\(\int \frac {a+b \text {arccosh}(c x)}{x (d+e x^2)} \, dx\) [392]

Optimal result

Integrand size = 21, antiderivative size = 472 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=-\frac {-2 b \text {arccosh}(c x)^2-2 b \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+b \text {arccosh}(c x) \log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+b \text {arccosh}(c x) \log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \text {arccosh}(c x) \log \left (1-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \text {arccosh}(c x) \log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 a \log (x)+a \log \left (d+e x^2\right )+b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+b \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+b \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{2 d} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)),x]
 

Output:

-1/2*(-2*b*ArcCosh[c*x]^2 - 2*b*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] 
+ b*ArcCosh[c*x]*Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2* 
d + e])] + b*ArcCosh[c*x]*Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) 
 + Sqrt[c^2*d + e])] + b*ArcCosh[c*x]*Log[1 - (I*Sqrt[e]*E^ArcCosh[c*x])/( 
c*Sqrt[d] + Sqrt[c^2*d + e])] + b*ArcCosh[c*x]*Log[1 + (I*Sqrt[e]*E^ArcCos 
h[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])] - 2*a*Log[x] + a*Log[d + e*x^2] + b 
*PolyLog[2, -E^(-2*ArcCosh[c*x])] + b*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x] 
)/(c*Sqrt[d] - Sqrt[c^2*d + e])] + b*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x]) 
/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + b*PolyLog[2, ((-I)*Sqrt[e]*E^ArcCosh[ 
c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])] + b*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c 
*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])/d
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6374, 2009}

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.

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)),x]
 

Output:

(a + b*ArcCosh[c*x])^2/(b*d) + ((a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh 
[c*x])])/d - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqr 
t[-d] - Sqrt[-(c^2*d) - e])])/(2*d) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[ 
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d) - ((a + b*Arc 
Cosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - 
e])])/(2*d) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sq 
rt[-d] + Sqrt[-(c^2*d) - e])])/(2*d) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])] 
)/(2*d) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^ 
2*d) - e]))])/(2*d) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - 
 Sqrt[-(c^2*d) - e])])/(2*d) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c 
*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*d) - (b*PolyLog[2, (Sqrt[e]*E^ArcCos 
h[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6374

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, 
 (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d 
 + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.56 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.81

method	result	size
parts	\(-\frac {a \ln \left (e \,x^{2}+d \right )}{2 d}+\frac {a \ln \left (x \right )}{d}+b \left (-\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {\operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {\operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}}{4 d}\right )\)	\(381\)
derivativedivides	\(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}\)	\(393\)
default	\(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}\)	\(393\)

Input:

int((a+b*arccosh(c*x))/x/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

-1/2*a/d*ln(e*x^2+d)+a/d*ln(x)+b*(-1/4*e/d*sum((_R1^2+1)/(_R1^2*e+2*c^2*d+ 
e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1- 
c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^ 
2+e))+1/d*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/d*arcco 
sh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/d*dilog(1+I*(c*x+(c*x- 
1)^(1/2)*(c*x+1)^(1/2)))+1/d*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))- 
1/4/d*sum((_R1^2*e+4*c^2*d+e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c* 
x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^( 
1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e)))
 

Fricas [F]

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

integrate((a+b*arccosh(c*x))/x/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*arccosh(c*x) + a)/(e*x^3 + d*x), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (d + e x^{2}\right )}\, dx \] Input:

integrate((a+b*acosh(c*x))/x/(e*x**2+d),x)
 

Output:

Integral((a + b*acosh(c*x))/(x*(d + e*x**2)), x)
 

Maxima [F]

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

integrate((a+b*arccosh(c*x))/x/(e*x^2+d),x, algorithm="maxima")
 

Output:

-1/2*a*(log(e*x^2 + d)/d - 2*log(x)/d) + b*integrate(log(c*x + sqrt(c*x + 
1)*sqrt(c*x - 1))/(e*x^3 + d*x), x)
 

Giac [F]

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

integrate((a+b*arccosh(c*x))/x/(e*x^2+d),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/((e*x^2 + d)*x), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a + b*acosh(c*x))/(x*(d + e*x^2)),x)
 

Output:

int((a + b*acosh(c*x))/(x*(d + e*x^2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d+e x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e \,x^{3}+d x}d x \right ) b d -\mathrm {log}\left (e \,x^{2}+d \right ) a +2 \,\mathrm {log}\left (x \right ) a}{2 d} \] Input:

int((a+b*acosh(c*x))/x/(e*x^2+d),x)
 

Output:

(2*int(acosh(c*x)/(d*x + e*x**3),x)*b*d - log(d + e*x**2)*a + 2*log(x)*a)/ 
(2*d)