Optimal. Leaf size=57 \[ -\frac {\sqrt {x+1}}{3 (1-x)}+\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4743, 627, 51, 63, 206} \[ -\frac {\sqrt {x+1}}{3 (1-x)}+\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 627
Rule 4743
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(x)}{(1-x)^{5/2}} \, dx &=\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {2}{3} \int \frac {1}{(1-x)^{3/2} \sqrt {1-x^2}} \, dx\\ &=\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {2}{3} \int \frac {1}{(1-x)^2 \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{3 (1-x)}+\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {1}{6} \int \frac {1}{(1-x) \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1+x}}{3 (1-x)}+\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\sqrt {1+x}}{3 (1-x)}+\frac {2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {2}}\right )}{3 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 61, normalized size = 1.07 \[ \frac {1}{6} \left (-\frac {2 \left (\sqrt {1-x^2}-2 \sin ^{-1}(x)\right )}{(1-x)^{3/2}}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2-2 x}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{-1}(x)}{(1-x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.14, size = 90, normalized size = 1.58 \[ \frac {\sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {2} \sqrt {-x^{2} + 1} \sqrt {-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right ) - 4 \, \sqrt {-x + 1} {\left (\sqrt {-x^{2} + 1} - 2 \, \arcsin \relax (x)\right )}}{12 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 58, normalized size = 1.02 \[ \frac {1}{12} \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {x + 1}}{\sqrt {2} + \sqrt {x + 1}}\right ) + \frac {\sqrt {x + 1}}{3 \, {\left (x - 1\right )}} - \frac {2 \, \arcsin \relax (x)}{3 \, {\left (x - 1\right )} \sqrt {-x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 70, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {2 \arcsin \relax (x )}{3 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}\, \left (\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}}{\sqrt {1+x}}\right ) \left (1-x \right )+2 \sqrt {1+x}\right )}{6 \sqrt {1-x}\, \sqrt {-\left (1-x \right )^{2}+2-2 x}}\) | \(70\) |
default | \(\frac {2 \arcsin \relax (x )}{3 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}\, \left (\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}}{\sqrt {1+x}}\right ) \left (1-x \right )+2 \sqrt {1+x}\right )}{6 \sqrt {1-x}\, \sqrt {-\left (1-x \right )^{2}+2-2 x}}\) | \(70\) |
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 \[ -\frac {2 \, {\left (\frac {1}{8} \, {\left (7 \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x + 1}}{\sqrt {2} + \sqrt {x + 1}}\right ) + 16 \, \sqrt {x + 1} - \frac {4 \, \sqrt {x + 1}}{x - 1}\right )} {\left (x - 1\right )} \sqrt {-x + 1} + \arctan \left (x, \sqrt {x + 1} \sqrt {-x + 1}\right )\right )}}{3 \, {\left (x - 1\right )} \sqrt {-x + 1}} \]
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 \[ \int \frac {\mathrm {asin}\relax (x)}{{\left (1-x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}{\relax (x )}}{\left (1 - x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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