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 = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, 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 92
Rule 94
Rule 96
Rule 206
Rule 6175
Rule 6180
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*}
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Mathematica [A] time = 0.05, size = 52, normalized size = 0.39 \[ \frac {15}{8} \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right )+\frac {1}{8} \sqrt {1-\frac {1}{x^2}} x \left (2 x^3+8 x^2+15 x+24\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 66, normalized size = 0.50 \[ \frac {1}{8} \, {\left (2 \, x^{4} + 10 \, x^{3} + 23 \, x^{2} + 39 \, x + 24\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {15}{8} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {15}{8} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 130, normalized size = 0.98 \[ -\frac {\frac {73 \, {\left (x - 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x + 1} - \frac {55 \, {\left (x - 1\right )}^{2} \sqrt {\frac {x - 1}{x + 1}}}{{\left (x + 1\right )}^{2}} + \frac {15 \, {\left (x - 1\right )}^{3} \sqrt {\frac {x - 1}{x + 1}}}{{\left (x + 1\right )}^{3}} - 49 \, \sqrt {\frac {x - 1}{x + 1}}}{4 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{4}} + \frac {15}{8} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {15}{8} \, \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 0.59 \[ \frac {\left (-1+x \right ) \left (2 x \left (x^{2}-1\right )^{\frac {3}{2}}+8 \left (\left (1+x \right ) \left (-1+x \right )\right )^{\frac {3}{2}}+17 x \sqrt {x^{2}-1}+32 \sqrt {x^{2}-1}+15 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{8 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 138, normalized size = 1.04 \[ \frac {15 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {7}{2}} - 55 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{2}} + 73 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} - 49 \, \sqrt {\frac {x - 1}{x + 1}}}{4 \, {\left (\frac {4 \, {\left (x - 1\right )}}{x + 1} - \frac {6 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {4 \, {\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {{\left (x - 1\right )}^{4}}{{\left (x + 1\right )}^{4}} - 1\right )}} + \frac {15}{8} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {15}{8} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 118, normalized size = 0.89 \[ \frac {15\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )}{4}+\frac {\frac {49\,\sqrt {\frac {x-1}{x+1}}}{4}-\frac {73\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{4}+\frac {55\,{\left (\frac {x-1}{x+1}\right )}^{5/2}}{4}-\frac {15\,{\left (\frac {x-1}{x+1}\right )}^{7/2}}{4}}{\frac {6\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {4\,\left (x-1\right )}{x+1}-\frac {4\,{\left (x-1\right )}^3}{{\left (x+1\right )}^3}+\frac {{\left (x-1\right )}^4}{{\left (x+1\right )}^4}+1} \]
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 \[ \int \frac {x \left (x + 1\right )^{2}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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