\(\int \frac {(d+e x^2)^2 (a+b \text {arccosh}(c x))}{x^3} \, dx\) [376]

Optimal result

Integrand size = 21, antiderivative size = 321 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {b c d^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/2*b*c*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x-1/4*b*e^2*x*(c*x-1)^(1/2)*(c*x+1 
)^(1/2)/c-1/4*b*e^2*arccosh(c*x)/c^2-1/2*d^2*(a+b*arccosh(c*x))/x^2+1/2*e^ 
2*x^2*(a+b*arccosh(c*x))-I*b*d*e*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2/(c*x-1)^ 
(1/2)/(c*x+1)^(1/2)+2*b*d*e*(-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)+2*d*e*(a+b*arccosh(c*x))*l 
n(x)-2*b*d*e*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1 
/2)-I*b*d*e*(-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 = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.54 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a d^2}{x^2}+2 a e^2 x^2+\frac {2 b d^2 \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}+\frac {b e^2 \left (-c x \sqrt {-1+c x} \sqrt {1+c x}+2 c^2 x^2 \text {arccosh}(c x)-2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{c^2}+8 a d e \log (x)+4 b d e \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )\right ) \] Input:

Integrate[((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^3,x]
 

Output:

((-2*a*d^2)/x^2 + 2*a*e^2*x^2 + (2*b*d^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] 
 - ArcCosh[c*x]))/x^2 + (b*e^2*(-(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 2*c^ 
2*x^2*ArcCosh[c*x] - 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2 + 8*a*d*e 
*Log[x] + 4*b*d*e*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c* 
x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/4
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.02, 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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx\)

\(\Big \downarrow \) 6373

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^3,x]
 

Output:

-1/2*(d^2*(a + b*ArcCosh[c*x]))/x^2 + (e^2*x^2*(a + b*ArcCosh[c*x]))/2 + 2 
*d*e*(a + b*ArcCosh[c*x])*Log[x] + (b*c*((d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] 
)/x - (e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) - (e^2*ArcCosh[c*x])/(2 
*c^3) - ((2*I)*d*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x]) + (4*d*e*Sqrt[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]) - (4*d*e*Sqrt[1 - c^2*x^2]*ArcSin 
[c*x]*Log[x])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((2*I)*d*e*Sqrt[1 - c^2*x 
^2]*PolyLog[2, E^((2*I)*ArcSin[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 = 0.52 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.61

method	result	size
parts	\(a \left (\frac {e^{2} x^{2}}{2}+2 d e \ln \left (x \right )-\frac {d^{2}}{2 x^{2}}\right )-b \operatorname {arccosh}\left (c x \right )^{2} d e +\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right ) x^{2}}{2}-\frac {b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{2}}+\frac {b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{2 x}-\frac {d^{2} b \,c^{2}}{2}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 x^{2}}+2 b e d \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+b e d \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\)	\(196\)
derivativedivides	\(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}-\frac {b d e \operatorname {arccosh}\left (c x \right )^{2}}{c^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} x}{4 c^{3}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{4}}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b}{2}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {2 b \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e \,\operatorname {arccosh}\left (c x \right )}{c^{2}}+\frac {b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e}{c^{2}}\right )\)	\(225\)
default	\(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}-\frac {b d e \operatorname {arccosh}\left (c x \right )^{2}}{c^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, e^{2} x}{4 c^{3}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{4}}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b}{2}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {2 b \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e \,\operatorname {arccosh}\left (c x \right )}{c^{2}}+\frac {b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e}{c^{2}}\right )\)	\(225\)

Input:

int((e*x^2+d)^2*(a+b*arccosh(c*x))/x^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((e*x^2+d)^2*(a+b*arccosh(c*x))/x^3,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^3, x)
 

Sympy [F]

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

integrate((e*x**2+d)**2*(a+b*acosh(c*x))/x**3,x)
 

Output:

Integral((a + b*acosh(c*x))*(d + e*x**2)**2/x**3, x)
 

Maxima [F]

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

integrate((e*x^2+d)^2*(a+b*arccosh(c*x))/x^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x^2+d)^2*(a+b*arccosh(c*x))/x^3,x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)^2*(b*arccosh(c*x) + a)/x^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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