Optimal. Leaf size=88 \[ \frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {(1-x)^2}{6 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}-\frac {1-x}{2 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6345, 12, 43} \[ \frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {(1-x)^2}{6 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}-\frac {1-x}{2 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 6345
Rubi steps
\begin {align*} \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \frac {x}{2 \sqrt {1-x}} \, dx}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \frac {x}{\sqrt {1-x}} \, dx}{4 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx}{4 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=-\frac {1-x}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {(1-x)^2}{6 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.64 \[ \frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \sqrt {\frac {1-\sqrt {x}}{\sqrt {x}+1}} \left (x^{3/2}+x+2 \sqrt {x}+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 45, normalized size = 0.51 \[ \frac {1}{2} \, x^{2} \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \frac {1}{6} \, {\left (x + 2\right )} \sqrt {x} \sqrt {-\frac {x - 1}{x}} \]
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 \[ \int x \operatorname {arsech}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 42, normalized size = 0.48 \[ \frac {x^{2} \mathrm {arcsech}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-\frac {-1+\sqrt {x}}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {1+\sqrt {x}}{\sqrt {x}}}\, \left (x +2\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 34, normalized size = 0.39 \[ \frac {1}{6} \, x^{\frac {3}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, x^{2} \operatorname {arsech}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x} \sqrt {\frac {1}{x} - 1} \]
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 \[ \int x\,\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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 \[ \int x \operatorname {asech}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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