\(\int \frac {(d+e x^2)^2 (a+b \text {arccosh}(c x))}{x} \, dx\) [374]

Optimal result

Integrand size = 21, antiderivative size = 342 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {b e \left (16 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b e \left (16 c^2 d+3 e\right ) \text {arccosh}(c x)}{32 c^4}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/32*b*e*(16*c^2*d+3*e)*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/16*b*e^2*x^3* 
(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/32*b*e*(16*c^2*d+3*e)*arccosh(c*x)/c^4+d*e 
*x^2*(a+b*arccosh(c*x))+1/4*e^2*x^4*(a+b*arccosh(c*x))-1/2*I*b*d^2*(-c^2*x 
^2+1)^(1/2)*arcsin(c*x)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*d^2*(-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^2*(a+b*arccosh(c*x))*ln(x)-b*d^2*(-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^2*(-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] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.67 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=a d e x^2+\frac {1}{4} a e^2 x^4+b d e x^2 \text {arccosh}(c x)+\frac {1}{4} b e^2 x^4 \text {arccosh}(c x)-\frac {b d e \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{2 c^2}-\frac {b e^2 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{32 c^4}+\frac {1}{2} b d^2 \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+a d^2 \log (x)-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \] Input:

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

Output:

a*d*e*x^2 + (a*e^2*x^4)/4 + b*d*e*x^2*ArcCosh[c*x] + (b*e^2*x^4*ArcCosh[c* 
x])/4 - (b*d*e*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c* 
x)/(1 + c*x)]]))/(2*c^2) - (b*e^2*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(3 + 3*c 
*x + 2*c^2*x^2 + 2*c^3*x^3) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(32* 
c^4) + (b*d^2*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) 
)/2 + a*d^2*Log[x] - (b*d^2*PolyLog[2, -E^(-2*ArcCosh[c*x])])/2
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.

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

\(\Big \downarrow \) 6373

\(\displaystyle -b c \int \frac {e^2 x^4+4 d e x^2+4 d^2 \log (x)}{4 \sqrt {c x-1} \sqrt {c x+1}}dx+d^2 \log (x) (a+b \text {arccosh}(c x))+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{4} b c \int \frac {e^2 x^4+4 d e x^2+4 d^2 \log (x)}{\sqrt {c x-1} \sqrt {c x+1}}dx+d^2 \log (x) (a+b \text {arccosh}(c x))+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle d^2 \log (x) (a+b \text {arccosh}(c x))+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 e^2 \text {arccosh}(c x)}{8 c^5}+\frac {2 d e \text {arccosh}(c x)}{c^3}+\frac {2 i d^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 d^2 \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 e^2 x \sqrt {c x-1} \sqrt {c x+1}}{8 c^4}+\frac {2 d e x \sqrt {c x-1} \sqrt {c x+1}}{c^2}+\frac {e^2 x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\)

Input:

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

Output:

d*e*x^2*(a + b*ArcCosh[c*x]) + (e^2*x^4*(a + b*ArcCosh[c*x]))/4 + d^2*(a + 
 b*ArcCosh[c*x])*Log[x] - (b*c*((2*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^2 
 + (3*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(8*c^4) + (e^2*x^3*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x])/(4*c^2) + (2*d*e*ArcCosh[c*x])/c^3 + (3*e^2*ArcCosh[c*x]) 
/(8*c^5) + ((2*I)*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*Sqrt[-1 + c*x]*S 
qrt[1 + c*x]) - (4*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcS 
in[c*x])])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*d^2*Sqrt[1 - c^2*x^2]*Arc 
Sin[c*x]*Log[x])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((2*I)*d^2*Sqrt[1 - c^ 
2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) 
))/4
 

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 = 0.44 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.65

method	result	size
parts	\(a \left (\frac {e^{2} x^{4}}{4}+d e \,x^{2}+d^{2} \ln \left (x \right )\right )-\frac {b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {3 b \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} x}{32 c^{3}}+b \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{4}}{4}-\frac {b d e \,\operatorname {arccosh}\left (c x \right )}{2 c^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, d e x}{2 c}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}\)	\(223\)
derivativedivides	\(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+b \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, d e x}{2 c}-\frac {b d e \,\operatorname {arccosh}\left (c x \right )}{2 c^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{4}}{4}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {3 b \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} x}{32 c^{3}}-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}\)	\(225\)
default	\(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+b \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, d e x}{2 c}-\frac {b d e \,\operatorname {arccosh}\left (c x \right )}{2 c^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{4}}{4}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {3 b \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} x}{32 c^{3}}-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}\)	\(225\)

Input:

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

Output:

a*(1/4*e^2*x^4+d*e*x^2+d^2*ln(x))-1/16*b*e^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)/c-3/32*b/c^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*e^2*x+b*arccosh(c*x)*d*e*x^2+1 
/4*b*arccosh(c*x)*e^2*x^4-1/2*b/c^2*d*e*arccosh(c*x)-1/2*b/c*(c*x+1)^(1/2) 
*(c*x-1)^(1/2)*d*e*x-1/2*d^2*b*arccosh(c*x)^2+1/2*b*d^2*polylog(2,-(c*x+(c 
*x-1)^(1/2)*(c*x+1)^(1/2))^2)+d^2*b*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*( 
c*x+1)^(1/2))^2)-3/32*b/c^4*e^2*arccosh(c*x)
 

Fricas [F]

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

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

Output:

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d 
^2)*arccosh(c*x))/x, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/4*a*e^2*x^4 + a*d*e*x^2 + a*d^2*log(x) + integrate(b*e^2*x^3*log(c*x + s 
qrt(c*x + 1)*sqrt(c*x - 1)) + 2*b*d*e*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 
 1)) + b*d^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {32 \mathit {acosh} \left (c x \right ) b \,c^{4} d e \,x^{2}+8 \mathit {acosh} \left (c x \right ) b \,c^{4} e^{2} x^{4}-16 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} d e x -2 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} e^{2} x^{3}-3 \sqrt {c^{2} x^{2}-1}\, b c \,e^{2} x +32 \left (\int \frac {\mathit {acosh} \left (c x \right )}{x}d x \right ) b \,c^{4} d^{2}-16 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{2} d e -3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,e^{2}+32 \,\mathrm {log}\left (x \right ) a \,c^{4} d^{2}+32 a \,c^{4} d e \,x^{2}+8 a \,c^{4} e^{2} x^{4}}{32 c^{4}} \] Input:

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

Output:

(32*acosh(c*x)*b*c**4*d*e*x**2 + 8*acosh(c*x)*b*c**4*e**2*x**4 - 16*sqrt(c 
**2*x**2 - 1)*b*c**3*d*e*x - 2*sqrt(c**2*x**2 - 1)*b*c**3*e**2*x**3 - 3*sq 
rt(c**2*x**2 - 1)*b*c*e**2*x + 32*int(acosh(c*x)/x,x)*b*c**4*d**2 - 16*log 
(sqrt(c**2*x**2 - 1) + c*x)*b*c**2*d*e - 3*log(sqrt(c**2*x**2 - 1) + c*x)* 
b*e**2 + 32*log(x)*a*c**4*d**2 + 32*a*c**4*d*e*x**2 + 8*a*c**4*e**2*x**4)/ 
(32*c**4)