Optimal. Leaf size=79 \[ \frac {1}{2} \left (\frac {1}{x}+1\right )^{3/2} \sqrt {\frac {x-1}{x}} x^2+\frac {3}{2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x+\frac {3}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \]
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Rubi [A] time = 0.05, 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 = {6175, 6180, 94, 92, 206} \[ \frac {1}{2} \left (\frac {1}{x}+1\right )^{3/2} \sqrt {\frac {x-1}{x}} x^2+\frac {3}{2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x+\frac {3}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 206
Rule 6175
Rule 6180
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\\ &=-\operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.51 \[ \frac {1}{2} \sqrt {1-\frac {1}{x^2}} x (x+4)+\frac {3}{2} \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 54, normalized size = 0.68 \[ \frac {1}{2} \, {\left (x^{2} + 5 \, x + 4\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 84, normalized size = 1.06 \[ -\frac {\frac {3 \, {\left (x - 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x + 1} - 5 \, \sqrt {\frac {x - 1}{x + 1}}}{{\left (\frac {x - 1}{x + 1} - 1\right )}^{2}} + \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {3}{2} \, \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.72 \[ \frac {\left (-1+x \right ) \left (x \sqrt {x^{2}-1}+4 \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 87, normalized size = 1.10 \[ \frac {3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} - 5 \, \sqrt {\frac {x - 1}{x + 1}}}{\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1} + \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {3}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 68, normalized size = 0.86 \[ 3\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )+\frac {5\,\sqrt {\frac {x-1}{x+1}}-3\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1} \]
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 \[ \int \frac {x + 1}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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