Optimal. Leaf size=46 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)} \]
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Rubi [A] time = 0.0219625, antiderivative size = 46, 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 = {4122, 195, 217, 206} \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (a \csc ^2(x)\right )^{3/2} \, dx &=-\left (a \operatorname{Subst}\left (\int \sqrt{a+a x^2} \, dx,x,\cot (x)\right )\right )\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+a x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}\right )\\ &=-\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right )-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0719564, size = 39, normalized size = 0.85 \[ -\frac{1}{2} a \sin (x) \sqrt{a \csc ^2(x)} \left (-\log \left (\sin \left (\frac{x}{2}\right )\right )+\log \left (\cos \left (\frac{x}{2}\right )\right )+\cot (x) \csc (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 53, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{4}\sin \left ( x \right ) }{4} \left ( \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +\cos \left ( x \right ) -\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \left ( -{\frac{a}{ \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 [B] time = 1.69075, size = 429, normalized size = 9.33 \begin{align*} -\frac{{\left (4 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \left (x\right ) -{\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \,{\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) +{\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \,{\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (a \sin \left (3 \, x\right ) + a \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (a \cos \left (3 \, x\right ) + a \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \sin \left (3 \, x\right ) + 2 \, a \sin \left (x\right )\right )} \sqrt{-a}}{2 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487157, size = 136, normalized size = 2.96 \begin{align*} -\frac{{\left (2 \, a \cos \left (x\right ) +{\left (a \cos \left (x\right )^{2} - a\right )} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}}}{4 \, \sin \left (x\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 (a \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 [B] time = 1.32157, size = 97, normalized size = 2.11 \begin{align*} \frac{1}{8} \,{\left (2 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) - \frac{{\left (\frac{2 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )}{\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac{{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1}\right )} a^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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