Optimal. Leaf size=53 \[ -\frac {2 b \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cos (x))}{a} \]
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Rubi [A]
time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2826, 3855,
2739, 632, 210} \begin {gather*} -\frac {2 b \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cos (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2826
Rule 3855
Rubi steps
\begin {align*} \int \frac {\csc (x)}{a+b \sin (x)} \, dx &=\frac {\int \csc (x) \, dx}{a}-\frac {b \int \frac {1}{a+b \sin (x)} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\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 {\tanh ^{-1}(\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 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cos (x))}{a}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 62, normalized size = 1.17 \begin {gather*} \frac {-\frac {2 b \tan ^{-1}\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 53, normalized size = 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 | \(\frac {i b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, a}-\frac {i b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, a}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 239, normalized size = 4.51 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc {\left (x \right )}}{a + b \sin {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 63, normalized size = 1.19 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.86, size = 122, normalized size = 2.30 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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