Optimal. Leaf size=33 \[ \frac{1}{2} \sqrt{-\cot ^2(x)} \cot (x)+\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.0272457, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4121, 3658, 3473, 3475} \[ \frac{1}{2} \sqrt{-\cot ^2(x)} \cot (x)+\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]
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 \left (-\cot ^2(x)\right )^{3/2} \, dx\\ &=-\left (\left (\sqrt{-\cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx\right )\\ &=\frac{1}{2} \cot (x) \sqrt{-\cot ^2(x)}+\left (\sqrt{-\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=\frac{1}{2} \cot (x) \sqrt{-\cot ^2(x)}+\sqrt{-\cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0168073, size = 26, normalized size = 0.79 \[ \frac{1}{2} \tan (x) \sqrt{-\cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.125, size = 91, normalized size = 2.8 \begin{align*}{\frac{\sin \left ( x \right ) \sqrt{4}}{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 [C] time = 1.51738, size = 28, normalized size = 0.85 \begin{align*} \frac{i}{2 \, \tan \left (x\right )^{2}} - \frac{1}{2} i \, \log \left (\tan \left (x\right )^{2} + 1\right ) + i \, \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 = 0.487482, size = 36, normalized size = 1.09 \begin{align*} x + \arctan \left (\frac{\cos \left (x\right )}{\sin \left (x\right )}\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 (1 - \csc ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.11824, size = 63, normalized size = 1.91 \begin{align*} \frac{1}{8} i \, \tan \left (\frac{1}{2} \, x\right )^{2} - \frac{4 i \, \tan \left (\frac{1}{2} \, x\right )^{2} - i}{8 \, \tan \left (\frac{1}{2} \, x\right )^{2}} - i \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + \frac{1}{2} i \, \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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