3.1.32 \(\int \frac {\sin ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx\) [32]

Optimal. Leaf size=21 \[ -\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\log (x) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4771, 29} \begin {gather*} \log (x)-\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/(x^2*Sqrt[1 - x^2]),x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4771

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

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\int \frac {1}{x} \, dx\\ &=-\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\log (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[ArcSin[x]/(x^2*Sqrt[1 - x^2]),x]')

[Out]

Timed out

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Maple [A]
time = 0.05, size = 20, normalized size = 0.95

method	result	size
default	\(\ln \left (x \right )-\frac {\arcsin \left (x \right ) \sqrt {-x^{2}+1}}{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)/x^2/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 19, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {-x^{2} + 1} \arcsin \left (x\right )}{x} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)*arcsin(x)/x + log(x)

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Fricas [A]
time = 0.34, size = 22, normalized size = 1.05 \begin {gather*} \frac {x \log \left (x\right ) - \sqrt {-x^{2} + 1} \arcsin \left (x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}{\left (x \right )}}{x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
time = 0.01, size = 47, normalized size = 2.24 \begin {gather*} \left (-\frac {x}{-2 \sqrt {-x^{2}+1}+2}+\frac {-2 \sqrt {-x^{2}+1}+2}{4 x}\right ) \arcsin x+\ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{x^2\,\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)/(x^2*(1 - x^2)^(1/2)),x)

[Out]

int(asin(x)/(x^2*(1 - x^2)^(1/2)), x)

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