Optimal. Leaf size=107 \[ \frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {6 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}} \]
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Rubi [A]
time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6311, 6316, 79,
47, 37} \begin {gather*} \frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x^2}{5 \sqrt {\frac {1}{x}+1}}+\frac {6 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{5 \sqrt {\frac {1}{x}+1}}+\frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{5 \sqrt {\frac {1}{x}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} x \sqrt {1+x} \, dx &=\frac {\sqrt {1+x} \int e^{\coth ^{-1}(x)} \sqrt {1+\frac {1}{x}} x^{3/2} \, dx}{\sqrt {1+\frac {1}{x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1+x}{\sqrt {1-x} x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1+\frac {1}{x}}}\\ &=\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}-\frac {\left (9 \sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 \sqrt {1+\frac {1}{x}}}\\ &=\frac {6 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}-\frac {\left (6 \sqrt {\frac {1}{x}} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 \sqrt {1+\frac {1}{x}}}\\ &=\frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {6 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 \sqrt {1+x}}{5 \sqrt {1+\frac {1}{x}}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.36 \begin {gather*} \frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} \left (6+3 x+x^2\right )}{5 \sqrt {1+\frac {1}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 30, normalized size = 0.28
method | result | size |
gosper | \(\frac {2 \left (-1+x \right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}}\) | \(30\) |
default | \(\frac {2 \left (-1+x \right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}}\) | \(30\) |
risch | \(\frac {2 \left (-1+x \right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 0.19 \begin {gather*} \frac {2 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x - 6\right )}}{5 \, \sqrt {x - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 26, normalized size = 0.24 \begin {gather*} \frac {2}{5} \, {\left (x^{2} + 3 \, x + 6\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.96, size = 129, normalized size = 1.21 \begin {gather*} - 2 \left (\begin {cases} \frac {x \sqrt {x - 1}}{3} + \frac {5 \sqrt {x - 1}}{3} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i x \sqrt {1 - x}}{3} + \frac {5 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}\right ) + 2 \left (\begin {cases} \frac {8 x \sqrt {x - 1}}{15} + \frac {\sqrt {x - 1} \left (x + 1\right )^{2}}{5} + \frac {8 \sqrt {x - 1}}{3} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {8 i x \sqrt {1 - x}}{15} + \frac {i \sqrt {1 - x} \left (x + 1\right )^{2}}{5} + \frac {8 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 38, normalized size = 0.36 \begin {gather*} \sqrt {\frac {x-1}{x+1}}\,\left (\frac {6\,x\,\sqrt {x+1}}{5}+\frac {12\,\sqrt {x+1}}{5}+\frac {2\,x^2\,\sqrt {x+1}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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