Optimal. Leaf size=17 \[ \frac {\sin (x)}{b (a \sin (x)+b \cos (x))} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3075} \[ \frac {\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
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Rule 3075
Rubi steps
\begin {align*} \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx &=\frac {\sin (x)}{b (b \cos (x)+a \sin (x))}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 17, normalized size = 1.00 \[ \frac {\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 39, normalized size = 2.29 \[ -\frac {a \cos \relax (x) - b \sin \relax (x)}{{\left (a^{2} b + b^{3}\right )} \cos \relax (x) + {\left (a^{3} + a b^{2}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 13, normalized size = 0.76 \[ -\frac {1}{{\left (a \tan \relax (x) + b\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 14, normalized size = 0.82 \[ -\frac {1}{\left (a \tan \relax (x )+b \right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 14, normalized size = 0.82 \[ -\frac {1}{a^{2} \tan \relax (x) + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 29, normalized size = 1.71 \[ \frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )}{b\,\left (-b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )+b\right )} \]
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 \[ \int \frac {1}{\left (a \sin {\relax (x )} + b \cos {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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