Optimal. Leaf size=63 \[ \frac{x^2}{12}+\frac{1}{4} x^4 \csc ^{-1}(x)^2+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{\log (x)}{3} \]
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Rubi [A] time = 0.0671453, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5223, 3758, 4185, 4184, 3475} \[ \frac{x^2}{12}+\frac{1}{4} x^4 \csc ^{-1}(x)^2+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{\log (x)}{3} \]
Antiderivative was successfully verified.
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Rule 5223
Rule 3758
Rule 4185
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x^3 \csc ^{-1}(x)^2 \, dx &=-\operatorname{Subst}\left (\int x^2 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{1}{4} x^4 \csc ^{-1}(x)^2-\frac{1}{2} \operatorname{Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{x^2}{12}+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{4} x^4 \csc ^{-1}(x)^2-\frac{1}{3} \operatorname{Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{x^2}{12}+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{4} x^4 \csc ^{-1}(x)^2-\frac{1}{3} \operatorname{Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac{x^2}{12}+\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \csc ^{-1}(x)+\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x^3 \csc ^{-1}(x)+\frac{1}{4} x^4 \csc ^{-1}(x)^2+\frac{\log (x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0475762, size = 42, normalized size = 0.67 \[ \frac{1}{12} \left (x^2+3 x^4 \csc ^{-1}(x)^2+2 \sqrt{1-\frac{1}{x^2}} \left (x^2+2\right ) x \csc ^{-1}(x)+4 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 56, normalized size = 0.9 \begin{align*}{\frac{{x}^{4} \left ({\rm arccsc} \left (x\right ) \right ) ^{2}}{4}}+{\frac{{\rm arccsc} \left (x\right ){x}^{3}}{6}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}}+{\frac{{x}^{2}}{12}}+{\frac{{\rm arccsc} \left (x\right )x}{3}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}}-{\frac{\ln \left ({x}^{-1} \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66807, size = 128, normalized size = 2.03 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccsc}\left (x\right )^{2} + \frac{2 \, x^{4} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) + 2 \, x^{2} \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right ) +{\left (x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt{x + 1} \sqrt{x - 1} - 4 \, \arctan \left (1, \sqrt{x + 1} \sqrt{x - 1}\right )}{12 \, \sqrt{x + 1} \sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62521, size = 115, normalized size = 1.83 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccsc}\left (x\right )^{2} + \frac{1}{6} \,{\left (x^{2} + 2\right )} \sqrt{x^{2} - 1} \operatorname{arccsc}\left (x\right ) + \frac{1}{12} \, x^{2} + \frac{1}{3} \, \log \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 x^{3} \operatorname{acsc}^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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