3.5.74 \(\int \frac {\sqrt {1-x^2}+x \text {ArcSin}(x)}{\text {ArcSin}(x)-x^2 \text {ArcSin}(x)} \, dx\) [474]

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

[Out]

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

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Rubi [F]
time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {1-x^2}+x \text {ArcSin}(x)}{\text {ArcSin}(x)-x^2 \text {ArcSin}(x)} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.07, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log \left (1-x^2\right )+\log (\text {ArcSin}(x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Log[1 - x^2] + Log[ArcSin[x]]

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Maple [A]
time = 0.14, size = 17, normalized size = 1.06

method	result	size
default	\(-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\ln \left (\arcsin \left (x \right )\right )\)	\(17\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.96, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \left (x\right )\right ) \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {1 - x^{2}}}{x^{2} \operatorname {asin}{\left (x \right )} - \operatorname {asin}{\left (x \right )}}\, dx - \int \frac {x \operatorname {asin}{\left (x \right )}}{x^{2} \operatorname {asin}{\left (x \right )} - \operatorname {asin}{\left (x \right )}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.48, size = 20, normalized size = 1.25 \begin {gather*} -\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 {gather*}

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)))

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Mupad [B]
time = 0.29, size = 12, normalized size = 0.75 \begin {gather*} \ln \left (\mathrm {asin}\left (x\right )\right )-\frac {\ln \left (x^2-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(asin(x)) - log(x^2 - 1)/2

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