Optimal. Leaf size=30 \[ -\frac{1}{2} \tan (x) \cot ^2(x)^{3/2}-\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.0218156, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4121, 3658, 3473, 3475} \[ -\frac{1}{2} \tan (x) \cot ^2(x)^{3/2}-\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx &=\int \cot ^2(x)^{3/2} \, dx\\ &=\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx\\ &=-\frac{1}{2} \cot ^2(x)^{3/2} \tan (x)-\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=-\frac{1}{2} \cot ^2(x)^{3/2} \tan (x)-\sqrt{\cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0143198, size = 24, normalized size = 0.8 \[ -\frac{1}{2} \tan (x) \sqrt{\cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.099, size = 92, normalized size = 3.1 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{8\, \left ( \cos \left ( x \right ) \right ) ^{3}} \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -4\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \right ) - \left ( \cos \left ( x \right ) \right ) ^{2}-4\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +4\,\ln \left ( 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \right ) -1 \right ) \left ( -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54427, size = 28, normalized size = 0.93 \begin{align*} -\frac{1}{2 \, \tan \left (x\right )^{2}} + \frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.488378, size = 80, normalized size = 2.67 \begin{align*} \frac{2 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 1}{2 \,{\left (\cos \left (x\right )^{2} - 1\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*} \int \left (\csc ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.61731, size = 63, normalized size = 2.1 \begin{align*} \frac{1}{8} \, \tan \left (\frac{1}{2} \, x\right )^{2} - \frac{4 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{8 \, \tan \left (\frac{1}{2} \, x\right )^{2}} - \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + \frac{1}{2} \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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