Optimal. Leaf size=36 \[ -\frac {\tanh ^{-1}\left (\frac {\sin (x) (a \cot (x)-b)}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3509, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sin (x) (a \cot (x)-b)}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3509
Rubi steps
\begin {align*} \int \frac {\csc (x)}{a+b \cot (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,(-b+a \cot (x)) \sin (x)\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {(-b+a \cot (x)) \sin (x)}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.06 \[ \frac {2 \tanh ^{-1}\left (\frac {b \tan \left (\frac {x}{2}\right )-a}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 98, normalized size = 2.72 \[ \frac {\log \left (-\frac {2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - a^{2} - 2 \, b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cos \relax (x) - b \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}}\right )}{2 \, \sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 61, normalized size = 1.69 \[ -\frac {\log \left (\frac {{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 35, normalized size = 0.97 \[ \frac {2 \arctanh \left (\frac {2 \tan \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.88, size = 61, normalized size = 1.69 \[ -\frac {\log \left (\frac {a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 31, normalized size = 0.86 \[ -\frac {2\,\mathrm {atanh}\left (\frac {a-b\,\mathrm {tan}\left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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 \[ \int \frac {\csc {\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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