Optimal. Leaf size=53 \[ -\frac{3 x}{2 a}+\frac{4 \cos ^3(x)}{3 a}-\frac{4 \cos (x)}{a}+\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin ^2(x) \cos (x)}{a \csc (x)+a} \]
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Rubi [A] time = 0.0670422, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2633, 2635, 8} \[ -\frac{3 x}{2 a}+\frac{4 \cos ^3(x)}{3 a}-\frac{4 \cos (x)}{a}+\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin ^2(x) \cos (x)}{a \csc (x)+a} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a+a \csc (x)} \, dx &=\frac{\cos (x) \sin ^2(x)}{a+a \csc (x)}-\frac{\int (-4 a+3 a \csc (x)) \sin ^3(x) \, dx}{a^2}\\ &=\frac{\cos (x) \sin ^2(x)}{a+a \csc (x)}-\frac{3 \int \sin ^2(x) \, dx}{a}+\frac{4 \int \sin ^3(x) \, dx}{a}\\ &=\frac{3 \cos (x) \sin (x)}{2 a}+\frac{\cos (x) \sin ^2(x)}{a+a \csc (x)}-\frac{3 \int 1 \, dx}{2 a}-\frac{4 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a}\\ &=-\frac{3 x}{2 a}-\frac{4 \cos (x)}{a}+\frac{4 \cos ^3(x)}{3 a}+\frac{3 \cos (x) \sin (x)}{2 a}+\frac{\cos (x) \sin ^2(x)}{a+a \csc (x)}\\ \end{align*}
Mathematica [A] time = 0.16353, size = 49, normalized size = 0.92 \[ \frac{-21 \cos (x)+\cos (3 x)+3 \left (-6 x+\sin (2 x)+\frac{8 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}\right )}{12 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 121, normalized size = 2.3 \begin{align*} -{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-2\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{4}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-8\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-{\frac{10}{3\,a} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-3\,{\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 = 1.46401, size = 243, normalized size = 4.58 \begin{align*} -\frac{\frac{7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{39 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{24 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{24 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{9 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{9 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 16}{3 \,{\left (a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{3 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{3 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac{3 \, \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 = 0.482549, size = 211, normalized size = 3.98 \begin{align*} \frac{2 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \,{\left (3 \, x + 5\right )} \cos \left (x\right ) - 12 \, \cos \left (x\right )^{2} +{\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 9 \, x - 9 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) - 9 \, x - 6}{6 \,{\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \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 ^{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 = 1.27658, size = 90, normalized size = 1.7 \begin{align*} -\frac{3 \, x}{2 \, a} - \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} - \frac{3 \, \tan \left (\frac{1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac{1}{2} \, x\right ) + 10}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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