Optimal. Leaf size=25 \[ -\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a} \]
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Rubi [A] time = 0.0468814, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3819, 3787, 2638, 8} \[ -\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 2638
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a+a \csc (x)} \, dx &=\frac{\cos (x)}{a+a \csc (x)}-\frac{\int (-2 a+a \csc (x)) \sin (x) \, dx}{a^2}\\ &=\frac{\cos (x)}{a+a \csc (x)}-\frac{\int 1 \, dx}{a}+\frac{2 \int \sin (x) \, dx}{a}\\ &=-\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a+a \csc (x)}\\ \end{align*}
Mathematica [A] time = 0.077861, size = 32, normalized size = 1.28 \[ -\frac{x+\cos (x)-\frac{2 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 40, normalized size = 1.6 \begin{align*} -2\,{\frac{1}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}-2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.45752, size = 105, normalized size = 4.2 \begin{align*} -\frac{2 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right )}}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}} - \frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.472957, size = 122, normalized size = 4.88 \begin{align*} -\frac{{\left (x + 2\right )} \cos \left (x\right ) + \cos \left (x\right )^{2} +{\left (x + \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + x + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \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*} \frac{\int \frac{\sin{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33014, size = 59, normalized size = 2.36 \begin{align*} -\frac{x}{a} - \frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 2\right )}}{{\left (\tan \left (\frac{1}{2} \, x\right )^{3} + \tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 1\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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