Optimal. Leaf size=60 \[ \frac {a}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}-\frac {b \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3154, 3074, 206} \[ \frac {a}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}-\frac {b \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3074
Rule 3154
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac {a}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {b \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}\\ &=\frac {a}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^2+b^2}\\ &=-\frac {b \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 62, normalized size = 1.03 \[ \frac {a}{\left (a^2+b^2\right ) (a \cos (x)+b \sin (x))}+\frac {2 b \tanh ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )-b}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 164, normalized size = 2.73 \[ \frac {2 \, a^{3} + 2 \, a b^{2} + {\left (a b \cos \relax (x) + b^{2} \sin \relax (x)\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \relax (x) - a \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}}\right )}{2 \, {\left ({\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \relax (x) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 103, normalized size = 1.72 \[ -\frac {b \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, x\right ) + a\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )} {\left (a^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 97, normalized size = 1.62 \[ \frac {8 \tan \left (\frac {x}{2}\right ) b +8 a}{\left (-4 a^{2}-4 b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) b -a \right )}-\frac {8 b \arctanh \left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (-4 a^{2}-4 b^{2}\right ) \sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 128, normalized size = 2.13 \[ -\frac {b \log \left (\frac {b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (a + \frac {b \sin \relax (x)}{\cos \relax (x) + 1}\right )}}{a^{3} + a b^{2} + \frac {2 \, {\left (a^{2} b + b^{3}\right )} \sin \relax (x)}{\cos \relax (x) + 1} - \frac {{\left (a^{3} + a b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 86, normalized size = 1.43 \[ \frac {\frac {2\,a}{a^2+b^2}+\frac {2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}}{-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a}-\frac {2\,b\,\mathrm {atanh}\left (\frac {2\,b-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,\sqrt {a^2+b^2}}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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