Optimal. Leaf size=93 \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {-\frac {1-x}{x}} x^2}{(x+1)^{3/2}}+\frac {\sqrt {2} \left (\frac {1}{x}+1\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {-\frac {1-x}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 93, 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, 96, 93, 203} \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {-\frac {1-x}{x}} x^2}{(x+1)^{3/2}}+\frac {\sqrt {2} \left (\frac {1}{x}+1\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {-\frac {1-x}{x}}}\right )}{\left (\frac {1}{x}\right )^{3/2} (x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 203
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)} 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} \sqrt {x}} \, dx}{(1+x)^{3/2}}\\ &=-\frac {\left (1+\frac {1}{x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2} (1+x)} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}}\\ &=\frac {2 \left (1+\frac {1}{x}\right )^{3/2} \sqrt {-\frac {1-x}{x}} x^2}{(1+x)^{3/2}}+\frac {\left (1+\frac {1}{x}\right )^{3/2} \operatorname {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 {2 \left (1+\frac {1}{x}\right )^{3/2} \sqrt {-\frac {1-x}{x}} x^2}{(1+x)^{3/2}}+\frac {\left (2 \left (1+\frac {1}{x}\right )^{3/2}\right ) \operatorname {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 {2 \left (1+\frac {1}{x}\right )^{3/2} \sqrt {-\frac {1-x}{x}} x^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 = 0.04, size = 65, normalized size = 0.70 \[ \frac {\sqrt {\frac {1}{x}+1} x \left (2 \sqrt {\frac {x-1}{x}}-\sqrt {2} \sqrt {\frac {1}{x}} \tan ^{-1}\left (\frac {\sqrt {\frac {x-1}{x^2}} x}{\sqrt {2}}\right )\right )}{\sqrt {x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.87, size = 46, normalized size = 0.49 \[ -\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}}\right ) + 2 \, \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 47, normalized size = 0.51 \[ \frac {\sqrt {-1+x}\, \left (2 \sqrt {-1+x}-\sqrt {2}\, \arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right )\right )}{\sqrt {\frac {-1+x}{1+x}}\, \sqrt {1+x}} \]
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 \[ \int \frac {x}{{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {\frac {x - 1}{x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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