Integrand size = 13, antiderivative size = 168 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=-\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2}}-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (x))}{2 a^4}+\frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))} \]
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Time = 0.39 (sec) , antiderivative size = 168, 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 = {2881, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2}}-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (x))}{2 a^4}+\frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )} \]
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Rule 210
Rule 632
Rule 2739
Rule 2881
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\int \frac {\csc ^3(x) \left (a^2-3 b^2-a b \sin (x)+2 b^2 \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{a \left (a^2-b^2\right )} \\ & = -\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\int \frac {\csc ^2(x) \left (-2 b \left (2 a^2-3 b^2\right )+a \left (a^2+b^2\right ) \sin (x)+b \left (a^2-3 b^2\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 a^2 \left (a^2-b^2\right )} \\ & = \frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\int \frac {\csc (x) \left (a^4+5 a^2 b^2-6 b^4+a b \left (a^2-3 b^2\right ) \sin (x)\right )}{a+b \sin (x)} \, dx}{2 a^3 \left (a^2-b^2\right )} \\ & = \frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac {\left (b^3 \left (4 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{a^4 \left (a^2-b^2\right )}+\frac {\left (a^2+6 b^2\right ) \int \csc (x) \, dx}{2 a^4} \\ & = -\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (x))}{2 a^4}+\frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac {\left (2 b^3 \left (4 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (x))}{2 a^4}+\frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\left (4 b^3 \left (4 a^2-3 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 )}{a^4 \left (a^2-b^2\right )} \\ & = -\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2}}-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (x))}{2 a^4}+\frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\frac {\frac {16 b^3 \left (-4 a^2+3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+8 a b \cot \left (\frac {x}{2}\right )-a^2 \csc ^2\left (\frac {x}{2}\right )-4 \left (a^2+6 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \left (a^2+6 b^2\right ) \log \left (\sin \left (\frac {x}{2}\right )\right )+a^2 \sec ^2\left (\frac {x}{2}\right )-\frac {8 a b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}-8 a b \tan \left (\frac {x}{2}\right )}{8 a^4} \]
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Time = 0.84 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\frac {a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-4 b \tan \left (\frac {x}{2}\right )}{4 a^{3}}-\frac {4 b^{3} \left (\frac {\frac {b^{2} \tan \left (\frac {x}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a}+\frac {\left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (2 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {x}{2}\right )}\) | \(181\) |
risch | \(\frac {2 a^{4} {\mathrm e}^{4 i x}+2 a^{2} b^{2} {\mathrm e}^{4 i x}-i a^{3} b \,{\mathrm e}^{5 i x}+3 i a \,b^{3} {\mathrm e}^{5 i x}+8 i a^{3} b \,{\mathrm e}^{3 i x}-12 i a \,b^{3} {\mathrm e}^{3 i x}+2 a^{4} {\mathrm e}^{2 i x}-10 a^{2} b^{2} {\mathrm e}^{2 i x}-7 i a^{3} b \,{\mathrm e}^{i x}+9 i a \,b^{3} {\mathrm e}^{i x}-6 b^{4} {\mathrm e}^{4 i x}+12 b^{4} {\mathrm e}^{2 i x}+4 a^{2} b^{2}-6 b^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left (a^{2}-b^{2}\right ) \left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right ) a^{3}}-\frac {4 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{2}}+\frac {3 i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{4}}+\frac {4 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{2}}-\frac {3 i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{4}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{a^{4}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{a^{4}}\) | \(572\) |
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Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (158) = 316\).
Time = 0.74 (sec) , antiderivative size = 1174, normalized size of antiderivative = 6.99 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=-\frac {2 \, {\left (4 \, a^{2} b^{3} - 3 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (b^{5} \tan \left (\frac {1}{2} \, x\right ) + a b^{4}\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}} + \frac {{\left (a^{2} + 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 36 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}}{8 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
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Time = 7.64 (sec) , antiderivative size = 1576, normalized size of antiderivative = 9.38 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\text {Too large to display} \]
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