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 = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 43
Rule 6346
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*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.47 \[ \frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right )+\frac {1}{45} \sqrt {\frac {1}{x}+1} \left (3 x^2-4 x+8\right ) \sqrt {x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 50, normalized size = 0.56 \[ \frac {1}{3} \, x^{3} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \frac {1}{45} \, {\left (3 \, x^{2} - 4 \, x + 8\right )} \sqrt {x} \sqrt {\frac {x + 1}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arcsch}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.43 \[ \frac {x^{3} \mathrm {arccsch}\left (\sqrt {x}\right )}{3}+\frac {\left (1+x \right ) \left (3 x^{2}-4 x +8\right )}{45 \sqrt {\frac {1+x}{x}}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 46, normalized size = 0.52 \[ \frac {1}{15} \, x^{\frac {5}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, x^{3} \operatorname {arcsch}\left (\sqrt {x}\right ) - \frac {2}{9} \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{3} \, \sqrt {x} \sqrt {\frac {1}{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acsch}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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