Optimal. Leaf size=79 \[ \frac {3}{2} \sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2+\frac {3}{2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6310, 6315, 96,
94, 212} \begin {gather*} \frac {1}{2} \left (\frac {1}{x}+1\right )^{3/2} \sqrt {\frac {x-1}{x}} x^2+\frac {3}{2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x+\frac {3}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 94
Rule 96
Rule 212
Rule 6310
Rule 6315
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} (1+x) \, dx &=\int e^{\coth ^{-1}(x)} \left (1+\frac {1}{x}\right ) x \, dx\\ &=-\text {Subst}\left (\int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2-\frac {3}{2} \text {Subst}\left (\int \frac {\sqrt {1+x}}{\sqrt {1-x} x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{2} \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2-\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{2} \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right )\\ &=\frac {3}{2} \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2+\frac {3}{2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 40, normalized size = 0.51 \begin {gather*} \frac {1}{2} \sqrt {1-\frac {1}{x^2}} x (4+x)+\frac {3}{2} \log \left (\left (1+\sqrt {1-\frac {1}{x^2}}\right ) x\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 57, normalized size = 0.72
method | result | size |
default | \(\frac {\left (-1+x \right ) \left (x \sqrt {x^{2}-1}+4 \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}}\) | \(57\) |
risch | \(\frac {\left (x +4\right ) \left (-1+x \right )}{2 \sqrt {\frac {-1+x}{1+x}}}+\frac {3 \ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right )}\) | \(58\) |
trager | \(\frac {\left (1+x \right ) \left (x +4\right ) \sqrt {-\frac {1-x}{1+x}}}{2}-\frac {3 \ln \left (-\sqrt {-\frac {1-x}{1+x}}\, x -\sqrt {-\frac {1-x}{1+x}}+x \right )}{2}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 87, normalized size = 1.10 \begin {gather*} \frac {3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} - 5 \, \sqrt {\frac {x - 1}{x + 1}}}{\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1} + \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 54, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + 5 \, x + 4\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 48, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 1} {\left (\frac {x}{\mathrm {sgn}\left (x + 1\right )} + \frac {4}{\mathrm {sgn}\left (x + 1\right )}\right )} - \frac {3 \, \log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{2 \, \mathrm {sgn}\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 68, normalized size = 0.86 \begin {gather*} 3\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )+\frac {5\,\sqrt {\frac {x-1}{x+1}}-3\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________