Optimal. Leaf size=133 \[ \frac {15}{8} \sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}} x+\frac {5}{8} \left (1+\frac {1}{x}\right )^{3/2} \sqrt {\frac {-1+x}{x}} x^2+\frac {1}{4} \left (1+\frac {1}{x}\right )^{5/2} \sqrt {\frac {-1+x}{x}} x^3+\frac {1}{4} \left (1+\frac {1}{x}\right )^{7/2} \sqrt {\frac {-1+x}{x}} x^4+\frac {15}{8} \tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right ) \]
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Rubi [A]
time = 0.07, 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 = {6310, 6315, 98,
96, 94, 212} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 94
Rule 96
Rule 98
Rule 212
Rule 6310
Rule 6315
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\\ &=-\text {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} \text {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} \text {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} \text {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} \text {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} \text {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 = 0.03, size = 52, normalized size = 0.39 \begin {gather*} \frac {1}{8} \sqrt {1-\frac {1}{x^2}} x \left (24+15 x+8 x^2+2 x^3\right )+\frac {15}{8} \log \left (\left (1+\sqrt {1-\frac {1}{x^2}}\right ) x\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 79, normalized size = 0.59
method | result | size |
risch | \(\frac {\left (2 x^{3}+8 x^{2}+15 x +24\right ) \left (-1+x \right )}{8 \sqrt {\frac {-1+x}{1+x}}}+\frac {15 \ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{8 \sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right )}\) | \(70\) |
trager | \(\frac {\left (1+x \right ) \left (2 x^{3}+8 x^{2}+15 x +24\right ) \sqrt {-\frac {1-x}{1+x}}}{8}-\frac {15 \ln \left (-\sqrt {-\frac {1-x}{1+x}}\, x -\sqrt {-\frac {1-x}{1+x}}+x \right )}{8}\) | \(74\) |
default | \(\frac {\left (-1+x \right ) \left (2 x \left (x^{2}-1\right )^{\frac {3}{2}}+8 \left (\left (1+x \right ) \left (-1+x \right )\right )^{\frac {3}{2}}+17 x \sqrt {x^{2}-1}+32 \sqrt {x^{2}-1}+15 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{8 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 138, normalized size = 1.04 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 66, normalized size = 0.50 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {x \left (x + 1\right )^{2}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 71, normalized size = 0.53 \begin {gather*} \frac {1}{8} \, {\left ({\left (2 \, x {\left (\frac {x}{\mathrm {sgn}\left (x + 1\right )} + \frac {4}{\mathrm {sgn}\left (x + 1\right )}\right )} + \frac {15}{\mathrm {sgn}\left (x + 1\right )}\right )} x + \frac {24}{\mathrm {sgn}\left (x + 1\right )}\right )} \sqrt {x^{2} - 1} - \frac {15 \, \log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{8 \, \mathrm {sgn}\left (x + 1\right )} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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