3.441 \(\int (-\frac{3 x}{8 (1-x^2) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{(1-x^2)^2}) \, dx\)

Optimal. Leaf size=42 \[ \frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}} \]

[Out]

(-3*x*Sqrt[ArcSin[x]])/(4*Sqrt[1 - x^2]) + ArcSin[x]^(3/2)/(2*(1 - x^2))

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Rubi [A]  time = 0.150535, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4677, 4651} \[ \frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(-3*x)/(8*(1 - x^2)*Sqrt[ArcSin[x]]) + (x*ArcSin[x]^(3/2))/(1 - x^2)^2,x]

[Out]

(-3*x*Sqrt[ArcSin[x]])/(4*Sqrt[1 - x^2]) + ArcSin[x]^(3/2)/(2*(1 - x^2))

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 4651

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSin[c
*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x
] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]

Rubi steps

\begin{align*} \int \left (-\frac{3 x}{8 \left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{\left (1-x^2\right )^2}\right ) \, dx &=-\left (\frac{3}{8} \int \frac{x}{\left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}} \, dx\right )+\int \frac{x \sin ^{-1}(x)^{3/2}}{\left (1-x^2\right )^2} \, dx\\ &=\frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}-\frac{3}{8} \int \frac{x}{\left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}} \, dx-\frac{3}{4} \int \frac{\sqrt{\sin ^{-1}(x)}}{\left (1-x^2\right )^{3/2}} \, dx\\ &=-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}}+\frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}\\ \end{align*}

Mathematica [F]  time = 3.52706, size = 0, normalized size = 0. \[ \int \left (-\frac{3 x}{8 \left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{\left (1-x^2\right )^2}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(-3*x)/(8*(1 - x^2)*Sqrt[ArcSin[x]]) + (x*ArcSin[x]^(3/2))/(1 - x^2)^2,x]

[Out]

Integrate[(-3*x)/(8*(1 - x^2)*Sqrt[ArcSin[x]]) + (x*ArcSin[x]^(3/2))/(1 - x^2)^2, x]

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Maple [F]  time = 0.218, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( -{x}^{2}+1 \right ) ^{2}} \left ( \arcsin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x}{-8\,{x}^{2}+8}{\frac{1}{\sqrt{\arcsin \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsin(x)^(3/2)/(-x^2+1)^2-3/8*x/(-x^2+1)/arcsin(x)^(1/2),x)

[Out]

int(x*arcsin(x)^(3/2)/(-x^2+1)^2-3/8*x/(-x^2+1)/arcsin(x)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x)^(3/2)/(-x^2+1)^2-3/8*x/(-x^2+1)/arcsin(x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x)^(3/2)/(-x^2+1)^2-3/8*x/(-x^2+1)/arcsin(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int - \frac{3 x}{x^{4} \sqrt{\operatorname{asin}{\left (x \right )}} - 2 x^{2} \sqrt{\operatorname{asin}{\left (x \right )}} + \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx + \int \frac{3 x^{3}}{x^{4} \sqrt{\operatorname{asin}{\left (x \right )}} - 2 x^{2} \sqrt{\operatorname{asin}{\left (x \right )}} + \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx + \int \frac{8 x \operatorname{asin}^{2}{\left (x \right )}}{x^{4} \sqrt{\operatorname{asin}{\left (x \right )}} - 2 x^{2} \sqrt{\operatorname{asin}{\left (x \right )}} + \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(x)**(3/2)/(-x**2+1)**2-3/8*x/(-x**2+1)/asin(x)**(1/2),x)

[Out]

(Integral(-3*x/(x**4*sqrt(asin(x)) - 2*x**2*sqrt(asin(x)) + sqrt(asin(x))), x) + Integral(3*x**3/(x**4*sqrt(as
in(x)) - 2*x**2*sqrt(asin(x)) + sqrt(asin(x))), x) + Integral(8*x*asin(x)**2/(x**4*sqrt(asin(x)) - 2*x**2*sqrt
(asin(x)) + sqrt(asin(x))), x))/8

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \arcsin \left (x\right )^{\frac{3}{2}}}{{\left (x^{2} - 1\right )}^{2}} + \frac{3 \, x}{8 \,{\left (x^{2} - 1\right )} \sqrt{\arcsin \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x)^(3/2)/(-x^2+1)^2-3/8*x/(-x^2+1)/arcsin(x)^(1/2),x, algorithm="giac")

[Out]

integrate(x*arcsin(x)^(3/2)/(x^2 - 1)^2 + 3/8*x/((x^2 - 1)*sqrt(arcsin(x))), x)