3.234 \(\int \cosh ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=50 \[ -\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0313341, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5901, 12, 323, 330, 52} \[ -\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[Sqrt[x]],x]

[Out]

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

Rule 5901

Int[ArcCosh[u_], x_Symbol] :> Simp[x*ArcCosh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(Sqrt[-1 + u]*Sqrt[1 +
u]), x], x] /; InverseFunctionFreeQ[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 323

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

Rule 330

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x]
, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.0233279, size = 64, normalized size = 1.28 \[ -\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[Sqrt[x]],x]

[Out]

-(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/2 + x*ArcCosh[Sqrt[x]] - ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt
[x])]]

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Maple [A]  time = 0.003, size = 49, normalized size = 1. \begin{align*} x{\rm arccosh} \left (\sqrt{x}\right )-{\frac{1}{2}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( \sqrt{x}\sqrt{-1+x}+\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01828, size = 45, normalized size = 0.9 \begin{align*} x \operatorname{arcosh}\left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x - 1} \sqrt{x} - \frac{1}{2} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.0325, size = 92, normalized size = 1.84 \begin{align*} \frac{1}{2} \,{\left (2 \, x - 1\right )} \log \left (\sqrt{x - 1} + \sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x - 1} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.343594, size = 29, normalized size = 0.58 \begin{align*} - \frac{\sqrt{x} \sqrt{x - 1}}{2} + x \operatorname{acosh}{\left (\sqrt{x} \right )} - \frac{\operatorname{acosh}{\left (\sqrt{x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x)*sqrt(x - 1)/2 + x*acosh(sqrt(x)) - acosh(sqrt(x))/2

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Giac [A]  time = 1.16816, size = 65, normalized size = 1.3 \begin{align*} x \log \left (\sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x - 1} \sqrt{x} + \frac{1}{2} \, \log \left ({\left | \sqrt{x - 1} - \sqrt{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + sqrt(x)) - 1/2*sqrt(x - 1)*sqrt(x) + 1/2*log(abs(sqrt(x - 1) - sqr
t(x)))