Optimal. Leaf size=107 \[ \frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x^2}{5 \sqrt {\frac {1}{x}+1}}+\frac {6 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{5 \sqrt {\frac {1}{x}+1}}+\frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{5 \sqrt {\frac {1}{x}+1}} \]
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Rubi [A] time = 0.11, 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 = {6176, 6181, 78, 45, 37} \[ \frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x^2}{5 \sqrt {\frac {1}{x}+1}}+\frac {6 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{5 \sqrt {\frac {1}{x}+1}}+\frac {12 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{5 \sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 6176
Rule 6181
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.02, size = 39, normalized size = 0.36 \[ \frac {2 \sqrt {\frac {x-1}{x}} \sqrt {x+1} \left (x^2+3 x+6\right )}{5 \sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 26, normalized size = 0.24 \[ \frac {2}{5} \, {\left (x^{2} + 3 \, x + 6\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.28 \[ \frac {2 \left (-1+x \right ) \left (x^{2}+3 x +6\right )}{5 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 20, normalized size = 0.19 \[ \frac {2 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x - 6\right )}}{5 \, \sqrt {x - 1}} \]
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 \[ \sqrt {\frac {x-1}{x+1}}\,\left (\frac {6\,x\,\sqrt {x+1}}{5}+\frac {12\,\sqrt {x+1}}{5}+\frac {2\,x^2\,\sqrt {x+1}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.93, size = 133, normalized size = 1.24 \[ - 2 \left (\begin {cases} \frac {x \sqrt {x - 1}}{3} + \frac {5 \sqrt {x - 1}}{3} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {i x \sqrt {1 - x}}{3} + \frac {5 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}\right ) + 2 \left (\begin {cases} \frac {8 x \sqrt {x - 1}}{15} + \frac {\sqrt {x - 1} \left (x + 1\right )^{2}}{5} + \frac {8 \sqrt {x - 1}}{3} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {8 i x \sqrt {1 - x}}{15} + \frac {i \sqrt {1 - x} \left (x + 1\right )^{2}}{5} + \frac {8 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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