Integrand size = 13, antiderivative size = 67 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=-\frac {2 b^2 \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {(b-a \cos (x)) \csc (x)}{a^2-b^2} \]
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Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2775, 12, 2738, 211} \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=\frac {\csc (x) (b-a \cos (x))}{a^2-b^2}-\frac {2 b^2 \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]
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Rule 12
Rule 211
Rule 2738
Rule 2775
Rubi steps \begin{align*} \text {integral}& = \frac {(b-a \cos (x)) \csc (x)}{a^2-b^2}+\frac {\int \frac {b^2}{a+b \cos (x)} \, dx}{-a^2+b^2} \\ & = \frac {(b-a \cos (x)) \csc (x)}{a^2-b^2}-\frac {b^2 \int \frac {1}{a+b \cos (x)} \, dx}{a^2-b^2} \\ & = \frac {(b-a \cos (x)) \csc (x)}{a^2-b^2}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2-b^2} \\ & = -\frac {2 b^2 \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {(b-a \cos (x)) \csc (x)}{a^2-b^2} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=-\frac {2 b^2 \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+\frac {(b-a \cos (x)) \csc (x)}{a^2-b^2} \]
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Time = 0.59 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\tan \left (\frac {x}{2}\right )}{2 a -2 b}-\frac {1}{2 \left (a +b \right ) \tan \left (\frac {x}{2}\right )}-\frac {2 b^{2} \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\) | \(78\) |
risch | \(-\frac {2 i \left (-{\mathrm e}^{i x} b +a \right )}{\left ({\mathrm e}^{2 i x}-1\right ) \left (a^{2}-b^{2}\right )}+\frac {b^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}-\frac {b^{2} \ln \left ({\mathrm e}^{i x}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}\) | \(186\) |
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Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 3.43 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=\left [\frac {\sqrt {-a^{2} + b^{2}} b^{2} \log \left (\frac {2 \, a b \cos \left (x\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right ) \sin \left (x\right ) + 2 \, a^{2} b - 2 \, b^{3} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (x\right )}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )}, -\frac {\sqrt {a^{2} - b^{2}} b^{2} \arctan \left (-\frac {a \cos \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x\right )}\right ) \sin \left (x\right ) - a^{2} b + b^{3} + {\left (a^{3} - a b^{2}\right )} \cos \left (x\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )}\right ] \]
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\[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=\int \frac {\csc ^{2}{\left (x \right )}}{a + b \cos {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.36 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x\right ) - b \tan \left (\frac {1}{2} \, x\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{2}}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, {\left (a - b\right )}} - \frac {1}{2 \, {\left (a + b\right )} \tan \left (\frac {1}{2} \, x\right )} \]
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Time = 14.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)} \, dx=\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a-2\,b}-\frac {2\,b^2\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2-b^2\right )}{{\left (a+b\right )}^{3/2}\,\sqrt {a-b}}\right )}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {a-b}{\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a+b\right )\,\left (2\,a-2\,b\right )} \]
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