Optimal. Leaf size=126 \[ \frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{(1-x)^3}{15 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}+\frac{2 (1-x)^2}{9 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{3 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
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Rubi [A] time = 0.0270286, antiderivative size = 126, 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 = {6345, 12, 43} \[ \frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{(1-x)^3}{15 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}+\frac{2 (1-x)^2}{9 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{3 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6345
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x^2 \text{sech}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x^2}{2 \sqrt{1-x}} \, dx}{3 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x^2}{\sqrt{1-x}} \, dx}{6 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \left (\frac{1}{\sqrt{1-x}}-2 \sqrt{1-x}+(1-x)^{3/2}\right ) \, dx}{6 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{1-x}{3 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{2 (1-x)^2}{9 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}-\frac{(1-x)^3}{15 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0395421, size = 72, normalized size = 0.57 \[ \frac{1}{3} x^3 \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1}{45} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (3 x^{5/2}+3 x^2+4 x^{3/2}+4 x+8 \sqrt{x}+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 49, normalized size = 0.4 \begin{align*}{\frac{{x}^{3}}{3}{\rm arcsech} \left (\sqrt{x}\right )}-{\frac{3\,{x}^{2}+4\,x+8}{45}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982506, size = 62, normalized size = 0.49 \begin{align*} -\frac{1}{15} \, x^{\frac{5}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{5}{2}} + \frac{1}{3} \, x^{3} \operatorname{arsech}\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 = 1.87177, size = 131, normalized size = 1.04 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{asech}{\left (\sqrt{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*} \int x^{2} \operatorname{arsech}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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