\(\int \frac {a+b \text {arccosh}(c x)}{x} \, dx\) [137]

Optimal result

Integrand size = 12, antiderivative size = 55 \[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{2 b}+(a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )+\frac {1}{2} b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=a \log (x)+\frac {1}{2} b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \] Input:

Integrate[(a + b*ArcCosh[c*x])/x,x]
 

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6297, 25, 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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x} \, dx\)

\(\Big \downarrow \) 6297

\(\displaystyle \frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcCosh[c*x])/x,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 6297

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33

method	result	size
parts	\(a \ln \left (x \right )+b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\)	\(73\)
derivativedivides	\(a \ln \left (c x \right )+b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\)	\(75\)
default	\(a \ln \left (c x \right )+b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\)	\(75\)

Input:

int((a+b*arccosh(c*x))/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((a+b*arccosh(c*x))/x,x, algorithm="fricas")
 

Output:

integral((b*arccosh(c*x) + a)/x, x)
 

Sympy [F]

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

integrate((a+b*acosh(c*x))/x,x)
 

Output:

Integral((a + b*acosh(c*x))/x, x)
 

Maxima [F]

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

integrate((a+b*arccosh(c*x))/x,x, algorithm="maxima")
 

Output:

b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x) + a*log(x)
 

Giac [F]

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

integrate((a+b*arccosh(c*x))/x,x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/x, x)
 

Mupad [F(-1)]

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

int((a + b*acosh(c*x))/x,x)
                                                                                    
                                                                                    
 

Output:

int((a + b*acosh(c*x))/x, x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x} \, dx=\left (\int \frac {\mathit {acosh} \left (c x \right )}{x}d x \right ) b +\mathrm {log}\left (x \right ) a \] Input:

int((a+b*acosh(c*x))/x,x)
 

Output:

int(acosh(c*x)/x,x)*b + log(x)*a