3.661 \(\int \frac{x \sin ^{-1}(x)}{(1-x^2)^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}}-\tanh ^{-1}(x) \]

[Out]

ArcSin[x]/Sqrt[1 - x^2] - ArcTanh[x]

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Rubi [A]  time = 0.0375043, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4677, 206} \[ \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}}-\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcSin[x])/(1 - x^2)^(3/2),x]

[Out]

ArcSin[x]/Sqrt[1 - x^2] - ArcTanh[x]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx &=\frac{\sin ^{-1}(x)}{\sqrt{1-x^2}}-\int \frac{1}{1-x^2} \, dx\\ &=\frac{\sin ^{-1}(x)}{\sqrt{1-x^2}}-\tanh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.010899, size = 19, normalized size = 1. \[ \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}}-\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(x*ArcSin[x])/(1 - x^2)^(3/2),x]

[Out]

ArcSin[x]/Sqrt[1 - x^2] - ArcTanh[x]

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Maple [B]  time = 0.035, size = 46, normalized size = 2.4 \begin{align*} -{\frac{\arcsin \left ( x \right ) }{{x}^{2}-1}\sqrt{-{x}^{2}+1}}-\ln \left ({\frac{1}{\sqrt{-{x}^{2}+1}}}+{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsin(x)/(-x^2+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.40527, size = 34, normalized size = 1.79 \begin{align*} \frac{\arcsin \left (x\right )}{\sqrt{-x^{2} + 1}} - \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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.53159, size = 123, normalized size = 6.47 \begin{align*} -\frac{{\left (x^{2} - 1\right )} \log \left (x + 1\right ) -{\left (x^{2} - 1\right )} \log \left (x - 1\right ) + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right )}{2 \,{\left (x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x)/(-x^2+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.89581, size = 20, normalized size = 1.05 \begin{align*} - \begin{cases} \operatorname{acoth}{\left (x \right )} & \text{for}\: x^{2} > 1 \\\operatorname{atanh}{\left (x \right )} & \text{for}\: x^{2} < 1 \end{cases} + \frac{\operatorname{asin}{\left (x \right )}}{\sqrt{1 - x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(x)/(-x**2+1)**(3/2),x)

[Out]

-Piecewise((acoth(x), x**2 > 1), (atanh(x), x**2 < 1)) + asin(x)/sqrt(1 - x**2)

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Giac [A]  time = 1.07836, size = 36, normalized size = 1.89 \begin{align*} \frac{\arcsin \left (x\right )}{\sqrt{-x^{2} + 1}} - \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x)/(-x^2+1)^(3/2),x, algorithm="giac")

[Out]

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