Optimal. Leaf size=40 \[ \frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin (x) \cos (x)}{a \csc (x)+a} \]
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Rubi [A] time = 0.0612801, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2638} \[ \frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin (x) \cos (x)}{a \csc (x)+a} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 2635
Rule 8
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sin ^2(x)}{a+a \csc (x)} \, dx &=\frac{\cos (x) \sin (x)}{a+a \csc (x)}-\frac{\int (-3 a+2 a \csc (x)) \sin ^2(x) \, dx}{a^2}\\ &=\frac{\cos (x) \sin (x)}{a+a \csc (x)}-\frac{2 \int \sin (x) \, dx}{a}+\frac{3 \int \sin ^2(x) \, dx}{a}\\ &=\frac{2 \cos (x)}{a}-\frac{3 \cos (x) \sin (x)}{2 a}+\frac{\cos (x) \sin (x)}{a+a \csc (x)}+\frac{3 \int 1 \, dx}{2 a}\\ &=\frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \cos (x) \sin (x)}{2 a}+\frac{\cos (x) \sin (x)}{a+a \csc (x)}\\ \end{align*}
Mathematica [A] time = 0.129952, size = 42, normalized size = 1.05 \[ -\frac{-6 x+\sin (2 x)-4 \cos (x)+\frac{8 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{4 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 100, normalized size = 2.5 \begin{align*}{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{1}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+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 = 1.45891, size = 173, normalized size = 4.32 \begin{align*} \frac{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 4}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{2 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}} + \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 = 0.474, size = 166, normalized size = 4.15 \begin{align*} \frac{\cos \left (x\right )^{3} + 3 \,{\left (x + 1\right )} \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{2} - 3 \, x - \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + 3 \, x + 2}{2 \,{\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 ^{2}{\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 = 1.36766, size = 76, normalized size = 1.9 \begin{align*} \frac{3 \, x}{2 \, a} + \frac{\tan \left (\frac{1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \tan \left (\frac{1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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