Optimal. Leaf size=31 \[ \frac {x}{3}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6037, 45}
\begin {gather*} \frac {2}{3} x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right )+\frac {x}{3}+\frac {1}{3} \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6037
Rubi steps
\begin {align*} \int \sqrt {x} \tanh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x}{1-x} \, dx\\ &=\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \left (-1+\frac {1}{1-x}\right ) \, dx\\ &=\frac {x}{3}+\frac {2}{3} x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.81 \begin {gather*} \frac {1}{3} \left (x+2 x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right )+\log (1-x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 30, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {3}{2}} \arctanh \left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (\sqrt {x}-1\right )}{3}+\frac {\ln \left (\sqrt {x}+1\right )}{3}\) | \(30\) |
default | \(\frac {2 x^{\frac {3}{2}} \arctanh \left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (\sqrt {x}-1\right )}{3}+\frac {\ln \left (\sqrt {x}+1\right )}{3}\) | \(30\) |
meijerg | \(\frac {x}{3}-\frac {x^{\frac {3}{2}} \left (\ln \left (1-\sqrt {x}\right )-\ln \left (\sqrt {x}+1\right )\right )}{3}+\frac {\ln \left (1-x \right )}{3}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 19, normalized size = 0.61 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {artanh}\left (\sqrt {x}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, x^{\frac {3}{2}} \log \left (-\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.55, size = 39, normalized size = 1.26 \begin {gather*} \frac {2 x^{\frac {3}{2}} \operatorname {atanh}{\left (\sqrt {x} \right )}}{3} + \frac {x}{3} + \frac {2 \log {\left (\sqrt {x} + 1 \right )}}{3} - \frac {2 \operatorname {atanh}{\left (\sqrt {x} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (21) = 42\).
time = 0.41, size = 121, normalized size = 3.90 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{3 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{3}} + \frac {4 \, {\left (\sqrt {x} + 1\right )}}{3 \, {\left (\sqrt {x} - 1\right )} {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{2}} + \frac {2}{3} \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - \frac {2}{3} \, \log \left ({\left | -\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {x}\,\mathrm {atanh}\left (\sqrt {x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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