Optimal. Leaf size=70 \[ \frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{3 \sqrt {\frac {1}{x}+1}}+\frac {10 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{3 \sqrt {\frac {1}{x}+1}} \]
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Rubi [A] time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6176, 6181, 78, 37} \[ \frac {2 \sqrt {-\frac {1-x}{x}} \sqrt {x+1} x}{3 \sqrt {\frac {1}{x}+1}}+\frac {10 \sqrt {-\frac {1-x}{x}} \sqrt {x+1}}{3 \sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} \sqrt {1+x} \, dx &=\frac {\sqrt {1+x} \int e^{\coth ^{-1}(x)} \sqrt {1+\frac {1}{x}} \sqrt {x} \, 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^{5/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1+\frac {1}{x}}}\\ &=\frac {2 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}}-\frac {\left (5 \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 )}{3 \sqrt {1+\frac {1}{x}}}\\ &=\frac {10 \sqrt {-\frac {1-x}{x}} \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}}+\frac {2 \sqrt {-\frac {1-x}{x}} x \sqrt {1+x}}{3 \sqrt {1+\frac {1}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.49 \[ \frac {2 \sqrt {\frac {x-1}{x}} \sqrt {x+1} (x+5)}{3 \sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 21, normalized size = 0.30 \[ \frac {2}{3} \, {\left (x + 5\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
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 = 25, normalized size = 0.36 \[ \frac {2 \left (-1+x \right ) \left (x +5\right )}{3 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 15, normalized size = 0.21 \[ \frac {2 \, {\left (x^{2} + 4 \, x - 5\right )}}{3 \, \sqrt {x - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 21, normalized size = 0.30 \[ \frac {2\,\sqrt {\frac {x-1}{x+1}}\,\sqrt {x+1}\,\left (x+5\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.83, size = 39, normalized size = 0.56 \[ 2 \left (\begin {cases} 2 \sqrt {2} \left (\frac {\sqrt {2} \left (x - 1\right )^{\frac {3}{2}}}{12} + \frac {\sqrt {2} \sqrt {x - 1}}{2}\right ) & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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