3.16 \(\int \frac{\sin ^{-1}(x)}{\left (1+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{x \sin ^{-1}(x)}{\sqrt{x^2+1}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0376588, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \sin ^{-1}(x)}{\sqrt{x^2+1}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x]/(1 + x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.52921, size = 19, normalized size = 0.86 \[ \frac{x \operatorname{asin}{\left (x \right )}}{\sqrt{x^{2} + 1}} - \frac{\operatorname{asin}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x)/(x**2+1)**(3/2),x)

[Out]

x*asin(x)/sqrt(x**2 + 1) - asin(x**2)/2

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Mathematica [A]  time = 0.0437711, size = 22, normalized size = 1. \[ \frac{x \sin ^{-1}(x)}{\sqrt{x^2+1}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[x]/(1 + x^2)^(3/2),x]

[Out]

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

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Maple [F]  time = 0.141, size = 0, normalized size = 0. \[ \int{\arcsin \left ( x \right ) \left ({x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)/(x^2+1)^(3/2),x)

[Out]

int(arcsin(x)/(x^2+1)^(3/2),x)

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Maxima [A]  time = 1.63158, size = 24, normalized size = 1.09 \[ \frac{x \arcsin \left (x\right )}{\sqrt{x^{2} + 1}} - \frac{1}{2} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(x^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

x*arcsin(x)/sqrt(x^2 + 1) - 1/2*arcsin(x^2)

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Fricas [A]  time = 0.264871, size = 68, normalized size = 3.09 \[ \frac{2 \, \sqrt{x^{2} + 1} x \arcsin \left (x\right ) -{\left (x^{2} + 1\right )} \arctan \left (\frac{x^{2}}{\sqrt{x^{2} + 1} \sqrt{-x^{2} + 1}}\right )}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(x^2 + 1)^(3/2),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(x^2 + 1)*x*arcsin(x) - (x^2 + 1)*arctan(x^2/(sqrt(x^2 + 1)*sqrt(-x^2
+ 1))))/(x^2 + 1)

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Sympy [A]  time = 21.0028, size = 78, normalized size = 3.55 \[ \frac{x \operatorname{asin}{\left (x \right )}}{\sqrt{x^{2} + 1}} + \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{x^{4}}} \right )}}{8 \pi ^{\frac{3}{2}}} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{4}}} \right )}}{8 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x)/(x**2+1)**(3/2),x)

[Out]

x*asin(x)/sqrt(x**2 + 1) + I*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1
/2, 3/4, 1, 0), ()), x**(-4))/(8*pi**(3/2)) - meijerg(((-1/2, -1/4, 0, 1/4, 1/2,
1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), exp_polar(-2*I*pi)/x**4)/(8*pi**(3/2))

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GIAC/XCAS [A]  time = 0.209626, size = 24, normalized size = 1.09 \[ \frac{x \arcsin \left (x\right )}{\sqrt{x^{2} + 1}} - \frac{1}{2} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)/(x^2 + 1)^(3/2),x, algorithm="giac")

[Out]

x*arcsin(x)/sqrt(x^2 + 1) - 1/2*arcsin(x^2)