3.299 \(\int \frac{\sinh ^{-1}(\sqrt{x})}{x^5} \, dx\)

Optimal. Leaf size=78 \[ -\frac{2 \sqrt{x+1}}{35 x^{3/2}}+\frac{3 \sqrt{x+1}}{70 x^{5/2}}-\frac{\sqrt{x+1}}{28 x^{7/2}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{4 \sqrt{x+1}}{35 \sqrt{x}} \]

[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]) - ArcSinh[Sqrt[x]]/(4*x^4)

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Rubi [A]  time = 0.0273138, antiderivative size = 78, 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 = {5902, 12, 45, 37} \[ -\frac{2 \sqrt{x+1}}{35 x^{3/2}}+\frac{3 \sqrt{x+1}}{70 x^{5/2}}-\frac{\sqrt{x+1}}{28 x^{7/2}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{4 \sqrt{x+1}}{35 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[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]) - ArcSinh[Sqrt[x]]/(4*x^4)

Rule 5902

Int[((a_.) + ArcSinh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin
h[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{\sinh ^{-1}\left (\sqrt{x}\right )}{x^5} \, dx &=-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{1}{4} \int \frac{1}{2 x^{9/2} \sqrt{1+x}} \, dx\\ &=-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{1}{8} \int \frac{1}{x^{9/2} \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1+x}}{28 x^{7/2}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{3}{28} \int \frac{1}{x^{7/2} \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1+x}}{28 x^{7/2}}+\frac{3 \sqrt{1+x}}{70 x^{5/2}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}+\frac{3}{35} \int \frac{1}{x^{5/2} \sqrt{1+x}} \, 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{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}-\frac{2}{35} \int \frac{1}{x^{3/2} \sqrt{1+x}} \, 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{\sinh ^{-1}\left (\sqrt{x}\right )}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0171397, size = 44, normalized size = 0.56 \[ \frac{\sqrt{x} \sqrt{x+1} \left (16 x^3-8 x^2+6 x-5\right )-35 \sinh ^{-1}\left (\sqrt{x}\right )}{140 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*Sqrt[1 + x]*(-5 + 6*x - 8*x^2 + 16*x^3) - 35*ArcSinh[Sqrt[x]])/(140*x^4)

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Maple [A]  time = 0.006, size = 51, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,{x}^{4}}{\it Arcsinh} \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(arcsinh(x^(1/2))/x^5,x)

[Out]

-1/4*arcsinh(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 = 1.80861, size = 68, normalized size = 0.87 \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{\operatorname{arsinh}\left (\sqrt{x}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(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*arcsinh(sqrt(x))/x^4

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Fricas [A]  time = 3.1853, size = 124, normalized size = 1.59 \begin{align*} \frac{{\left (16 \, x^{3} - 8 \, x^{2} + 6 \, x - 5\right )} \sqrt{x + 1} \sqrt{x} - 35 \, \log \left (\sqrt{x + 1} + \sqrt{x}\right )}{140 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/140*((16*x^3 - 8*x^2 + 6*x - 5)*sqrt(x + 1)*sqrt(x) - 35*log(sqrt(x + 1) + 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(asinh(x**(1/2))/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.35595, size = 111, normalized size = 1.42 \begin{align*} -\frac{\log \left (\sqrt{x + 1} + \sqrt{x}\right )}{4 \, x^{4}} + \frac{8 \,{\left (35 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{6} - 21 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{4} + 7 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} - 1\right )}}{35 \,{\left ({\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(sqrt(x + 1) + sqrt(x))/x^4 + 8/35*(35*(sqrt(x + 1) - sqrt(x))^6 - 21*(sqrt(x + 1) - sqrt(x))^4 + 7*(s
qrt(x + 1) - sqrt(x))^2 - 1)/((sqrt(x + 1) - sqrt(x))^2 - 1)^7