\frac {2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a^{5} \sqrt {-a^{2}+b^{2}}}+\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+\frac {1}{2} a^{2} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{7}+a \,b^{3} \tan \left (\frac {x}{2}\right )^{6}+\left (\frac {1}{2} a^{2} b^{2}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {x}{2}\right )^{5}+\left (2 a^{3} b +3 a \,b^{3}\right ) \tan \left (\frac {x}{2}\right )^{4}+\left (-\frac {1}{2} a^{2} b^{2}-\frac {11}{8} a^{4}\right ) \tan \left (\frac {x}{2}\right )^{3}+\left (3 a \,b^{3}+\frac {8}{3} a^{3} b \right ) \tan \left (\frac {x}{2}\right )^{2}+\left (-\frac {3}{8} a^{4}-\frac {1}{2} a^{2} b^{2}\right ) \tan \left (\frac {x}{2}\right )+\frac {2 a^{3} b}{3}+a \,b^{3}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4}+4 a^{2} b^{2}+8 b^{4}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{4}}{a^{5}}\) | \(234\) |
risch | \(\frac {3 x}{8 a}+\frac {x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}+\frac {3 b \,{\mathrm e}^{i x}}{8 a^{2}}+\frac {b^{3} {\mathrm e}^{i x}}{2 a^{4}}+\frac {3 b \,{\mathrm e}^{-i x}}{8 a^{2}}+\frac {b^{3} {\mathrm e}^{-i x}}{2 a^{4}}+\frac {b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, a^{5}}-\frac {b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, a^{5}}+\frac {\sin \left (4 x \right )}{32 a}-\frac {b \cos \left (3 x \right )}{12 a^{2}}-\frac {\sin \left (2 x \right )}{4 a}-\frac {\sin \left (2 x \right ) b^{2}}{4 a^{3}}\) | \(242\) |
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Time = 0.28 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.85 \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\left [\frac {12 \, \sqrt {a^{2} - b^{2}} b^{5} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (x\right ) \sin \left (x\right ) + a \cos \left (x\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (x\right )^{3} + 3 \, {\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} x + 24 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (x\right ) + 3 \, {\left (2 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (x\right )^{3} - {\left (5 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )}}, \frac {24 \, \sqrt {-a^{2} + b^{2}} b^{5} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (x\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (x\right )}\right ) - 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (x\right )^{3} + 3 \, {\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} x + 24 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (x\right ) + 3 \, {\left (2 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (x\right )^{3} - {\left (5 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )}}\right ] \]
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\[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\int \frac {\sin ^{4}{\left (x \right )}}{a + b \csc {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.75 \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=-\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} b^{5}}{\sqrt {-a^{2} + b^{2}} a^{5}} + \frac {{\left (3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac {9 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{7} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{7} + 24 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{6} + 33 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{5} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{5} + 48 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{4} + 72 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{4} - 33 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 64 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} + 72 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, x\right ) - 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) + 16 \, a^{2} b + 24 \, b^{3}}{12 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{4} a^{4}} \]
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Time = 19.32 (sec) , antiderivative size = 1639, normalized size of antiderivative = 11.38 \[ \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx=\text {Too large to display} \]
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