\(\int \frac {\text {arccosh}(\sqrt {x})}{x} \, dx\) [144]

Optimal result

Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=-\text {arccosh}\left (\sqrt {x}\right )^2+2 \text {arccosh}\left (\sqrt {x}\right ) \log \left (1+e^{2 \text {arccosh}\left (\sqrt {x}\right )}\right )+\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\sqrt {x}\right )}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\text {arccosh}\left (\sqrt {x}\right ) \left (\text {arccosh}\left (\sqrt {x}\right )+2 \log \left (1+e^{-2 \text {arccosh}\left (\sqrt {x}\right )}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\sqrt {x}\right )}\right ) \] Input:

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

Output:

ArcCosh[Sqrt[x]]*(ArcCosh[Sqrt[x]] + 2*Log[1 + E^(-2*ArcCosh[Sqrt[x]])]) - 
 PolyLog[2, -E^(-2*ArcCosh[Sqrt[x]])]
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.41, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6426, 3042, 26, 4201, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx\)

\(\Big \downarrow \) 6426

\(\displaystyle 2 \int \frac {\sqrt {\frac {\sqrt {x}-1}{\sqrt {x}+1}} \left (\sqrt {x}+1\right ) \text {arccosh}\left (\sqrt {x}\right )}{\sqrt {x}}d\text {arccosh}\left (\sqrt {x}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int -i \text {arccosh}\left (\sqrt {x}\right ) \tan \left (i \text {arccosh}\left (\sqrt {x}\right )\right )d\text {arccosh}\left (\sqrt {x}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -2 i \int \text {arccosh}\left (\sqrt {x}\right ) \tan \left (i \text {arccosh}\left (\sqrt {x}\right )\right )d\text {arccosh}\left (\sqrt {x}\right )\)

\(\Big \downarrow \) 4201

\(\displaystyle -2 i \left (2 i \int \frac {e^{2 \text {arccosh}\left (\sqrt {x}\right )} \text {arccosh}\left (\sqrt {x}\right )}{1+e^{2 \text {arccosh}\left (\sqrt {x}\right )}}d\text {arccosh}\left (\sqrt {x}\right )-\frac {1}{2} i \text {arccosh}\left (\sqrt {x}\right )^2\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -2 i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (\sqrt {x}\right ) \log \left (e^{2 \text {arccosh}\left (\sqrt {x}\right )}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}\left (\sqrt {x}\right )}\right )d\text {arccosh}\left (\sqrt {x}\right )\right )-\frac {1}{2} i \text {arccosh}\left (\sqrt {x}\right )^2\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -2 i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (\sqrt {x}\right ) \log \left (e^{2 \text {arccosh}\left (\sqrt {x}\right )}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}\left (\sqrt {x}\right )} \log \left (1+e^{2 \text {arccosh}\left (\sqrt {x}\right )}\right )de^{2 \text {arccosh}\left (\sqrt {x}\right )}\right )-\frac {1}{2} i \text {arccosh}\left (\sqrt {x}\right )^2\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -2 i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\sqrt {x}\right )}\right )+\frac {1}{2} \text {arccosh}\left (\sqrt {x}\right ) \log \left (e^{2 \text {arccosh}\left (\sqrt {x}\right )}+1\right )\right )-\frac {1}{2} i \text {arccosh}\left (\sqrt {x}\right )^2\right )\)

Input:

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

Output:

(-2*I)*((-1/2*I)*ArcCosh[Sqrt[x]]^2 + (2*I)*((ArcCosh[Sqrt[x]]*Log[1 + E^( 
2*ArcCosh[Sqrt[x]])])/2 + PolyLog[2, -E^(2*ArcCosh[Sqrt[x]])]/4))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6426

Int[ArcCosh[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Simp[1/p   Subst[Int[ 
x^n*Tanh[x], x], x, ArcCosh[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(-\operatorname {arccosh}\left (\sqrt {x}\right )^{2}+2 \,\operatorname {arccosh}\left (\sqrt {x}\right ) \ln \left (1+\left (\sqrt {x}+\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (\sqrt {x}+\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\right )^{2}\right )\)	\(65\)
default	\(-\operatorname {arccosh}\left (\sqrt {x}\right )^{2}+2 \,\operatorname {arccosh}\left (\sqrt {x}\right ) \ln \left (1+\left (\sqrt {x}+\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (\sqrt {x}+\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\right )^{2}\right )\)	\(65\)

Input:

int(arccosh(x^(1/2))/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{x} \,d x } \] Input:

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

Output:

integral(arccosh(sqrt(x))/x, x)
 

Sympy [F]

\[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{x}\, dx \] Input:

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

Output:

Integral(acosh(sqrt(x))/x, x)
 

Maxima [F]

\[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{x} \,d x } \] Input:

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

Output:

integrate(arccosh(sqrt(x))/x, x)
 

Giac [F]

\[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{x} \,d x } \] Input:

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

Output:

integrate(arccosh(sqrt(x))/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\mathrm {acosh}\left (\sqrt {x}\right )}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\mathit {acosh} \left (\sqrt {x}\right )}{x}d x \] Input:

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

Output:

int(acosh(sqrt(x))/x,x)