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]

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

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

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 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]

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*}

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Mathematica [A]  time = 0.02, 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]

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

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fricas [A]  time = 0.83, size = 28, normalized size = 0.56 \[ \frac {1}{2} \, {\left (2 \, x - 1\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} \]

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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giac [A]  time = 1.46, size = 47, normalized size = 0.94 \[ x \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} + \frac {1}{2} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \]

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(-sqrt(x - 1) + sqrt(x
))

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maple [A]  time = 0.00, size = 49, normalized size = 0.98 \[ x \,\mathrm {arccosh}\left (\sqrt {x}\right )-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{2 \sqrt {-1+x}} \]

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 = 0.62, size = 33, normalized size = 0.66 \[ x \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} - \frac {1}{2} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]

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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mupad [B]  time = 1.43, size = 40, normalized size = 0.80 \[ -2\,\sqrt {x}\,\mathrm {acosh}\left (\sqrt {x}\right )\,\left (\frac {1}{4\,\sqrt {x}}-\frac {\sqrt {x}}{2}\right )-\frac {\sqrt {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2*x^(1/2)*acosh(x^(1/2))*(1/(4*x^(1/2)) - x^(1/2)/2) - (x^(1/2)*(x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2))/2

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sympy [A]  time = 0.26, size = 29, normalized size = 0.58 \[ - \frac {\sqrt {x} \sqrt {x - 1}}{2} + x \operatorname {acosh}{\left (\sqrt {x} \right )} - \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{2} \]

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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