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 = 0.0826707, antiderivative size = 45, normalized size of antiderivative = 1., 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 6175
Rule 6180
Rule 96
Rule 92
Rule 206
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*}
Mathematica [A] time = 0.0429387, size = 36, normalized size = 0.8 \[ \log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right )-\frac{\sqrt{1-\frac{1}{x^2}} x}{x+1} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.123, size = 110, normalized size = 2.4 \begin{align*}{\frac{-1+x}{2\, \left ( 1+x \right ) ^{2}} \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 ){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00741, size = 59, normalized size = 1.31 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62333, size = 122, normalized size = 2.71 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\frac{x - 1}{x + 1}} \left (x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13298, size = 61, normalized size = 1.36 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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