3.367 \(\int \frac{\sin ^{-1}(\sqrt{x})}{x^4} \, dx\)

Optimal. Leaf size=68 \[ -\frac{4 \sqrt{1-x}}{45 x^{3/2}}-\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{8 \sqrt{1-x}}{45 \sqrt{x}} \]

[Out]

-Sqrt[1 - x]/(15*x^(5/2)) - (4*Sqrt[1 - x])/(45*x^(3/2)) - (8*Sqrt[1 - x])/(45*Sqrt[x]) - ArcSin[Sqrt[x]]/(3*x
^3)

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Rubi [A]  time = 0.0223059, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4842, 12, 45, 37} \[ -\frac{4 \sqrt{1-x}}{45 x^{3/2}}-\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{8 \sqrt{1-x}}{45 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sqrt[x]]/x^4,x]

[Out]

-Sqrt[1 - x]/(15*x^(5/2)) - (4*Sqrt[1 - x])/(45*x^(3/2)) - (8*Sqrt[1 - x])/(45*Sqrt[x]) - ArcSin[Sqrt[x]]/(3*x
^3)

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^4} \, dx &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{1}{3} \int \frac{1}{2 \sqrt{1-x} x^{7/2}} \, dx\\ &=-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{1}{6} \int \frac{1}{\sqrt{1-x} x^{7/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{2}{15} \int \frac{1}{\sqrt{1-x} x^{5/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{4 \sqrt{1-x}}{45 x^{3/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{4}{45} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{4 \sqrt{1-x}}{45 x^{3/2}}-\frac{8 \sqrt{1-x}}{45 \sqrt{x}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0254635, size = 44, normalized size = 0.65 \[ 2 \left (-\frac{\sqrt{1-x} \left (8 x^2+4 x+3\right )}{90 x^{5/2}}-\frac{\sin ^{-1}\left (\sqrt{x}\right )}{6 x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[Sqrt[x]]/x^4,x]

[Out]

2*(-(Sqrt[1 - x]*(3 + 4*x + 8*x^2))/(90*x^(5/2)) - ArcSin[Sqrt[x]]/(6*x^3))

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Maple [A]  time = 0.004, size = 47, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,{x}^{3}}\arcsin \left ( \sqrt{x} \right ) }-{\frac{1}{15}\sqrt{1-x}{x}^{-{\frac{5}{2}}}}-{\frac{4}{45}\sqrt{1-x}{x}^{-{\frac{3}{2}}}}-{\frac{8}{45}\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^4,x)

[Out]

-1/3*arcsin(x^(1/2))/x^3-1/15*(1-x)^(1/2)/x^(5/2)-4/45*(1-x)^(1/2)/x^(3/2)-8/45*(1-x)^(1/2)/x^(1/2)

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Maxima [A]  time = 1.41558, size = 62, normalized size = 0.91 \begin{align*} -\frac{8 \, \sqrt{-x + 1}}{45 \, \sqrt{x}} - \frac{4 \, \sqrt{-x + 1}}{45 \, x^{\frac{3}{2}}} - \frac{\sqrt{-x + 1}}{15 \, x^{\frac{5}{2}}} - \frac{\arcsin \left (\sqrt{x}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x^(1/2))/x^4,x, algorithm="maxima")

[Out]

-8/45*sqrt(-x + 1)/sqrt(x) - 4/45*sqrt(-x + 1)/x^(3/2) - 1/15*sqrt(-x + 1)/x^(5/2) - 1/3*arcsin(sqrt(x))/x^3

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Fricas [A]  time = 2.2572, size = 99, normalized size = 1.46 \begin{align*} -\frac{{\left (8 \, x^{2} + 4 \, x + 3\right )} \sqrt{x} \sqrt{-x + 1} + 15 \, \arcsin \left (\sqrt{x}\right )}{45 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x^(1/2))/x^4,x, algorithm="fricas")

[Out]

-1/45*((8*x^2 + 4*x + 3)*sqrt(x)*sqrt(-x + 1) + 15*arcsin(sqrt(x)))/x^3

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Sympy [A]  time = 97.7187, size = 58, normalized size = 0.85 \begin{align*} \frac{\begin{cases} - \frac{\sqrt{1 - x}}{\sqrt{x}} - \frac{2 \left (1 - x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} - \frac{\left (1 - x\right )^{\frac{5}{2}}}{5 x^{\frac{5}{2}}} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{3} - \frac{\operatorname{asin}{\left (\sqrt{x} \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x**(1/2))/x**4,x)

[Out]

Piecewise((-sqrt(1 - x)/sqrt(x) - 2*(1 - x)**(3/2)/(3*x**(3/2)) - (1 - x)**(5/2)/(5*x**(5/2)), (x >= 0) & (x <
1)))/3 - asin(sqrt(x))/(3*x**3)

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Giac [B]  time = 1.22546, size = 143, normalized size = 2.1 \begin{align*} -\frac{{\left (\sqrt{-x + 1} - 1\right )}^{5}}{480 \, x^{\frac{5}{2}}} - \frac{5 \,{\left (\sqrt{-x + 1} - 1\right )}^{3}}{288 \, x^{\frac{3}{2}}} - \frac{5 \,{\left (\sqrt{-x + 1} - 1\right )}}{48 \, \sqrt{x}} + \frac{{\left (\frac{150 \,{\left (\sqrt{-x + 1} - 1\right )}^{4}}{x^{2}} + \frac{25 \,{\left (\sqrt{-x + 1} - 1\right )}^{2}}{x} + 3\right )} x^{\frac{5}{2}}}{1440 \,{\left (\sqrt{-x + 1} - 1\right )}^{5}} - \frac{\arcsin \left (\sqrt{x}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x^(1/2))/x^4,x, algorithm="giac")

[Out]

-1/480*(sqrt(-x + 1) - 1)^5/x^(5/2) - 5/288*(sqrt(-x + 1) - 1)^3/x^(3/2) - 5/48*(sqrt(-x + 1) - 1)/sqrt(x) + 1
/1440*(150*(sqrt(-x + 1) - 1)^4/x^2 + 25*(sqrt(-x + 1) - 1)^2/x + 3)*x^(5/2)/(sqrt(-x + 1) - 1)^5 - 1/3*arcsin
(sqrt(x))/x^3