Optimal. Leaf size=66 \[ -\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}+\frac {b^2 \tanh ^{-1}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 66, 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 = {3511, 3486, 2638, 3509, 206} \[ -\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}+\frac {b^2 \tanh ^{-1}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2638
Rule 3486
Rule 3509
Rule 3511
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a+b \cot (x)} \, dx &=\frac {\int (a-b \cot (x)) \sin (x) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {\csc (x)}{a+b \cot (x)} \, dx}{a^2+b^2}\\ &=-\frac {b \sin (x)}{a^2+b^2}+\frac {a \int \sin (x) \, dx}{a^2+b^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,(-b+a \cot (x)) \sin (x)\right )}{a^2+b^2}\\ &=\frac {b^2 \tanh ^{-1}\left (\frac {(b-a \cot (x)) \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \cos (x)}{a^2+b^2}-\frac {b \sin (x)}{a^2+b^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 62, normalized size = 0.94 \[ \frac {2 b^2 \tanh ^{-1}\left (\frac {b \tan \left (\frac {x}{2}\right )-a}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \cos (x)+b \sin (x)}{a^2+b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 146, normalized size = 2.21 \[ \frac {\sqrt {a^{2} + b^{2}} b^{2} \log \left (-\frac {2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - a^{2} - 2 \, b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cos \relax (x) - b \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}}\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \cos \relax (x) - 2 \, {\left (a^{2} b + b^{3}\right )} \sin \relax (x)}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 94, normalized size = 1.42 \[ -\frac {b^{2} \log \left (\frac {{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, x\right ) + a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 84, normalized size = 1.27 \[ \frac {8 b^{2} \arctanh \left (\frac {2 \tan \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}+\frac {-2 \tan \left (\frac {x}{2}\right ) b -2 a}{\left (a^{2}+b^{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 106, normalized size = 1.61 \[ -\frac {b^{2} \log \left (\frac {a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a + \frac {b \sin \relax (x)}{\cos \relax (x) + 1}\right )}}{a^{2} + b^{2} + \frac {{\left (a^{2} + b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 94, normalized size = 1.42 \[ -\frac {\frac {2\,a}{a^2+b^2}+\frac {2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {2\,a\,b^2+2\,a^3-2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+b^2\right )}{2\,{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]
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 \[ \int \frac {\sin {\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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