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

Optimal. Leaf size=16 \[ \log \left (\sin ^{-1}(x)\right )-\frac{1}{2} \log \left (1-x^2\right ) \]

[Out]

-Log[1 - x^2]/2 + Log[ArcSin[x]]

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Rubi [F]  time = 0.156641, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{1-x^2}+x \sin ^{-1}(x)}{\sin ^{-1}(x)-x^2 \sin ^{-1}(x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}+x \sin ^{-1}(x)}{\sin ^{-1}(x)-x^2 \sin ^{-1}(x)} \, dx &=\int \frac{\sqrt{1-x^2}+x \sin ^{-1}(x)}{\left (1-x^2\right ) \sin ^{-1}(x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0927873, size = 16, normalized size = 1. \[ \log \left (\sin ^{-1}(x)\right )-\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x^2]/2 + Log[ArcSin[x]]

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Maple [A]  time = 0.004, size = 17, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( x-1 \right ) }{2}}-{\frac{\ln \left ( x+1 \right ) }{2}}+\ln \left ( \arcsin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*arcsin(x)+(-x^2+1)^(1/2))/(arcsin(x)-x^2*arcsin(x)),x, algorithm="maxima")

[Out]

-integrate(sqrt(x + 1)*sqrt(-x + 1)/((x^2 - 1)*arctan2(x, sqrt(x + 1)*sqrt(-x + 1))), x) - 1/2*log(x + 1) - 1/
2*log(x - 1)

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Fricas [A]  time = 1.98462, size = 50, normalized size = 3.12 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*arcsin(x)+(-x^2+1)^(1/2))/(arcsin(x)-x^2*arcsin(x)),x, algorithm="fricas")

[Out]

-1/2*log(x^2 - 1) + log(-arcsin(x))

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Sympy [A]  time = 4.30566, size = 14, normalized size = 0.88 \begin{align*} - \frac{\log{\left (2 x^{2} - 2 \right )}}{2} + \log{\left (\operatorname{asin}{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(2*x**2 - 2)/2 + log(asin(x))

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Giac [A]  time = 1.22327, size = 27, normalized size = 1.69 \begin{align*} -\log \left (2\right ) - \frac{1}{2} \, \log \left ({\left | -x^{2} + 1 \right |}\right ) + \log \left ({\left | \arcsin \left (x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*arcsin(x)+(-x^2+1)^(1/2))/(arcsin(x)-x^2*arcsin(x)),x, algorithm="giac")

[Out]

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