Optimal. Leaf size=55 \[ -\frac {4 \left (\frac {1}{x}+1\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {\frac {5}{x}+3}{3 \sqrt {1-\frac {1}{x^2}}}+\tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {6175, 6178, 852, 1805, 823, 12, 266, 63, 206} \[ -\frac {4 \left (\frac {1}{x}+1\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {\frac {5}{x}+3}{3 \sqrt {1-\frac {1}{x^2}}}+\tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 206
Rule 266
Rule 823
Rule 852
Rule 1805
Rule 6175
Rule 6178
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 {\sqrt {1-x^2}}{(1-x)^3 x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {(1+x)^3}{x \left (1-x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-3-5 x}{x \left (1-x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\frac {1}{3} \operatorname {Subst}\left (\int -\frac {3}{x \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}-\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x^2}}\right )\\ &=-\frac {4 \left (1+\frac {1}{x}\right )}{3 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {3+\frac {5}{x}}{3 \sqrt {1-\frac {1}{x^2}}}+\tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 43, normalized size = 0.78 \[ \frac {\sqrt {1-\frac {1}{x^2}} (5-7 x) x}{3 (x-1)^2}+\log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 84, normalized size = 1.53 \[ \frac {3 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - 3 \, {\left (x^{2} - 2 \, x + 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) - {\left (7 \, x^{2} + 2 \, x - 5\right )} \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 65, normalized size = 1.18 \[ -\frac {{\left (x + 1\right )} {\left (\frac {6 \, {\left (x - 1\right )}}{x + 1} + 1\right )}}{3 \, {\left (x - 1\right )} \sqrt {\frac {x - 1}{x + 1}}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 146, normalized size = 2.65 \[ -\frac {3 x \left (x^{2}-1\right )^{\frac {3}{2}}-3 x^{3} \sqrt {x^{2}-1}-3 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{3}-2 \left (x^{2}-1\right )^{\frac {3}{2}}+9 x^{2} \sqrt {x^{2}-1}+9 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{2}-9 x \sqrt {x^{2}-1}-9 \ln \left (x +\sqrt {x^{2}-1}\right ) x +3 \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )}{3 \left (-1+x \right )^{2} \sqrt {\left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 56, normalized size = 1.02 \[ -\frac {\frac {6 \, {\left (x - 1\right )}}{x + 1} + 1}{3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}}} + \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 40, normalized size = 0.73 \[ 2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {\frac {2\,\left (x-1\right )}{x+1}+\frac {1}{3}}{{\left (\frac {x-1}{x+1}\right )}^{3/2}} \]
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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