3.14 \(\int \frac{\sin ^{-1}(x)}{1+\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=45 \[ -\log \left (\sqrt{1-x^2}+1\right )-\frac{x \sin ^{-1}(x)}{\sqrt{1-x^2}+1}+\frac{1}{2} \sin ^{-1}(x)^2 \]

[Out]

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

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Rubi [A]  time = 0.205189, antiderivative size = 51, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611 \[ \frac{\sqrt{1-x^2} \sin ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\log (x)+\frac{1}{2} \sin ^{-1}(x)^2-\frac{\sin ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*asin(x)*atan((sqrt(-x**2 + 1) - 1)/x) - Integral((-2*x*atan((sqrt(-x**2 + 1)
- 1)/x) + sqrt(-x**2 + 1) - 1)/(x*sqrt(-x**2 + 1)), x) - (-sqrt(-x**2 + 1) + 1)*
asin(x)/x

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Mathematica [A]  time = 0.0400507, size = 44, normalized size = 0.98 \[ -\log \left (\sqrt{1-x^2}+1\right )+\frac{\left (\sqrt{1-x^2}-1\right ) \sin ^{-1}(x)}{x}+\frac{1}{2} \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \[ \int{\arcsin \left ( x \right ) \left ( 1+\sqrt{-{x}^{2}+1} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)/(1+(-x^2+1)^(1/2)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\arcsin \left (x\right )}{\sqrt{-x^{2} + 1} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(arcsin(x)/(sqrt(-x^2 + 1) + 1), x)

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Fricas [A]  time = 0.25443, size = 85, normalized size = 1.89 \[ \frac{x \arcsin \left (x\right )^{2} - 2 \, x \log \left (x\right ) - x \log \left (\sqrt{-x^{2} + 1} + 1\right ) + x \log \left (\sqrt{-x^{2} + 1} - 1\right ) + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, \arcsin \left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219925, size = 77, normalized size = 1.71 \[ \frac{1}{2} \, \arcsin \left (x\right )^{2} - \frac{x \arcsin \left (x\right )}{\sqrt{-x^{2} + 1} + 1} -{\rm ln}\left (4\right ) +{\rm ln}\left (2 \, \sqrt{-x^{2} + 1} + 2\right ) - 2 \,{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arcsin(x)^2 - x*arcsin(x)/(sqrt(-x^2 + 1) + 1) - ln(4) + ln(2*sqrt(-x^2 + 1)
+ 2) - 2*ln(sqrt(-x^2 + 1) + 1)