3.297 \(\int \frac{\sinh ^{-1}(\sqrt{x})}{x^3} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\sqrt{x+1}}{6 x^{3/2}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\sqrt{x+1}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0170543, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5902, 12, 45, 37} \[ -\frac{\sqrt{x+1}}{6 x^{3/2}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\sqrt{x+1}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[Sqrt[x]]/x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.011902, size = 34, normalized size = 0.74 \[ \frac{\sqrt{x} \sqrt{x+1} (2 x-1)-3 \sinh ^{-1}\left (\sqrt{x}\right )}{6 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[Sqrt[x]]/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 31, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\it Arcsinh} \left ( \sqrt{x} \right ) }-{\frac{1}{6}\sqrt{1+x}{x}^{-{\frac{3}{2}}}}+{\frac{1}{3}\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^3,x)

[Out]

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

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Maxima [A]  time = 1.65973, size = 41, normalized size = 0.89 \begin{align*} \frac{\sqrt{x + 1}}{3 \, \sqrt{x}} - \frac{\sqrt{x + 1}}{6 \, x^{\frac{3}{2}}} - \frac{\operatorname{arsinh}\left (\sqrt{x}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x^(1/2))/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.84893, size = 97, normalized size = 2.11 \begin{align*} \frac{{\left (2 \, x - 1\right )} \sqrt{x + 1} \sqrt{x} - 3 \, \log \left (\sqrt{x + 1} + \sqrt{x}\right )}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x^(1/2))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (\sqrt{x} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(x**(1/2))/x**3,x)

[Out]

Integral(asinh(sqrt(x))/x**3, x)

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Giac [A]  time = 1.35205, size = 70, normalized size = 1.52 \begin{align*} -\frac{\log \left (\sqrt{x + 1} + \sqrt{x}\right )}{2 \, x^{2}} + \frac{2 \,{\left (3 \,{\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} - 1\right )}}{3 \,{\left ({\left (\sqrt{x + 1} - \sqrt{x}\right )}^{2} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x^(1/2))/x^3,x, algorithm="giac")

[Out]

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