3.237 \(\int \frac{\cosh ^{-1}(\sqrt{x})}{x^3} \, dx\)

Optimal. Leaf size=76 \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{6 x^{3/2}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0406338, antiderivative size = 76, 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 = {5903, 12, 272, 265} \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{6 x^{3/2}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5903

Int[((a_.) + ArcCosh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcCos
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(Sqrt[-1 + u]*S
qrt[1 + u]), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !Function
OfQ[(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 272

Int[(x_)^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a
1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*(m + 1)), x] - Dist[(b1*b2*(m + 2*n*(p + 1) + 1))/(a1*a2*(m
+ 1)), Int[x^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x], x] /; FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a
2*b1 + a1*b2, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p + 1], 0] && NeQ[m, -1]

Rule 265

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*
x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(a1*a2*c*(m + 1)), x] /; FreeQ[{a1, b1, a2, b2, c, m,
n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0223006, size = 49, normalized size = 0.64 \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} (2 x+1)-3 \cosh ^{-1}\left (\sqrt{x}\right )}{6 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 35, normalized size = 0.5 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\rm arccosh} \left (\sqrt{x}\right )}+{\frac{1+2\,x}{6}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(x^(1/2))/x^3,x)

[Out]

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

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Maxima [A]  time = 1.53534, size = 41, normalized size = 0.54 \begin{align*} \frac{\sqrt{x - 1}}{3 \, \sqrt{x}} + \frac{\sqrt{x - 1}}{6 \, x^{\frac{3}{2}}} - \frac{\operatorname{arcosh}\left (\sqrt{x}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(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*arccosh(sqrt(x))/x^2

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Fricas [A]  time = 1.99472, size = 97, normalized size = 1.28 \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(arccosh(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{acosh}{\left (\sqrt{x} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(sqrt(x))/x**3, x)

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Giac [A]  time = 1.13663, size = 84, normalized size = 1.11 \begin{align*} -\frac{\log \left (\sqrt{\sqrt{x} + 1} \sqrt{\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(arccosh(x^(1/2))/x^3,x, algorithm="giac")

[Out]

-1/2*log(sqrt(sqrt(x) + 1)*sqrt(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