\(\int \frac {x^2 (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [410]

Optimal result

Integrand size = 21, antiderivative size = 1234 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:

-1/16*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-d)^(1/2)/e/(c^2*d+e)/((-d)^(1/2)-e 
^(1/2)*x)-1/16*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-d)^(1/2)/e/(c^2*d+e)/((-d 
)^(1/2)+e^(1/2)*x)-1/16*(a+b*arccosh(c*x))/(-d)^(1/2)/e^(3/2)/((-d)^(1/2)- 
e^(1/2)*x)^2-1/16*(a+b*arccosh(c*x))/d/e^(3/2)/((-d)^(1/2)-e^(1/2)*x)+1/16 
*(a+b*arccosh(c*x))/(-d)^(1/2)/e^(3/2)/((-d)^(1/2)+e^(1/2)*x)^2+1/16*(a+b* 
arccosh(c*x))/d/e^(3/2)/((-d)^(1/2)+e^(1/2)*x)+1/8*b*c^3*arctanh((c*(-d)^( 
1/2)-e^(1/2))^(1/2)*(c*x+1)^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)/(c*x-1)^(1/ 
2))/(c*(-d)^(1/2)-e^(1/2))^(3/2)/(c*(-d)^(1/2)+e^(1/2))^(3/2)/e^(3/2)+1/8* 
b*c*arctanh((c*(-d)^(1/2)-e^(1/2))^(1/2)*(c*x+1)^(1/2)/(c*(-d)^(1/2)+e^(1/ 
2))^(1/2)/(c*x-1)^(1/2))/d/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1 
/2))^(1/2)/e^(3/2)-1/8*b*c^3*arctanh((c*(-d)^(1/2)+e^(1/2))^(1/2)*(c*x+1)^ 
(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*x-1)^(1/2))/(c*(-d)^(1/2)-e^(1/2))^( 
3/2)/(c*(-d)^(1/2)+e^(1/2))^(3/2)/e^(3/2)-1/8*b*c*arctanh((c*(-d)^(1/2)+e^ 
(1/2))^(1/2)*(c*x+1)^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*x-1)^(1/2))/d/( 
c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)/e^(3/2)-1/16*(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)^(3/2)/e^(3/2)+1/16*(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) 
^(3/2)/e^(3/2)-1/16*(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)^(3/2)/e^(3/2)+1/16*(a...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.87 (sec) , antiderivative size = 1143, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((-8*a*Sqrt[e]*x)/(d + e*x^2)^2 + (4*a*Sqrt[e]*x)/(d^2 + d*e*x^2) + (4*a*A 
rcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*b*(ArcCosh[c*x]/((-I)*Sqrt[d] + S 
qrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]* 
Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]* 
x))])/Sqrt[-(c^2*d) - e]))/d + (2*b*(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x) 
+ (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[ 
-(c^2*d) - e]))/d - (2*I)*b*((c*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(Sqr 
t[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + ArcCosh[c*x]/(Sqrt[d]*(Sqrt 
[d] + I*Sqrt[e]*x)^2) + (c^3*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e 
] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d 
 + I*Sqrt[d]*Sqrt[e]*x))]))/(c^2*d + e)^(3/2)) + (2*I)*b*((c*Sqrt[e]*Sqrt[ 
-1 + c*x]*Sqrt[1 + c*x])/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) + A 
rcCosh[c*x]/(Sqrt[d]*(Sqrt[d] - I*Sqrt[e]*x)^2) - (c^3*(Log[4] + Log[(e*Sq 
rt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c* 
x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(c^2*d + e)^(3/2)) + 
 (I*b*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/( 
I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c 
*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/ 
((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcC...
 

Rubi [A] (verified)

Time = 4.14 (sec) , antiderivative size = 1234, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) c^3}{8 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} e^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) c^3}{8 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} e^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) c}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) c}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{3/2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \text {arccosh}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 d e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \text {arccosh}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \text {arccosh}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}\)

Input:

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

Output:

-1/16*(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] 
 - Sqrt[e]*x)) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*Sqrt[-d]*e*(c^2*d 
+ e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(16*Sqrt[-d]*e^(3/2)*( 
Sqrt[-d] - Sqrt[e]*x)^2) - (a + b*ArcCosh[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - 
Sqrt[e]*x)) + (a + b*ArcCosh[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] + Sqrt[e 
]*x)^2) + (a + b*ArcCosh[c*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b* 
c^3*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + 
Sqrt[e]]*Sqrt[-1 + c*x])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + S 
qrt[e])^(3/2)*e^(3/2)) + (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + 
 c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*Sqrt[c*Sqrt[-d] 
- Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) - (b*c^3*ArcTanh[(Sqrt[c*Sq 
rt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x 
])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*e^(3/2)) 
 - (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d 
] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt 
[-d] + Sqrt[e]]*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCos 
h[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a 
+ b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^ 
2*d) - e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt 
[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(3/2)*...
 

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 = 11.79 (sec) , antiderivative size = 1222, normalized size of antiderivative = 0.99

method	result	size
parts	\(\text {Expression too large to display}\)	\(1222\)
derivativedivides	\(\text {Expression too large to display}\)	\(1237\)
default	\(\text {Expression too large to display}\)	\(1237\)

Input:

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

Output:

a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/e/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1 
/2)))+b/c^3*(1/8*c^4*(arccosh(c*x)*d*c^5*e*x^3-arccosh(c*x)*d^2*c^5*x+c^4* 
d*e*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+c^4*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+ar 
ccosh(c*x)*e^2*c^3*x^3-arccosh(c*x)*d*c^3*e*x)/e/(c^2*e*x^2+c^2*d)^2/d/(c^ 
2*d+e)+1/16/d/(c^2*d+e)*c^4*sum(_R1/(_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))+1/16/e/(c^2*d+e 
)*c^6*sum(_R1/(_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))+1/8*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/ 
2)+e)*e)^(1/2)*(-2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(c^2* 
d*(c^2*d+e))^(1/2)*e)*c^4*arctan(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((2*c 
^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/d/e^3-1/8*((2*c^2* 
d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2) 
+e)*arctan(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e 
))^(1/2)+e)*e)^(1/2))*c^4/d/(c^2*d+e)/e^3+1/8*(-(2*c^2*d-2*(c^2*d*(c^2*d+e 
))^(1/2)+e)*e)^(1/2)*(2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e+ 
(c^2*d*(c^2*d+e))^(1/2)*e)*c^4*arctanh(e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) 
/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/d/e^3-1/8*( 
-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d+2*(c^2*d*(c^2*...
 

Fricas [F]

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

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

Output:

integral((b*x^2*arccosh(c*x) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 
 + d^3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+2 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{4} e^{2}+16 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3} x^{2}+8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e^{4} x^{4}-a \,d^{2} e x +a d \,e^{2} x^{3}}{8 d^{2} e^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 2*sqrt(e)*sqrt(d)* 
atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + sqrt(e)*sqrt(d)*atan((e*x)/(sqr 
t(e)*sqrt(d)))*a*e**2*x**4 + 8*int((acosh(c*x)*x**2)/(d**3 + 3*d**2*e*x**2 
 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**4*e**2 + 16*int((acosh(c*x)*x**2)/(d 
**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3*x**2 + 8*i 
nt((acosh(c*x)*x**2)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x) 
*b*d**2*e**4*x**4 - a*d**2*e*x + a*d*e**2*x**3)/(8*d**2*e**2*(d**2 + 2*d*e 
*x**2 + e**2*x**4))