Optimal. Leaf size=106 \[ \frac {1}{3} \left (\frac {1}{x}+1\right )^{5/2} \sqrt {\frac {x-1}{x}} x^3+\frac {5}{6} \left (\frac {1}{x}+1\right )^{3/2} \sqrt {\frac {x-1}{x}} x^2+\frac {5}{2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x+\frac {5}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6175, 6180, 94, 92, 206} \[ \frac {1}{3} \left (\frac {1}{x}+1\right )^{5/2} \sqrt {\frac {x-1}{x}} x^3+\frac {5}{6} \left (\frac {1}{x}+1\right )^{3/2} \sqrt {\frac {x-1}{x}} x^2+\frac {5}{2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}} x+\frac {5}{2} \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 206
Rule 6175
Rule 6180
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} (1+x)^2 \, dx &=\int e^{\coth ^{-1}(x)} \left (1+\frac {1}{x}\right )^2 x^2 \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(1+x)^{5/2}}{\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 {5}{3} \operatorname {Subst}\left (\int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{6} \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-\frac {5}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{\sqrt {1-x} x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{2} \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x+\frac {5}{6} \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-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{2} \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x+\frac {5}{6} \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+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right )\\ &=\frac {5}{2} \sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}} x+\frac {5}{6} \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+\frac {5}{2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {-\frac {1-x}{x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.44 \[ \frac {1}{6} \sqrt {1-\frac {1}{x^2}} x \left (2 x^2+9 x+22\right )+\frac {5}{2} \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 61, normalized size = 0.58 \[ \frac {1}{6} \, {\left (2 \, x^{3} + 11 \, x^{2} + 31 \, x + 22\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {5}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {5}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 107, normalized size = 1.01 \[ \frac {\frac {40 \, {\left (x - 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x + 1} - \frac {15 \, {\left (x - 1\right )}^{2} \sqrt {\frac {x - 1}{x + 1}}}{{\left (x + 1\right )}^{2}} - 33 \, \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {5}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {5}{2} \, \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 69, normalized size = 0.65 \[ \frac {\left (-1+x \right ) \left (2 \left (\left (1+x \right ) \left (-1+x \right )\right )^{\frac {3}{2}}+9 x \sqrt {x^{2}-1}+24 \sqrt {x^{2}-1}+15 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{6 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 112, normalized size = 1.06 \[ -\frac {15 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{2}} - 40 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} + 33 \, \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {5}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {5}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 94, normalized size = 0.89 \[ 5\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {11\,\sqrt {\frac {x-1}{x+1}}-\frac {40\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{3}+5\,{\left (\frac {x-1}{x+1}\right )}^{5/2}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1} \]
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 \[ \int \frac {\left (x + 1\right )^{2}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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