3.701 \(\int \frac{\sin ^{-1}(x)}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\sqrt{x+1}}{3 (1-x)}+\frac{2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

[Out]

-Sqrt[1 + x]/(3*(1 - x)) + (2*ArcSin[x])/(3*(1 - x)^(3/2)) - ArcTanh[Sqrt[1 + x]
/Sqrt[2]]/(3*Sqrt[2])

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Rubi [A]  time = 0.0701697, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\sqrt{x+1}}{3 (1-x)}+\frac{2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x]/(1 - x)^(5/2),x]

[Out]

-Sqrt[1 + x]/(3*(1 - x)) + (2*ArcSin[x])/(3*(1 - x)^(3/2)) - ArcTanh[Sqrt[1 + x]
/Sqrt[2]]/(3*Sqrt[2])

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Rubi in Sympy [A]  time = 5.25042, size = 46, normalized size = 0.81 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{6} - \frac{\sqrt{x + 1}}{3 \left (- x + 1\right )} + \frac{2 \operatorname{asin}{\left (x \right )}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x)/(1-x)**(5/2),x)

[Out]

-sqrt(2)*atanh(sqrt(2)*sqrt(x + 1)/2)/6 - sqrt(x + 1)/(3*(-x + 1)) + 2*asin(x)/(
3*(-x + 1)**(3/2))

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Mathematica [A]  time = 0.136826, 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.

[In]  Integrate[ArcSin[x]/(1 - x)^(5/2),x]

[Out]

((-2*(Sqrt[1 - x^2] - 2*ArcSin[x]))/(1 - x)^(3/2) - Sqrt[2]*ArcTanh[Sqrt[1 - x^2
]/Sqrt[2 - 2*x]])/6

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Maple [A]  time = 0.01, size = 70, normalized size = 1.2 \[{\frac{2\,\arcsin \left ( x \right ) }{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{6}\sqrt{1+x} \left ( \sqrt{2}{\it Artanh} \left ({\sqrt{2}{\frac{1}{\sqrt{1+x}}}} \right ) \left ( 1-x \right ) +2\,\sqrt{1+x} \right ){\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{- \left ( 1-x \right ) ^{2}+2-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)/(1-x)^(5/2),x)

[Out]

2/3*arcsin(x)/(1-x)^(3/2)-1/6/(1-x)^(1/2)*(1+x)^(1/2)*(2^(1/2)*arctanh(2^(1/2)/(
1+x)^(1/2))*(1-x)+2*(1+x)^(1/2))/(-(1-x)^2+2-2*x)^(1/2)

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Maxima [A]  time = 1.52608, size = 77, normalized size = 1.35 \[ -\frac{1}{6} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{x + 1}}{\sqrt{-x + 1}} + \frac{4}{\sqrt{-x + 1}}\right ) + \frac{\sqrt{x + 1}}{3 \,{\left (x - 1\right )}} + \frac{2 \, \arcsin \left (x\right )}{3 \,{\left (-x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(-x + 1)^(5/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(2)*log(2*sqrt(2)*sqrt(x + 1)/sqrt(-x + 1) + 4/sqrt(-x + 1)) + 1/3*sqrt
(x + 1)/(x - 1) + 2/3*arcsin(x)/(-x + 1)^(3/2)

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Fricas [A]  time = 0.227102, size = 135, normalized size = 2.37 \[ \frac{\sqrt{2}{\left ({\left (x^{2} - 2 \, x + 1\right )} \log \left (-\frac{\sqrt{2}{\left (x^{2} + 2 \, x - 3\right )} + 4 \, \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{x^{2} - 2 \, x + 1}\right ) + 2 \,{\left (2 \, \sqrt{2} \arcsin \left (x\right ) - \sqrt{2} \sqrt{-x^{2} + 1}\right )} \sqrt{-x + 1}\right )}}{12 \,{\left (x^{2} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(-x + 1)^(5/2),x, algorithm="fricas")

[Out]

1/12*sqrt(2)*((x^2 - 2*x + 1)*log(-(sqrt(2)*(x^2 + 2*x - 3) + 4*sqrt(-x^2 + 1)*s
qrt(-x + 1))/(x^2 - 2*x + 1)) + 2*(2*sqrt(2)*arcsin(x) - sqrt(2)*sqrt(-x^2 + 1))
*sqrt(-x + 1))/(x^2 - 2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\operatorname{asin}{\left (x \right )}}{\left (- x + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x)/(1-x)**(5/2),x)

[Out]

Integral(asin(x)/(-x + 1)**(5/2), x)

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GIAC/XCAS [A]  time = 0.216891, size = 78, normalized size = 1.37 \[ \frac{1}{12} \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2} - \sqrt{x + 1}}{\sqrt{2} + \sqrt{x + 1}}\right ) + \frac{\sqrt{x + 1}}{3 \,{\left (x - 1\right )}} - \frac{2 \, \arcsin \left (x\right )}{3 \,{\left (x - 1\right )} \sqrt{-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(-x + 1)^(5/2),x, algorithm="giac")

[Out]

1/12*sqrt(2)*ln((sqrt(2) - sqrt(x + 1))/(sqrt(2) + sqrt(x + 1))) + 1/3*sqrt(x +
1)/(x - 1) - 2/3*arcsin(x)/((x - 1)*sqrt(-x + 1))