Optimal. Leaf size=12 \[ -\frac{\cos (x)}{a \sin (x)+b} \]
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Rubi [A] time = 0.0288894, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2754, 8} \[ -\frac{\cos (x)}{a \sin (x)+b} \]
Antiderivative was successfully verified.
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Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \sin (x)}{(b+a \sin (x))^2} \, dx &=-\frac{\cos (x)}{b+a \sin (x)}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=-\frac{\cos (x)}{b+a \sin (x)}\\ \end{align*}
Mathematica [A] time = 0.0300018, size = 12, normalized size = 1. \[ -\frac{\cos (x)}{a \sin (x)+b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 34, normalized size = 2.8 \begin{align*} 2\,{\frac{1}{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}b+2\,a\tan \left ( x/2 \right ) +b} \left ( -{\frac{a\tan \left ( x/2 \right ) }{b}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36419, size = 32, normalized size = 2.67 \begin{align*} -\frac{\cos \left (x\right )}{a \sin \left (x\right ) + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25038, size = 43, normalized size = 3.58 \begin{align*} -\frac{2 \,{\left (a \tan \left (\frac{1}{2} \, x\right ) + b\right )}}{{\left (b \tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, a \tan \left (\frac{1}{2} \, x\right ) + b\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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