3.83 \(\int \frac{\sin ^{-1}(x)}{x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0399035, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sin ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.84138, size = 15, normalized size = 0.68 \[ - \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} - \frac{\operatorname{asin}{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x)/x**2,x)

[Out]

-atanh(sqrt(-x**2 + 1)) - asin(x)/x

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Mathematica [A]  time = 0.00766711, size = 26, normalized size = 1.18 \[ -\log \left (\sqrt{1-x^2}+1\right )+\log (x)-\frac{\sin ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[x]/x^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 21, normalized size = 1. \[ -{\frac{\arcsin \left ( x \right ) }{x}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)/x^2,x)

[Out]

-arcsin(x)/x-arctanh(1/(-x^2+1)^(1/2))

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Maxima [A]  time = 1.53027, size = 45, normalized size = 2.05 \[ -\frac{\arcsin \left (x\right )}{x} - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.274934, size = 53, normalized size = 2.41 \[ -\frac{x \log \left (\sqrt{-x^{2} + 1} + 1\right ) - x \log \left (\sqrt{-x^{2} + 1} - 1\right ) + 2 \, \arcsin \left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.36536, size = 22, normalized size = 1. \[ \begin{cases} - \operatorname{acosh}{\left (\frac{1}{x} \right )} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\i \operatorname{asin}{\left (\frac{1}{x} \right )} & \text{otherwise} \end{cases} - \frac{\operatorname{asin}{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x)/x**2,x)

[Out]

Piecewise((-acosh(1/x), Abs(x**(-2)) > 1), (I*asin(1/x), True)) - asin(x)/x

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GIAC/XCAS [A]  time = 0.211658, size = 51, normalized size = 2.32 \[ -\frac{\arcsin \left (x\right )}{x} - \frac{1}{2} \,{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (-\sqrt{-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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