Optimal. Leaf size=133 \[ \frac{1}{4} \left (\frac{1}{x}+1\right )^{7/2} \sqrt{\frac{x-1}{x}} x^4+\frac{1}{4} \left (\frac{1}{x}+1\right )^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{5}{8} \left (\frac{1}{x}+1\right )^{3/2} \sqrt{\frac{x-1}{x}} x^2+\frac{15}{8} \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\frac{15}{8} \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right ) \]
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Rubi [A] time = 0.111144, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {6175, 6180, 96, 94, 92, 206} \[ \frac{1}{4} \left (\frac{1}{x}+1\right )^{7/2} \sqrt{\frac{x-1}{x}} x^4+\frac{1}{4} \left (\frac{1}{x}+1\right )^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{5}{8} \left (\frac{1}{x}+1\right )^{3/2} \sqrt{\frac{x-1}{x}} x^2+\frac{15}{8} \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\frac{15}{8} \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right ) \]
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)^2 \, dx &=\int e^{\coth ^{-1}(x)} \left (1+\frac{1}{x}\right )^2 x^3 \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(1+x)^{5/2}}{\sqrt{1-x} x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} \left (1+\frac{1}{x}\right )^{7/2} \sqrt{\frac{-1+x}{x}} x^4-\frac{3}{4} \operatorname{Subst}\left (\int \frac{(1+x)^{5/2}}{\sqrt{1-x} x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{-\frac{1-x}{x}} x^3+\frac{1}{4} \left (1+\frac{1}{x}\right )^{7/2} \sqrt{\frac{-1+x}{x}} x^4-\frac{5}{4} \operatorname{Subst}\left (\int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{8} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{4} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{-\frac{1-x}{x}} x^3+\frac{1}{4} \left (1+\frac{1}{x}\right )^{7/2} \sqrt{\frac{-1+x}{x}} x^4-\frac{15}{8} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{\sqrt{1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{15}{8} \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x+\frac{5}{8} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{4} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{-\frac{1-x}{x}} x^3+\frac{1}{4} \left (1+\frac{1}{x}\right )^{7/2} \sqrt{\frac{-1+x}{x}} x^4-\frac{15}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{15}{8} \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x+\frac{5}{8} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{4} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{-\frac{1-x}{x}} x^3+\frac{1}{4} \left (1+\frac{1}{x}\right )^{7/2} \sqrt{\frac{-1+x}{x}} x^4+\frac{15}{8} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}}\right )\\ &=\frac{15}{8} \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x+\frac{5}{8} \left (1+\frac{1}{x}\right )^{3/2} \sqrt{-\frac{1-x}{x}} x^2+\frac{1}{4} \left (1+\frac{1}{x}\right )^{5/2} \sqrt{-\frac{1-x}{x}} x^3+\frac{1}{4} \left (1+\frac{1}{x}\right )^{7/2} \sqrt{\frac{-1+x}{x}} x^4+\frac{15}{8} \tanh ^{-1}\left (\sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0441036, size = 52, normalized size = 0.39 \[ \frac{1}{8} \sqrt{1-\frac{1}{x^2}} x \left (2 x^3+8 x^2+15 x+24\right )+\frac{15}{8} \log \left (\left (\sqrt{1-\frac{1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.107, size = 79, normalized size = 0.6 \begin{align*}{\frac{-1+x}{8} \left ( 2\,x \left ({x}^{2}-1 \right ) ^{3/2}+8\, \left ( \left ( 1+x \right ) \left ( -1+x \right ) \right ) ^{3/2}+17\,x\sqrt{{x}^{2}-1}+32\,\sqrt{{x}^{2}-1}+15\,\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 = 1.03094, size = 186, normalized size = 1.4 \begin{align*} \frac{15 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{7}{2}} - 55 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{2}} + 73 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}} - 49 \, \sqrt{\frac{x - 1}{x + 1}}}{4 \,{\left (\frac{4 \,{\left (x - 1\right )}}{x + 1} - \frac{6 \,{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{4 \,{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac{{\left (x - 1\right )}^{4}}{{\left (x + 1\right )}^{4}} - 1\right )}} + \frac{15}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{15}{8} \, \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 = 1.88403, size = 190, normalized size = 1.43 \begin{align*} \frac{1}{8} \,{\left (2 \, x^{4} + 10 \, x^{3} + 23 \, x^{2} + 39 \, x + 24\right )} \sqrt{\frac{x - 1}{x + 1}} + \frac{15}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{15}{8} \, \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 )^{2}}{\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 = 1.14158, size = 176, normalized size = 1.32 \begin{align*} -\frac{\frac{73 \,{\left (x - 1\right )} \sqrt{\frac{x - 1}{x + 1}}}{x + 1} - \frac{55 \,{\left (x - 1\right )}^{2} \sqrt{\frac{x - 1}{x + 1}}}{{\left (x + 1\right )}^{2}} + \frac{15 \,{\left (x - 1\right )}^{3} \sqrt{\frac{x - 1}{x + 1}}}{{\left (x + 1\right )}^{3}} - 49 \, \sqrt{\frac{x - 1}{x + 1}}}{4 \,{\left (\frac{x - 1}{x + 1} - 1\right )}^{4}} + \frac{15}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{15}{8} \, \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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