3.1 \(\int \sin ^{-1}(x) \log (x) \, dx\)

Optimal. Leaf size=51 \[ -2 \sqrt{1-x^2}+\sqrt{1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt{1-x^2}\right )-x (1-\log (x)) \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0615817, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.4 \[ -2 \sqrt{1-x^2}+\sqrt{1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt{1-x^2}\right )-x \sin ^{-1}(x)+x \log (x) \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[ArcSin[x]*Log[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.09152, size = 44, normalized size = 0.86 \[ x \log{\left (x \right )} \operatorname{asin}{\left (x \right )} - x \operatorname{asin}{\left (x \right )} + \sqrt{- x^{2} + 1} \log{\left (x \right )} - 2 \sqrt{- x^{2} + 1} + \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x)*ln(x),x)

[Out]

x*log(x)*asin(x) - x*asin(x) + sqrt(-x**2 + 1)*log(x) - 2*sqrt(-x**2 + 1) + atan
h(sqrt(-x**2 + 1))

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Mathematica [A]  time = 0.0355757, size = 52, normalized size = 1.02 \[ -2 \sqrt{1-x^2}+\left (\sqrt{1-x^2}-1\right ) \log (x)+\log \left (\sqrt{1-x^2}+1\right )+x (\log (x)-1) \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcSin[x]*Log[x],x]

[Out]

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

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Maple [B]  time = 0.067, size = 93, normalized size = 1.8 \[ 2\,{\frac{1}{ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}+1} \left ( \arcsin \left ( x \right ) \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \ln \left ( 2\,{\frac{\tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) }{ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}+1}} \right ) - \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}\ln \left ( 2\,{\frac{\tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) }{ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}+1}} \right ) -\arcsin \left ( x \right ) \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) -2 \right ) }-\ln \left ( \left ( \tan \left ({\frac{\arcsin \left ( x \right ) }{2}} \right ) \right ) ^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x)*ln(x),x)

[Out]

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

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Maxima [A]  time = 1.59251, size = 78, normalized size = 1.53 \[{\left (x \log \left (x\right ) - x\right )} \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \log \left (x\right ) - 2 \, \sqrt{-x^{2} + 1} + \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)*log(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.284946, size = 73, normalized size = 1.43 \[ x \arcsin \left (x\right ) \log \left (x\right ) - x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1}{\left (\log \left (x\right ) - 2\right )} + \frac{1}{2} \, \log \left (\sqrt{-x^{2} + 1} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{-x^{2} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)*log(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.22967, size = 102, normalized size = 2. \[ x \log{\left (x \right )} \operatorname{asin}{\left (x \right )} - x \operatorname{asin}{\left (x \right )} + \sqrt{- x^{2} + 1} \log{\left (x \right )} - \sqrt{- x^{2} + 1} - \begin{cases} - \frac{x}{\sqrt{-1 + \frac{1}{x^{2}}}} - \operatorname{acosh}{\left (\frac{1}{x} \right )} + \frac{1}{x \sqrt{-1 + \frac{1}{x^{2}}}} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\\frac{i x}{\sqrt{1 - \frac{1}{x^{2}}}} + i \operatorname{asin}{\left (\frac{1}{x} \right )} - \frac{i}{x \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(asin(x)*ln(x),x)

[Out]

x*log(x)*asin(x) - x*asin(x) + sqrt(-x**2 + 1)*log(x) - sqrt(-x**2 + 1) - Piecew
ise((-x/sqrt(-1 + x**(-2)) - acosh(1/x) + 1/(x*sqrt(-1 + x**(-2))), Abs(x**(-2))
> 1), (I*x/sqrt(1 - 1/x**2) + I*asin(1/x) - I/(x*sqrt(1 - 1/x**2)), True))

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GIAC/XCAS [A]  time = 0.239209, size = 367, normalized size = 7.2 \[ x \arcsin \left (x\right ){\rm ln}\left (x\right ) + \sqrt{-x^{2} + 1}{\rm ln}\left (x\right ) - \frac{2 \, x \arcsin \left (x\right )}{{\left (\sqrt{-x^{2} + 1} + 1\right )}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac{x^{2}{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right )}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac{{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right )}{\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1} - \frac{x^{2}{\rm ln}\left ({\left | x \right |}\right )}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac{{\rm ln}\left ({\left | x \right |}\right )}{\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1} + \frac{2 \, x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac{2}{\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(x)*log(x),x, algorithm="giac")

[Out]

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