Integrand size = 6, antiderivative size = 50 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=-\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {\text {arccosh}\left (\sqrt {x}\right )}{2}+x \text {arccosh}\left (\sqrt {x}\right ) \] Output:
-1/2*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*x^(1/2)-1/2*arccosh(x^(1/2))+x*a rccosh(x^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(273\) vs. \(2(50)=100\).
Time = 4.60 (sec) , antiderivative size = 273, normalized size of antiderivative = 5.46 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=-\frac {2 \left (4 \sqrt {1+\sqrt {x}} \left (-12-24 \sqrt {x}+x+5 x^{3/2}\right )+\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \left (-84-10 \sqrt {x}+28 x+7 x^{3/2}\right )+\sqrt {3} \left (28+70 \sqrt {x}+18 x-14 x^{3/2}-4 x^2-4 \sqrt {-1+\sqrt {x}} \left (-12-8 \sqrt {x}+5 x+3 x^{3/2}\right )\right )\right )}{56-16 \sqrt {3} \sqrt {1+\sqrt {x}} \left (2+3 \sqrt {x}\right )+\sqrt {-1+\sqrt {x}} \left (96-8 \sqrt {3} \sqrt {1+\sqrt {x}} \left (7+2 \sqrt {x}\right )+80 \sqrt {x}\right )+112 \sqrt {x}+28 x}+x \text {arccosh}\left (\sqrt {x}\right )+2 \text {arctanh}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \] Input:
Integrate[ArcCosh[Sqrt[x]],x]
Output:
(-2*(4*Sqrt[1 + Sqrt[x]]*(-12 - 24*Sqrt[x] + x + 5*x^(3/2)) + Sqrt[-1 + Sq rt[x]]*Sqrt[1 + Sqrt[x]]*(-84 - 10*Sqrt[x] + 28*x + 7*x^(3/2)) + Sqrt[3]*( 28 + 70*Sqrt[x] + 18*x - 14*x^(3/2) - 4*x^2 - 4*Sqrt[-1 + Sqrt[x]]*(-12 - 8*Sqrt[x] + 5*x + 3*x^(3/2)))))/(56 - 16*Sqrt[3]*Sqrt[1 + Sqrt[x]]*(2 + 3* Sqrt[x]) + Sqrt[-1 + Sqrt[x]]*(96 - 8*Sqrt[3]*Sqrt[1 + Sqrt[x]]*(7 + 2*Sqr t[x]) + 80*Sqrt[x]) + 112*Sqrt[x] + 28*x) + x*ArcCosh[Sqrt[x]] + 2*ArcTanh [(-1 + Sqrt[-1 + Sqrt[x]])/(Sqrt[3] - Sqrt[1 + Sqrt[x]])]
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6431, 27, 845, 852, 43}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \text {arccosh}\left (\sqrt {x}\right ) \, dx\) |
\(\Big \downarrow \) 6431 |
\(\displaystyle x \text {arccosh}\left (\sqrt {x}\right )-\int \frac {\sqrt {x}}{2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \text {arccosh}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}dx\) |
\(\Big \downarrow \) 845 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}}dx-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+x \text {arccosh}\left (\sqrt {x}\right )\) |
\(\Big \downarrow \) 852 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {1}{\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}d\sqrt {x}-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+x \text {arccosh}\left (\sqrt {x}\right )\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {1}{2} \left (-\text {arccosh}\left (\sqrt {x}\right )-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}\right )+x \text {arccosh}\left (\sqrt {x}\right )\) |
Input:
Int[ArcCosh[Sqrt[x]],x]
Output:
(-(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]) - ArcCosh[Sqrt[x]])/2 + x *ArcCosh[Sqrt[x]]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
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] - Simp[a1*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] && Eq Q[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]
Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^ (n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[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] && FractionQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]
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]
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(x \,\operatorname {arccosh}\left (\sqrt {x}\right )-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{2 \sqrt {-1+x}}\) | \(49\) |
default | \(x \,\operatorname {arccosh}\left (\sqrt {x}\right )-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{2 \sqrt {-1+x}}\) | \(49\) |
parts | \(x \,\operatorname {arccosh}\left (\sqrt {x}\right )-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{2 \sqrt {-1+x}}\) | \(49\) |
Input:
int(arccosh(x^(1/2)),x,method=_RETURNVERBOSE)
Output:
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)
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.56 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, {\left (2 \, x - 1\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} \] Input:
integrate(arccosh(x^(1/2)),x, algorithm="fricas")
Output:
1/2*(2*x - 1)*log(sqrt(x - 1) + sqrt(x)) - 1/2*sqrt(x - 1)*sqrt(x)
\[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=\int \operatorname {acosh}{\left (\sqrt {x} \right )}\, dx \] Input:
integrate(acosh(x**(1/2)),x)
Output:
Integral(acosh(sqrt(x)), x)
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=x \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} - \frac {1}{2} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \] Input:
integrate(arccosh(x^(1/2)),x, algorithm="maxima")
Output:
x*arccosh(sqrt(x)) - 1/2*sqrt(x - 1)*sqrt(x) - 1/2*log(2*sqrt(x - 1) + 2*s qrt(x))
Time = 0.41 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=x \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} + \frac {1}{2} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \] Input:
integrate(arccosh(x^(1/2)),x, algorithm="giac")
Output:
x*log(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + sqrt(x)) - 1/2*sqrt(x - 1)*sqr t(x) + 1/2*log(-sqrt(x - 1) + sqrt(x))
Time = 4.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=-2\,\sqrt {x}\,\mathrm {acosh}\left (\sqrt {x}\right )\,\left (\frac {1}{4\,\sqrt {x}}-\frac {\sqrt {x}}{2}\right )-\frac {\sqrt {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}}{2} \] Input:
int(acosh(x^(1/2)),x)
Output:
- 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
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.48 \[ \int \text {arccosh}\left (\sqrt {x}\right ) \, dx=\mathit {acosh} \left (\sqrt {x}\right ) x -\frac {\sqrt {x}\, \sqrt {x -1}}{2}-\frac {\mathrm {log}\left (\sqrt {x -1}+\sqrt {x}\right )}{2} \] Input:
int(acosh(x^(1/2)),x)
Output:
(2*acosh(sqrt(x))*x - sqrt(x)*sqrt(x - 1) - log(sqrt(x - 1) + sqrt(x)))/2