Optimal. Leaf size=99 \[ \frac{1}{3} \left (\frac{1}{x}+1\right )^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{1}{3} \left (\frac{1}{x}+1\right )^{3/2} \sqrt{\frac{x-1}{x}} x^2+\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right ) \]
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Rubi [A] time = 0.0697388, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6175, 6180, 96, 94, 92, 206} \[ \frac{1}{3} \left (\frac{1}{x}+1\right )^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{1}{3} \left (\frac{1}{x}+1\right )^{3/2} \sqrt{\frac{x-1}{x}} x^2+\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 96
Rule 94
Rule 92
Rule 206
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} x (1+x) \, dx &=\int e^{\coth ^{-1}(x)} \left (1+\frac{1}{x}\right ) x^2 \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{\frac{-1+x}{x}} x^3-\frac{2}{3} \operatorname{Subst}\left (\int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{3} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{\frac{-1+x}{x}} x^3-\operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{\sqrt{1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}} x+\frac{1}{3} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{3} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{\frac{-1+x}{x}} x^3-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}} x+\frac{1}{3} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{3} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{\frac{-1+x}{x}} x^3+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}}\right )\\ &=\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}} x+\frac{1}{3} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{3} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{\frac{-1+x}{x}} x^3+\tanh ^{-1}\left (\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0419161, size = 41, normalized size = 0.41 \[ \frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \left (x^2+3 x+5\right )+\log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.106, size = 67, normalized size = 0.7 \begin{align*}{\frac{-1+x}{3} \left ( \left ( \left ( 1+x \right ) \left ( -1+x \right ) \right ) ^{{\frac{3}{2}}}+3\,x\sqrt{{x}^{2}-1}+6\,\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 = 0.995801, size = 149, normalized size = 1.51 \begin{align*} -\frac{2 \,{\left (3 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{2}} - 8 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}} + 9 \, \sqrt{\frac{x - 1}{x + 1}}\right )}}{3 \,{\left (\frac{3 \,{\left (x - 1\right )}}{x + 1} - \frac{3 \,{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \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 = 1.73753, size = 158, normalized size = 1.6 \begin{align*} \frac{1}{3} \,{\left (x^{3} + 4 \, x^{2} + 8 \, x + 5\right )} \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 \left (x + 1\right )}{\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 = 1.1533, size = 142, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (\frac{8 \,{\left (x - 1\right )} \sqrt{\frac{x - 1}{x + 1}}}{x + 1} - \frac{3 \,{\left (x - 1\right )}^{2} \sqrt{\frac{x - 1}{x + 1}}}{{\left (x + 1\right )}^{2}} - 9 \, \sqrt{\frac{x - 1}{x + 1}}\right )}}{3 \,{\left (\frac{x - 1}{x + 1} - 1\right )}^{3}} + \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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