Optimal. Leaf size=42 \[ \frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6098, 50, 63, 206} \[ \frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 206
Rule 6098
Rubi steps
\begin {align*} \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^{3/2}}{1-x} \, dx\\ &=\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {\sqrt {x}}{1-x} \, dx\\ &=\frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x}} \, dx\\ &=\frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 52, normalized size = 1.24 \[ \frac {1}{12} \left (2 x^{3/2}+6 x^2 \coth ^{-1}\left (\sqrt {x}\right )+6 \sqrt {x}+3 \log \left (1-\sqrt {x}\right )-3 \log \left (\sqrt {x}+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 31, normalized size = 0.74 \[ \frac {1}{4} \, {\left (x^{2} - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{6} \, {\left (x + 3\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 37, normalized size = 0.88 \[ \frac {x^{2} \mathrm {arccoth}\left (\sqrt {x}\right )}{2}+\frac {x^{\frac {3}{2}}}{6}+\frac {\sqrt {x}}{2}+\frac {\ln \left (-1+\sqrt {x}\right )}{4}-\frac {\ln \left (1+\sqrt {x}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 36, normalized size = 0.86 \[ \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{6} \, x^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {x} - \frac {1}{4} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.26, size = 26, normalized size = 0.62 \[ \frac {x^2\,\mathrm {acoth}\left (\sqrt {x}\right )}{2}-\frac {\mathrm {acoth}\left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acoth}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________