Integrand size = 11, antiderivative size = 53 \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=-\frac {2 b \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\text {arctanh}(\cos (x))}{a} \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2826, 3855, 2739, 632, 210} \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=-\frac {2 b \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\text {arctanh}(\cos (x))}{a} \]
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Rule 210
Rule 632
Rule 2739
Rule 2826
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc (x) \, dx}{a}-\frac {b \int \frac {1}{a+b \sin (x)} \, dx}{a} \\ & = -\frac {\text {arctanh}(\cos (x))}{a}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a} \\ & = -\frac {\text {arctanh}(\cos (x))}{a}+\frac {(4 b) \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} \\ & = -\frac {2 b \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\text {arctanh}(\cos (x))}{a} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.17 \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )}{a} \]
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Time = 0.45 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {2 b \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a \sqrt {a^{2}-b^{2}}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a}\) | \(53\) |
risch | \(\frac {i b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, a}-\frac {i b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, a}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a}\) | \(155\) |
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none
Time = 0.36 (sec) , antiderivative size = 239, normalized size of antiderivative = 4.51 \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} b \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}, \frac {2 \, \sqrt {a^{2} - b^{2}} b \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left (a^{2} - b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}\right ] \]
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\[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=\int \frac {\csc {\left (x \right )}}{a + b \sin {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=-\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b}{\sqrt {a^{2} - b^{2}} a} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a} \]
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Time = 6.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.30 \[ \int \frac {\csc (x)}{a+b \sin (x)} \, dx=\frac {\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{a}+\frac {2\,b\,\mathrm {atanh}\left (\frac {\sqrt {b^2-a^2}\,\left (-1{}\mathrm {i}\,\sin \left (\frac {x}{2}\right )\,a^2+2{}\mathrm {i}\,\cos \left (\frac {x}{2}\right )\,a\,b+4{}\mathrm {i}\,\sin \left (\frac {x}{2}\right )\,b^2\right )}{1{}\mathrm {i}\,\cos \left (\frac {x}{2}\right )\,a^3+3{}\mathrm {i}\,\sin \left (\frac {x}{2}\right )\,a^2\,b-2{}\mathrm {i}\,\cos \left (\frac {x}{2}\right )\,a\,b^2-4{}\mathrm {i}\,\sin \left (\frac {x}{2}\right )\,b^3}\right )}{a\,\sqrt {b^2-a^2}} \]
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