Optimal. Leaf size=15 \[ \frac {1}{2 a (a \cot (x)+b)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3087, 37} \[ \frac {\tan ^2(x)}{2 a (a+b \tan (x))^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3087
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(a \cos (x)+b \sin (x))^3} \, dx &=\operatorname {Subst}\left (\int \frac {x}{(a+b x)^3} \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^2(x)}{2 a (a+b \tan (x))^2}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 47, normalized size = 3.13 \[ \frac {a (a+b \sin (2 x))+2 b^2 \sin ^2(x)}{2 a \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 116, normalized size = 7.73 \[ -\frac {4 \, a b^{2} \cos \relax (x)^{2} - a^{3} - 3 \, a b^{2} - 2 \, {\left (a^{2} b - b^{3}\right )} \cos \relax (x) \sin \relax (x)}{2 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} \cos \relax (x)^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \relax (x) \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 20, normalized size = 1.33 \[ -\frac {2 \, b \tan \relax (x) + a}{2 \, {\left (b \tan \relax (x) + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.69, size = 29, normalized size = 1.93 \[ \frac {a}{2 b^{2} \left (a +b \tan \relax (x )\right )^{2}}-\frac {1}{b^{2} \left (a +b \tan \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 84, normalized size = 5.60 \[ \frac {2 \, \sin \relax (x)^{2}}{{\left (a^{3} + \frac {4 \, a^{2} b \sin \relax (x)}{\cos \relax (x) + 1} - \frac {4 \, a^{2} b \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {a^{3} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {2 \, {\left (a^{3} - 2 \, a b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )} {\left (\cos \relax (x) + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 48, normalized size = 3.20 \[ \frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a-\frac {2\,a^2-4\,b^2}{2\,a}\right )}{b^2\,{\left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )}^2} \]
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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