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

Optimal. Leaf size=17 \[ -\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)} \]

[Out]

-(1/(Sqrt[1 - x^2]*ArcSin[x]))

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Rubi [A]  time = 0.12504, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {4659} \[ -\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(Sqrt[1 - x^2]*ArcSin[x]))

Rule 4659

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*
(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^Frac
Part[p])/(b*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(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] && LtQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 0.150456, size = 17, normalized size = 1. \[ -\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(Sqrt[1 - x^2]*ArcSin[x]))

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Maple [F]  time = 1.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( \arcsin \left ( x \right ) \right ) ^{2} \left ( -{x}^{2}+1 \right ) }}-{\frac{x}{\arcsin \left ( x \right ) } \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 3.10865, size = 50, normalized size = 2.94 \begin{align*} \frac{\sqrt{x + 1} \sqrt{-x + 1}}{{\left (x^{2} - 1\right )} \arctan \left (x, \sqrt{x + 1} \sqrt{-x + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.67678, size = 51, normalized size = 3. \begin{align*} \frac{\sqrt{-x^{2} + 1}}{{\left (x^{2} - 1\right )} \arcsin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-x^2 + 1)/((x^2 - 1)*arcsin(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - 1\right ) \left (x + 1\right ) \left (x \operatorname{asin}{\left (x \right )} - \sqrt{1 - x^{2}}\right )}{\left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac{5}{2}} \operatorname{asin}^{2}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 1)*(x + 1)*(x*asin(x) - sqrt(1 - x**2))/((-(x - 1)*(x + 1))**(5/2)*asin(x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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