\(\int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\) [134]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {1-a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {i \sqrt {1-a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {i \sqrt {1-a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}} \] Output:

-2*(-a*x+1)^(1/2)*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a* 
x-1)^(1/2)+I*(-a*x+1)^(1/2)*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)) 
)/(a*x-1)^(1/2)-I*(-a*x+1)^(1/2)*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1 
/2)))/(a*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\frac {i \sqrt {-((-1+a x) (1+a x))} \left (\text {arccosh}(a x) \left (\log \left (1-i e^{-\text {arccosh}(a x)}\right )-\log \left (1+i e^{-\text {arccosh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )}{\sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:

Integrate[ArcCosh[a*x]/(x*Sqrt[1 - a^2*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.64, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6361, 3042, 4668, 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.

\(\displaystyle \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6361

\(\displaystyle \frac {\sqrt {a x-1} \int \frac {\text {arccosh}(a x)}{a x}d\text {arccosh}(a x)}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCosh[a*x]/(x*Sqrt[1 - a^2*x^2]),x]
 

Output:

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

Defintions of rubi rules used

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 4668

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

rule 6361

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x 
]/Sqrt[d + e*x^2])]   Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] 
, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int 
egerQ[m]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (126 ) = 252\).

Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.62

method	result	size
default	\(\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{a^{2} x^{2}-1}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{a^{2} x^{2}-1}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{a^{2} x^{2}-1}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{a^{2} x^{2}-1}\)	\(270\)

Input:

int(arccosh(a*x)/x/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

I*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*arccosh(a*x)* 
ln(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-I*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/ 
2)*(a*x+1)^(1/2)/(a^2*x^2-1)*arccosh(a*x)*ln(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1 
)^(1/2)))+I*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*dil 
og(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-I*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/ 
2)*(a*x+1)^(1/2)/(a^2*x^2-1)*dilog(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))
 

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

integrate(arccosh(a*x)/x/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)/(a^2*x^3 - x), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(acosh(a*x)/x/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(acosh(a*x)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

integrate(arccosh(a*x)/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(arccosh(a*x)/(sqrt(-a^2*x^2 + 1)*x), x)
 

Giac [F]

\[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

integrate(arccosh(a*x)/x/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(arccosh(a*x)/(sqrt(-a^2*x^2 + 1)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{x\,\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(acosh(a*x)/(x*(1 - a^2*x^2)^(1/2)),x)
 

Output:

int(acosh(a*x)/(x*(1 - a^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}(a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}\, x}d x \] Input:

int(acosh(a*x)/x/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(acosh(a*x)/(sqrt( - a**2*x**2 + 1)*x),x)