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

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Rubi [A]  time = 0.13, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, 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*}

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Mathematica [A]  time = 0.17, size = 17, normalized size = 1.00 \[ -\frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)} \]

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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fricas [A]  time = 0.39, size = 21, normalized size = 1.24 \[ \frac {\sqrt {-x^{2} + 1}}{{\left (x^{2} - 1\right )} \arcsin \relax (x)} \]

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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giac [B]  time = 0.65, size = 70, normalized size = 4.12 \[ \frac {1}{\frac {x^{2} \arcsin \relax (x)}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} - \arcsin \relax (x)} + \frac {x^{2}}{{\left (\frac {x^{2} \arcsin \relax (x)}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} - \arcsin \relax (x)\right )} {\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} \]

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]

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

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maple [F]  time = 2.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x^{2}+1\right ) \arcsin \relax (x )^{2}}-\frac {x}{\left (-x^{2}+1\right )^{\frac {3}{2}} \arcsin \relax (x )}\, dx \]

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 = 1.71, size = 37, normalized size = 2.18 \[ \frac {\sqrt {x + 1} \sqrt {-x + 1}}{{\left (x^{2} - 1\right )} \arctan \left (x, \sqrt {x + 1} \sqrt {-x + 1}\right )} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ -\int \frac {1}{{\mathrm {asin}\relax (x)}^2\,\left (x^2-1\right )}+\frac {x}{\mathrm {asin}\relax (x)\,{\left (1-x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x \operatorname {asin}{\relax (x )} - \sqrt {1 - x^{2}}\right )}{\left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {5}{2}} \operatorname {asin}^{2}{\relax (x )}}\, dx \]

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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