Optimal. Leaf size=23 \[ \frac{\sin ^2(x)}{2 a}-\frac{\sin ^3(x)}{3 a} \]
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Rubi [A] time = 0.100773, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3872, 2835, 2564, 30} \[ \frac{\sin ^2(x)}{2 a}-\frac{\sin ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2835
Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \frac{\cos ^3(x)}{a+a \csc (x)} \, dx &=\int \frac{\cos ^3(x) \sin (x)}{a+a \sin (x)} \, dx\\ &=\frac{\int \cos (x) \sin (x) \, dx}{a}-\frac{\int \cos (x) \sin ^2(x) \, dx}{a}\\ &=\frac{\operatorname{Subst}(\int x \, dx,x,\sin (x))}{a}-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\sin (x)\right )}{a}\\ &=\frac{\sin ^2(x)}{2 a}-\frac{\sin ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0167823, size = 17, normalized size = 0.74 \[ \frac{(3-2 \sin (x)) \sin ^2(x)}{6 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 18, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ( -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96026, size = 24, normalized size = 1.04 \begin{align*} -\frac{2 \, \sin \left (x\right )^{3} - 3 \, \sin \left (x\right )^{2}}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.466228, size = 63, normalized size = 2.74 \begin{align*} -\frac{3 \, \cos \left (x\right )^{2} - 2 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{6 \, 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{\cos ^{3}{\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 = 1.33055, size = 24, normalized size = 1.04 \begin{align*} -\frac{2 \, \sin \left (x\right )^{3} - 3 \, \sin \left (x\right )^{2}}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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