\(\int \frac {(d+e x^2) (a+b \text {arccosh}(c x))}{x} \, dx\) [365]

Optimal result

Integrand size = 19, antiderivative size = 264 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/4*b*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/4*b*e*arccosh(c*x)/c^2+1/2*e*x^ 
2*(a+b*arccosh(c*x))-1/2*I*b*d*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2/(c*x-1)^(1 
/2)/(c*x+1)^(1/2)+b*d*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2 
+1)^(1/2))^2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+d*(a+b*arccosh(c*x))*ln(x)-b*d*( 
-c^2*x^2+1)^(1/2)*arcsin(c*x)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*I*b*d* 
(-c^2*x^2+1)^(1/2)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/(c*x-1)^(1/2)/( 
c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.49 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{2} a e x^2-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+\frac {1}{2} b e x^2 \text {arccosh}(c x)-\frac {b e \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{2 c^2}+a d \log (x)+\frac {1}{2} b d \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \] Input:

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

Output:

(a*e*x^2)/2 - (b*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) + (b*e*x^2*ArcCos 
h[c*x])/2 - (b*e*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/(2*c^2) + a*d*Log[ 
x] + (b*d*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - 
PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6373, 27, 7293, 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.

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

\(\Big \downarrow \) 6373

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle d \log (x) (a+b \text {arccosh}(c x))+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {e \text {arccosh}(c x)}{2 c^3}+\frac {i d \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {i d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 d \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {e x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\)

Input:

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

Output:

(e*x^2*(a + b*ArcCosh[c*x]))/2 + d*(a + b*ArcCosh[c*x])*Log[x] - (b*c*((e* 
x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + (e*ArcCosh[c*x])/(2*c^3) + (I*d* 
Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d*S 
qrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/(c*Sqrt[-1 + 
c*x]*Sqrt[1 + c*x]) + (2*d*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(c*Sqrt[- 
1 + c*x]*Sqrt[1 + c*x]) + (I*d*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSi 
n[c*x])])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 6373

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x 
_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim 
p[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le 
Q[m + p, 0]))
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.48

method	result	size
parts	\(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {b e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b d \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(128\)
derivativedivides	\(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {b e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b d \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(130\)
default	\(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {b e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b d \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(130\)

Input:

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

Output:

1/2*a*e*x^2+a*d*ln(x)-1/2*d*b*arccosh(c*x)^2+1/2*b*arccosh(c*x)*e*x^2-1/4* 
b*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/4*b*e*arccosh(c*x)/c^2+d*b*arccosh(c 
*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+1/2*b*d*polylog(2,-(c*x+(c*x 
-1)^(1/2)*(c*x+1)^(1/2))^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\frac {2 \mathit {acosh} \left (c x \right ) b \,c^{2} e \,x^{2}-\sqrt {c^{2} x^{2}-1}\, b c e x +4 \left (\int \frac {\mathit {acosh} \left (c x \right )}{x}d x \right ) b \,c^{2} d -\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b e +4 \,\mathrm {log}\left (x \right ) a \,c^{2} d +2 a \,c^{2} e \,x^{2}}{4 c^{2}} \] Input:

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

Output:

(2*acosh(c*x)*b*c**2*e*x**2 - sqrt(c**2*x**2 - 1)*b*c*e*x + 4*int(acosh(c* 
x)/x,x)*b*c**2*d - log(sqrt(c**2*x**2 - 1) + c*x)*b*e + 4*log(x)*a*c**2*d 
+ 2*a*c**2*e*x**2)/(4*c**2)