3.236 \(\int \frac{\cosh ^{-1}(\sqrt{x})}{x^2} \, dx\)

Optimal. Leaf size=40 \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x} \]

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x] - ArcCosh[Sqrt[x]]/x

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Rubi [A]  time = 0.0256731, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5903, 12, 265} \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[Sqrt[x]]/x^2,x]

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x] - ArcCosh[Sqrt[x]]/x

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

Mathematica [A]  time = 0.0135517, size = 40, normalized size = 1. \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[Sqrt[x]]/x^2,x]

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x] - ArcCosh[Sqrt[x]]/x

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Maple [A]  time = 0.003, size = 29, normalized size = 0.7 \begin{align*} -{\frac{1}{x}{\rm arccosh} \left (\sqrt{x}\right )}+{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.54244, size = 26, normalized size = 0.65 \begin{align*} \frac{\sqrt{x - 1}}{\sqrt{x}} - \frac{\operatorname{arcosh}\left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x - 1)/sqrt(x) - arccosh(sqrt(x))/x

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Fricas [A]  time = 2.17386, size = 73, normalized size = 1.82 \begin{align*} \frac{\sqrt{x - 1} \sqrt{x} - \log \left (\sqrt{x - 1} + \sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x - 1)*sqrt(x) - log(sqrt(x - 1) + sqrt(x)))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}{\left (\sqrt{x} \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(sqrt(x))/x**2, x)

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Giac [A]  time = 1.12095, size = 61, normalized size = 1.52 \begin{align*} -\frac{\log \left (\sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \sqrt{x}\right )}{x} + \frac{2}{{\left (\sqrt{x - 1} - \sqrt{x}\right )}^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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