Optimal. Leaf size=89 \[ \frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )+\frac{(-x-1)^{5/2} \sqrt{x}}{15 \sqrt{-x}}+\frac{2 (-x-1)^{3/2} \sqrt{x}}{9 \sqrt{-x}}+\frac{\sqrt{-x-1} \sqrt{x}}{3 \sqrt{-x}} \]
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Rubi [A] time = 0.0253573, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6346, 12, 43} \[ \frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )+\frac{(-x-1)^{5/2} \sqrt{x}}{15 \sqrt{-x}}+\frac{2 (-x-1)^{3/2} \sqrt{x}}{9 \sqrt{-x}}+\frac{\sqrt{-x-1} \sqrt{x}}{3 \sqrt{-x}} \]
Antiderivative was successfully verified.
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Rule 6346
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x^2 \text{csch}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{x^2}{2 \sqrt{-1-x}} \, dx}{3 \sqrt{-x}}\\ &=\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{x^2}{\sqrt{-1-x}} \, dx}{6 \sqrt{-x}}\\ &=\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \left (\frac{1}{\sqrt{-1-x}}+2 \sqrt{-1-x}+(-1-x)^{3/2}\right ) \, dx}{6 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x} \sqrt{x}}{3 \sqrt{-x}}+\frac{2 (-1-x)^{3/2} \sqrt{x}}{9 \sqrt{-x}}+\frac{(-1-x)^{5/2} \sqrt{x}}{15 \sqrt{-x}}+\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0302423, size = 42, normalized size = 0.47 \[ \frac{1}{45} \sqrt{\frac{1}{x}+1} \left (3 x^2-4 x+8\right ) \sqrt{x}+\frac{1}{3} x^3 \text{csch}^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 38, normalized size = 0.4 \begin{align*}{\frac{{x}^{3}}{3}{\rm arccsch} \left (\sqrt{x}\right )}+{\frac{ \left ( 1+x \right ) \left ( 3\,{x}^{2}-4\,x+8 \right ) }{45}{\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99561, size = 62, normalized size = 0.7 \begin{align*} \frac{1}{15} \, x^{\frac{5}{2}}{\left (\frac{1}{x} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \, x^{3} \operatorname{arcsch}\left (\sqrt{x}\right ) - \frac{2}{9} \, x^{\frac{3}{2}}{\left (\frac{1}{x} + 1\right )}^{\frac{3}{2}} + \frac{1}{3} \, \sqrt{x} \sqrt{\frac{1}{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63231, size = 128, normalized size = 1.44 \begin{align*} \frac{1}{3} \, x^{3} \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\right ) + \frac{1}{45} \,{\left (3 \, x^{2} - 4 \, x + 8\right )} \sqrt{x} \sqrt{\frac{x + 1}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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*} \int x^{2} \operatorname{arcsch}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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