3.368 \(\int \frac{\sin ^{-1}(\sqrt{x})}{x^5} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2 \sqrt{1-x}}{35 x^{3/2}}-\frac{3 \sqrt{1-x}}{70 x^{5/2}}-\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{4 \sqrt{1-x}}{35 \sqrt{x}} \]

[Out]

-Sqrt[1 - x]/(28*x^(7/2)) - (3*Sqrt[1 - x])/(70*x^(5/2)) - (2*Sqrt[1 - x])/(35*x^(3/2)) - (4*Sqrt[1 - x])/(35*
Sqrt[x]) - ArcSin[Sqrt[x]]/(4*x^4)

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Rubi [A]  time = 0.0290164, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4842, 12, 45, 37} \[ -\frac{2 \sqrt{1-x}}{35 x^{3/2}}-\frac{3 \sqrt{1-x}}{70 x^{5/2}}-\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{4 \sqrt{1-x}}{35 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sqrt[x]]/x^5,x]

[Out]

-Sqrt[1 - x]/(28*x^(7/2)) - (3*Sqrt[1 - x])/(70*x^(5/2)) - (2*Sqrt[1 - x])/(35*x^(3/2)) - (4*Sqrt[1 - x])/(35*
Sqrt[x]) - ArcSin[Sqrt[x]]/(4*x^4)

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

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*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}\left (\sqrt{x}\right )}{x^5} \, dx &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{1}{4} \int \frac{1}{2 \sqrt{1-x} x^{9/2}} \, dx\\ &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{1}{8} \int \frac{1}{\sqrt{1-x} x^{9/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{3}{28} \int \frac{1}{\sqrt{1-x} x^{7/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{3 \sqrt{1-x}}{70 x^{5/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{3}{35} \int \frac{1}{\sqrt{1-x} x^{5/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{3 \sqrt{1-x}}{70 x^{5/2}}-\frac{2 \sqrt{1-x}}{35 x^{3/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{2}{35} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{3 \sqrt{1-x}}{70 x^{5/2}}-\frac{2 \sqrt{1-x}}{35 x^{3/2}}-\frac{4 \sqrt{1-x}}{35 \sqrt{x}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0284445, size = 49, normalized size = 0.57 \[ 2 \left (-\frac{\sqrt{1-x} \left (16 x^3+8 x^2+6 x+5\right )}{280 x^{7/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{8 x^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[Sqrt[x]]/x^5,x]

[Out]

2*(-(Sqrt[1 - x]*(5 + 6*x + 8*x^2 + 16*x^3))/(280*x^(7/2)) - ArcSin[Sqrt[x]]/(8*x^4))

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Maple [A]  time = 0.005, size = 59, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,{x}^{4}}\arcsin \left ( \sqrt{x} \right ) }-{\frac{1}{28}\sqrt{1-x}{x}^{-{\frac{7}{2}}}}-{\frac{3}{70}\sqrt{1-x}{x}^{-{\frac{5}{2}}}}-{\frac{2}{35}\sqrt{1-x}{x}^{-{\frac{3}{2}}}}-{\frac{4}{35}\sqrt{1-x}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*arcsin(x^(1/2))/x^4-1/28*(1-x)^(1/2)/x^(7/2)-3/70*(1-x)^(1/2)/x^(5/2)-2/35*(1-x)^(1/2)/x^(3/2)-4/35*(1-x)
^(1/2)/x^(1/2)

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Maxima [A]  time = 1.4238, size = 78, normalized size = 0.91 \begin{align*} -\frac{4 \, \sqrt{-x + 1}}{35 \, \sqrt{x}} - \frac{2 \, \sqrt{-x + 1}}{35 \, x^{\frac{3}{2}}} - \frac{3 \, \sqrt{-x + 1}}{70 \, x^{\frac{5}{2}}} - \frac{\sqrt{-x + 1}}{28 \, x^{\frac{7}{2}}} - \frac{\arcsin \left (\sqrt{x}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/35*sqrt(-x + 1)/sqrt(x) - 2/35*sqrt(-x + 1)/x^(3/2) - 3/70*sqrt(-x + 1)/x^(5/2) - 1/28*sqrt(-x + 1)/x^(7/2)
- 1/4*arcsin(sqrt(x))/x^4

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Fricas [A]  time = 2.29405, size = 112, normalized size = 1.3 \begin{align*} -\frac{{\left (16 \, x^{3} + 8 \, x^{2} + 6 \, x + 5\right )} \sqrt{x} \sqrt{-x + 1} + 35 \, \arcsin \left (\sqrt{x}\right )}{140 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/140*((16*x^3 + 8*x^2 + 6*x + 5)*sqrt(x)*sqrt(-x + 1) + 35*arcsin(sqrt(x)))/x^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.13864, size = 186, normalized size = 2.16 \begin{align*} -\frac{{\left (\sqrt{-x + 1} - 1\right )}^{7}}{3584 \, x^{\frac{7}{2}}} - \frac{7 \,{\left (\sqrt{-x + 1} - 1\right )}^{5}}{2560 \, x^{\frac{5}{2}}} - \frac{7 \,{\left (\sqrt{-x + 1} - 1\right )}^{3}}{512 \, x^{\frac{3}{2}}} - \frac{35 \,{\left (\sqrt{-x + 1} - 1\right )}}{512 \, \sqrt{x}} + \frac{{\left (\frac{1225 \,{\left (\sqrt{-x + 1} - 1\right )}^{6}}{x^{3}} + \frac{245 \,{\left (\sqrt{-x + 1} - 1\right )}^{4}}{x^{2}} + \frac{49 \,{\left (\sqrt{-x + 1} - 1\right )}^{2}}{x} + 5\right )} x^{\frac{7}{2}}}{17920 \,{\left (\sqrt{-x + 1} - 1\right )}^{7}} - \frac{\arcsin \left (\sqrt{x}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3584*(sqrt(-x + 1) - 1)^7/x^(7/2) - 7/2560*(sqrt(-x + 1) - 1)^5/x^(5/2) - 7/512*(sqrt(-x + 1) - 1)^3/x^(3/2
) - 35/512*(sqrt(-x + 1) - 1)/sqrt(x) + 1/17920*(1225*(sqrt(-x + 1) - 1)^6/x^3 + 245*(sqrt(-x + 1) - 1)^4/x^2
+ 49*(sqrt(-x + 1) - 1)^2/x + 5)*x^(7/2)/(sqrt(-x + 1) - 1)^7 - 1/4*arcsin(sqrt(x))/x^4