3.7.63 \(\int \frac {\sin ^{-1}(x)}{(1-x^2)^{5/2}} \, dx\) [663]

Optimal. Leaf size=62 \[ -\frac {1}{6 \left (1-x^2\right )}+\frac {x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(x)}{3 \sqrt {1-x^2}}+\frac {1}{3} \log \left (1-x^2\right ) \]

[Out]

-1/6/(-x^2+1)+1/3*x*arcsin(x)/(-x^2+1)^(3/2)+1/3*ln(-x^2+1)+2/3*x*arcsin(x)/(-x^2+1)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4747, 4745, 266, 267} \begin {gather*} \frac {2 x \text {ArcSin}(x)}{3 \sqrt {1-x^2}}+\frac {x \text {ArcSin}(x)}{3 \left (1-x^2\right )^{3/2}}-\frac {1}{6 \left (1-x^2\right )}+\frac {1}{3} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/(1 - x^2)^(5/2),x]

[Out]

-1/6*1/(1 - x^2) + (x*ArcSin[x])/(3*(1 - x^2)^(3/2)) + (2*x*ArcSin[x])/(3*Sqrt[1 - x^2]) + Log[1 - x^2]/3

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4745

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/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcSin
[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4747

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

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(x)}{\left (1-x^2\right )^{5/2}} \, dx &=\frac {x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}-\frac {1}{3} \int \frac {x}{\left (1-x^2\right )^2} \, dx+\frac {2}{3} \int \frac {\sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{6 \left (1-x^2\right )}+\frac {x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(x)}{3 \sqrt {1-x^2}}-\frac {2}{3} \int \frac {x}{1-x^2} \, dx\\ &=-\frac {1}{6 \left (1-x^2\right )}+\frac {x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(x)}{3 \sqrt {1-x^2}}+\frac {1}{3} \log \left (1-x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 45, normalized size = 0.73 \begin {gather*} \frac {1}{6} \left (\frac {1}{-1+x^2}-\frac {2 x \left (-3+2 x^2\right ) \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}}+2 \log \left (1-x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x]/(1 - x^2)^(5/2),x]

[Out]

((-1 + x^2)^(-1) - (2*x*(-3 + 2*x^2)*ArcSin[x])/(1 - x^2)^(3/2) + 2*Log[1 - x^2])/6

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Maple [A]
time = 0.09, size = 63, normalized size = 1.02

method	result	size
default	\(\frac {1}{6 x^{2}-6}+\frac {x \arcsin \left (x \right ) \sqrt {-x^{2}+1}}{3 \left (x^{2}-1\right )^{2}}+\frac {\ln \left (-x^{2}+1\right )}{3}-\frac {2 \sqrt {-x^{2}+1}\, \arcsin \left (x \right ) x}{3 \left (x^{2}-1\right )}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)/(-x^2+1)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6/(x^2-1)+1/3*x*arcsin(x)*(-x^2+1)^(1/2)/(x^2-1)^2+1/3*ln(-x^2+1)-2/3*(-x^2+1)^(1/2)/(x^2-1)*arcsin(x)*x

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Maxima [A]
time = 1.25, size = 48, normalized size = 0.77 \begin {gather*} \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-x^{2} + 1}} + \frac {x}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} \arcsin \left (x\right ) + \frac {1}{6 \, {\left (x^{2} - 1\right )}} + \frac {1}{3} \, \log \left (-3 \, x^{2} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/(-x^2+1)^(5/2),x, algorithm="maxima")

[Out]

1/3*(2*x/sqrt(-x^2 + 1) + x/(-x^2 + 1)^(3/2))*arcsin(x) + 1/6/(x^2 - 1) + 1/3*log(-3*x^2 + 3)

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Fricas [A]
time = 0.47, size = 61, normalized size = 0.98 \begin {gather*} -\frac {2 \, {\left (2 \, x^{3} - 3 \, x\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - x^{2} - 2 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x^{2} - 1\right ) + 1}{6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/(-x^2+1)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(2*(2*x^3 - 3*x)*sqrt(-x^2 + 1)*arcsin(x) - x^2 - 2*(x^4 - 2*x^2 + 1)*log(x^2 - 1) + 1)/(x^4 - 2*x^2 + 1)

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Sympy [A]
time = 15.64, size = 78, normalized size = 1.26 \begin {gather*} \left (\begin {cases} \frac {x^{3}}{3 \left (1 - x^{2}\right )^{\frac {3}{2}}} + \frac {x}{\sqrt {1 - x^{2}}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \operatorname {asin}{\left (x \right )} - \begin {cases} \text {NaN} & \text {for}\: x < -1 \\- \frac {2 x^{2} \log {\left (1 - x^{2} \right )}}{6 x^{2} - 6} - \frac {x^{2}}{6 x^{2} - 6} + \frac {2 \log {\left (1 - x^{2} \right )}}{6 x^{2} - 6} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3/(3*(1 - x**2)**(3/2)) + x/sqrt(1 - x**2), (x > -1) & (x < 1)))*asin(x) - Piecewise((nan, x < -
1), (-2*x**2*log(1 - x**2)/(6*x**2 - 6) - x**2/(6*x**2 - 6) + 2*log(1 - x**2)/(6*x**2 - 6), x < 1), (nan, True
))

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Giac [A]
time = 1.10, size = 54, normalized size = 0.87 \begin {gather*} -\frac {{\left (2 \, x^{2} - 3\right )} \sqrt {-x^{2} + 1} x \arcsin \left (x\right )}{3 \, {\left (x^{2} - 1\right )}^{2}} - \frac {2 \, x^{2} - 3}{6 \, {\left (x^{2} - 1\right )}} + \frac {1}{3} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/(-x^2+1)^(5/2),x, algorithm="giac")

[Out]

-1/3*(2*x^2 - 3)*sqrt(-x^2 + 1)*x*arcsin(x)/(x^2 - 1)^2 - 1/6*(2*x^2 - 3)/(x^2 - 1) + 1/3*log(abs(x^2 - 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{{\left (1-x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)/(1 - x^2)^(5/2),x)

[Out]

int(asin(x)/(1 - x^2)^(5/2), x)

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