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 = 0.046782, antiderivative size = 79, normalized size of antiderivative = 1., 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 6175
Rule 6180
Rule 94
Rule 92
Rule 206
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*}
Mathematica [A] time = 0.0240794, 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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Maple [A] time = 0.104, size = 57, normalized size = 0.7 \begin{align*}{\frac{-1+x}{2} \left ( x\sqrt{{x}^{2}-1}+4\,\sqrt{{x}^{2}-1}+3\,\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 = 1.04258, size = 117, normalized size = 1.48 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91277, size = 158, normalized size = 2. \begin{align*} \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{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 + 1}{\sqrt{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11125, size = 113, normalized size = 1.43 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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