Optimal. Leaf size=20 \[ \frac{\sin (x)}{a}-\frac{b \log (a \sin (x)+b)}{a^2} \]
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Rubi [A] time = 0.0631133, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac{\sin (x)}{a}-\frac{b \log (a \sin (x)+b)}{a^2} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (x)}{a+b \csc (x)} \, dx &=\int \frac{\cos (x) \sin (x)}{b+a \sin (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (b+x)} \, dx,x,a \sin (x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{b+x} \, dx,x,a \sin (x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{b}{b+x}\right ) \, dx,x,a \sin (x)\right )}{a^2}\\ &=-\frac{b \log (b+a \sin (x))}{a^2}+\frac{\sin (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0120148, size = 19, normalized size = 0.95 \[ \frac{a \sin (x)-b \log (a \sin (x)+b)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 31, normalized size = 1.6 \begin{align*} -{\frac{b\ln \left ( a+b\csc \left ( x \right ) \right ) }{{a}^{2}}}+{\frac{1}{a\csc \left ( x \right ) }}+{\frac{b\ln \left ( \csc \left ( x \right ) \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955774, size = 27, normalized size = 1.35 \begin{align*} -\frac{b \log \left (a \sin \left (x\right ) + b\right )}{a^{2}} + \frac{\sin \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489971, size = 53, normalized size = 2.65 \begin{align*} -\frac{b \log \left (a \sin \left (x\right ) + b\right ) - a \sin \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (x \right )}}{a + b \csc{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34851, size = 28, normalized size = 1.4 \begin{align*} -\frac{b \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{2}} + \frac{\sin \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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