3.701 \(\int \frac{\sin ^{-1}(x)}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\sqrt{x+1}}{3 (1-x)}+\frac{2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0316633, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4743, 627, 51, 63, 206} \[ -\frac{\sqrt{x+1}}{3 (1-x)}+\frac{2 \sin ^{-1}(x)}{3 (1-x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/(1 - x)^(5/2),x]

[Out]

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

Rule 4743

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.111364, size = 61, normalized size = 1.07 \[ \frac{1}{6} \left (-\frac{2 \left (\sqrt{1-x^2}-2 \sin ^{-1}(x)\right )}{(1-x)^{3/2}}-\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2-2 x}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x]/(1 - x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 70, normalized size = 1.2 \begin{align*}{\frac{2\,\arcsin \left ( x \right ) }{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{6}\sqrt{1+x} \left ( \sqrt{2}{\it Artanh} \left ({\sqrt{2}{\frac{1}{\sqrt{1+x}}}} \right ) \left ( 1-x \right ) +2\,\sqrt{1+x} \right ){\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{- \left ( 1-x \right ) ^{2}+2-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)/(1-x)^(5/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (\frac{1}{8} \,{\left (7 \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x + 1}}{\sqrt{2} + \sqrt{x + 1}}\right ) + 16 \, \sqrt{x + 1} - \frac{4 \, \sqrt{x + 1}}{x - 1}\right )}{\left (x - 1\right )} \sqrt{-x + 1} + \arctan \left (x, \sqrt{x + 1} \sqrt{-x + 1}\right )\right )}}{3 \,{\left (x - 1\right )} \sqrt{-x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*(x - 1)*sqrt(-x + 1)*integrate(1/3*sqrt(x + 1)*x^2/(x^5 - x^4 - x^3 + x^2 + (x^3 - x^2 - x + 1)*e^(log
(x + 1) + log(-x + 1))), x) + arctan2(x, sqrt(x + 1)*sqrt(-x + 1)))/((x - 1)*sqrt(-x + 1))

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Fricas [B]  time = 2.71643, size = 235, normalized size = 4.12 \begin{align*} \frac{\sqrt{2}{\left (x^{2} - 2 \, x + 1\right )} \log \left (-\frac{x^{2} + 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right ) - 4 \, \sqrt{-x + 1}{\left (\sqrt{-x^{2} + 1} - 2 \, \arcsin \left (x\right )\right )}}{12 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(sqrt(2)*(x^2 - 2*x + 1)*log(-(x^2 + 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(-x + 1) + 2*x - 3)/(x^2 - 2*x + 1)) -
4*sqrt(-x + 1)*(sqrt(-x^2 + 1) - 2*arcsin(x)))/(x^2 - 2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(x)/(1 - x)**(5/2), x)

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Giac [A]  time = 1.0983, size = 78, normalized size = 1.37 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{\sqrt{2} - \sqrt{x + 1}}{\sqrt{2} + \sqrt{x + 1}}\right ) + \frac{\sqrt{x + 1}}{3 \,{\left (x - 1\right )}} - \frac{2 \, \arcsin \left (x\right )}{3 \,{\left (x - 1\right )} \sqrt{-x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(2)*log((sqrt(2) - sqrt(x + 1))/(sqrt(2) + sqrt(x + 1))) + 1/3*sqrt(x + 1)/(x - 1) - 2/3*arcsin(x)/((
x - 1)*sqrt(-x + 1))