Optimal. Leaf size=58 \[ -\frac {\sqrt {2} \left (1+\frac {1}{x}\right )^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {-\frac {1-x}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 58, 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 = {6311, 6316, 95,
209} \begin {gather*} -\frac {\sqrt {2} \left (\frac {1}{x}+1\right )^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {-\frac {1-x}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 209
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^{3/2}} \, dx &=\frac {\left (\left (1+\frac {1}{x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{\coth ^{-1}(x)}}{\left (1+\frac {1}{x}\right )^{3/2} x^{3/2}} \, dx}{(1+x)^{3/2}}\\ &=-\frac {\left (1+\frac {1}{x}\right )^{3/2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {x} (1+x)} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}}\\ &=-\frac {\left (2 \left (1+\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {\frac {-1+x}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}}\\ &=-\frac {\sqrt {2} \left (1+\frac {1}{x}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {-\frac {1-x}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.71 \begin {gather*} \sqrt {2} \sqrt {\frac {1}{1+x}} \sqrt {1+x} \text {ArcTan}\left (\frac {\sqrt {\frac {-1+x}{x^2}} x}{\sqrt {2}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 37, normalized size = 0.64
method | result | size |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {-1+x}}{\sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 0.45 \begin {gather*} \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 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 {1}{\sqrt {\frac {x - 1}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}}\, dx \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: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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