Optimal. Leaf size=45 \[ \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right )-\frac {\sqrt {\frac {x-1}{x}}}{\sqrt {\frac {1}{x}+1}} \]
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Rubi [A] time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6175, 6180, 96, 92, 206} \[ \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right )-\frac {\sqrt {\frac {x-1}{x}}}{\sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 92
Rule 96
Rule 206
Rule 6175
Rule 6180
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)} x}{(1+x)^2} \, dx &=\int \frac {e^{\coth ^{-1}(x)}}{\left (1+\frac {1}{x}\right )^2 x} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x (1+x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right )\\ &=-\frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}}+\tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 36, normalized size = 0.80 \[ \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right )-\frac {\sqrt {1-\frac {1}{x^2}} x}{x+1} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 44, normalized size = 0.98 \[ -\sqrt {\frac {x - 1}{x + 1}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 45, normalized size = 1.00 \[ -\sqrt {\frac {x - 1}{x + 1}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 110, normalized size = 2.44 \[ \frac {\left (-1+x \right ) \left (\left (x^{2}-1\right )^{\frac {3}{2}}-x^{2} \sqrt {x^{2}-1}+2 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{2}-2 x \sqrt {x^{2}-1}+4 \ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}+2 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}\, \left (1+x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 44, normalized size = 0.98 \[ -\sqrt {\frac {x - 1}{x + 1}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 28, normalized size = 0.62 \[ 2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\sqrt {\frac {x-1}{x+1}} \]
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 \[ \int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \left (x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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