Optimal. Leaf size=104 \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {22 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {44 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac {1}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6176, 6181, 78, 45, 37} \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {22 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {44 \left (\frac {1}{x}+1\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac {1}{x}\right )^{3/2}} \]
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)} (1-x)^{3/2} x \, dx &=\frac {(1-x)^{3/2} \int e^{\coth ^{-1}(x)} \left (1-\frac {1}{x}\right )^{3/2} x^{5/2} \, dx}{\left (1-\frac {1}{x}\right )^{3/2} x^{3/2}}\\ &=-\frac {\left ((1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {(1-x) \sqrt {1+x}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{x}\right )^{3/2}}\\ &=\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {\left (11 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 \left (1-\frac {1}{x}\right )^{3/2}}\\ &=-\frac {22 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {\left (22 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 \left (1-\frac {1}{x}\right )^{3/2}}\\ &=\frac {44 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2}}{105 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {22 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x}{35 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} (1-x)^{3/2} x^2}{7 \left (1-\frac {1}{x}\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.44 \[ -\frac {2 \sqrt {\frac {1}{x}+1} \sqrt {1-x} (x+1) \left (15 x^2-33 x+22\right )}{105 \sqrt {\frac {x-1}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 45, normalized size = 0.43 \[ -\frac {2 \, {\left (15 \, x^{4} - 3 \, x^{3} - 29 \, x^{2} + 11 \, x + 22\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{105 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 58, normalized size = 0.56 \[ \frac {1}{105} \, {\left (16 i \, \sqrt {2} + \frac {2 \, {\left (15 \, {\left (x + 1\right )}^{3} \sqrt {-x - 1} - 63 \, {\left (x + 1\right )}^{2} \sqrt {-x - 1} - 70 \, {\left (-x - 1\right )}^{\frac {3}{2}}\right )}}{\mathrm {sgn}\left (-x - 1\right )}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 0.33 \[ -\frac {2 \left (1+x \right ) \left (15 x^{2}-33 x +22\right ) \sqrt {1-x}}{105 \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.99, size = 22, normalized size = 0.21 \[ -\frac {1}{105} \, {\left (30 i \, x^{3} - 36 i \, x^{2} - 22 i \, x + 44 i\right )} \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 35, normalized size = 0.34 \[ \frac {2\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2\,\left (15\,x^2-33\,x+22\right )}{105\,\sqrt {1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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