Optimal. Leaf size=20 \[ 2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right )+\log (1-x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6038, 31}
\begin {gather*} \log (1-x)+2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 6038
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right )-\int \frac {1}{1-x} \, dx\\ &=2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right )+\log (1-x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} 2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right )+\log (1-x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 15, normalized size = 0.75
method | result | size |
derivativedivides | \(2 \,\mathrm {arccoth}\left (\sqrt {x}\right ) \sqrt {x}+\ln \left (-1+x \right )\) | \(15\) |
default | \(2 \,\mathrm {arccoth}\left (\sqrt {x}\right ) \sqrt {x}+\ln \left (-1+x \right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \sqrt {x} \operatorname {arcoth}\left (\sqrt {x}\right ) + \log \left (-x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 24, normalized size = 1.20 \begin {gather*} \sqrt {x} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (17) = 34\).
time = 0.22, size = 87, normalized size = 4.35 \begin {gather*} \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} - \frac {2 \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} + \frac {2 x \log {\left (\sqrt {x} + 1 \right )}}{x - 1} - \frac {2 x \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} - \frac {2 \log {\left (\sqrt {x} + 1 \right )}}{x - 1} + \frac {2 \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (16) = 32\).
time = 0.40, size = 70, normalized size = 3.50 \begin {gather*} \frac {2 \, \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1} + 2 \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - 2 \, \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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.29, size = 14, normalized size = 0.70 \begin {gather*} \ln \left (x-1\right )+2\,\sqrt {x}\,\mathrm {acoth}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________