\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [409]

Optimal result

Integrand size = 21, antiderivative size = 1224 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:

-1/16*b*c*(-d)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/e^2/(c^2*d+e)/((-d)^(1/2) 
-e^(1/2)*x)-1/16*b*c*(-d)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/e^2/(c^2*d+e)/ 
((-d)^(1/2)+e^(1/2)*x)-1/16*(-d)^(1/2)*(a+b*arccosh(c*x))/e^(5/2)/((-d)^(1 
/2)-e^(1/2)*x)^2+5/16*(a+b*arccosh(c*x))/e^(5/2)/((-d)^(1/2)-e^(1/2)*x)+1/ 
16*(-d)^(1/2)*(a+b*arccosh(c*x))/e^(5/2)/((-d)^(1/2)+e^(1/2)*x)^2-5/16*(a+ 
b*arccosh(c*x))/e^(5/2)/((-d)^(1/2)+e^(1/2)*x)-1/8*b*c^3*d*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^(5/2)-5/ 
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))/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1 
/2))^(1/2)/e^(5/2)+1/8*b*c^3*d*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^(5/2)+5/8*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))/( 
c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)/e^(5/2)+3/16*(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)^(1/2)/e^(5/2)-3/16*(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) 
^(1/2)/e^(5/2)+3/16*(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)^(1/2)/e^(5/2)-3/16*(a...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.12 (sec) , antiderivative size = 1185, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]
 

Output:

(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[ 
(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*((-5*(ArcCosh[c*x]/((-I)*Sqr 
t[d] + Sqrt[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]))/(16*e^(5/2)) + (5*(-(ArcCosh[c*x]/(I*Sq 
rt[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]))/(16*e^(5/2)) + ((I/16)*Sqrt[d]*((c*Sqr 
t[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcC 
osh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + L 
og[(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))]))/(Sqrt[e]*(c^ 
2*d + e)^(3/2))))/e^2 - ((I/16)*Sqrt[d]*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/ 
((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + 
 Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(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))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/e^2 + (((3*I)/3 
2)*(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*Sq 
rt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/...
 

Rubi [A] (verified)

Time = 5.22 (sec) , antiderivative size = 1224, 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.

\(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b d \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^{5/2}}+\frac {b d \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^{5/2}}-\frac {5 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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}+\frac {5 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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {b \sqrt {-d} \sqrt {c x-1} \sqrt {c x+1} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {-d} \sqrt {c x-1} \sqrt {c x+1} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {5 (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {5 (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 (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 \sqrt {-d} e^{5/2}}+\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}\)

Input:

Int[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]
 

Output:

-1/16*(b*c*Sqrt[-d]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(e^2*(c^2*d + e)*(Sqrt[- 
d] - Sqrt[e]*x)) - (b*c*Sqrt[-d]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*e^2*(c^ 
2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (Sqrt[-d]*(a + b*ArcCosh[c*x]))/(16*e^( 
5/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (5*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqr 
t[-d] - Sqrt[e]*x)) + (Sqrt[-d]*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d 
] + Sqrt[e]*x)^2) - (5*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[ 
e]*x)) - (b*c^3*d*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] + Sqrt[e])^(3/2)*e^(5/2)) - (5*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - S 
qrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*Sq 
rt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (b*c^3*d*Ar 
cTanh[(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] + Sqrt[e] 
)^(3/2)*e^(5/2)) + (5*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x 
])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*Sqrt[c*Sqrt[-d] - Sqrt 
[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (3*(a + b*ArcCosh[c*x])*Log[1 - 
 (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*Sqrt[-d] 
*e^(5/2)) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sq 
rt[-d] - Sqrt[-(c^2*d) - e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcCosh[c 
*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])...
 

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 = 19.97 (sec) , antiderivative size = 1752, normalized size of antiderivative = 1.43

method	result	size
parts	\(\text {Expression too large to display}\)	\(1752\)
derivativedivides	\(\text {Expression too large to display}\)	\(1754\)
default	\(\text {Expression too large to display}\)	\(1754\)

Input:

int(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**4*(a+b*acosh(c*x))/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^4*(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]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((x^4*(a + b*acosh(c*x)))/(d + e*x^2)^3,x)
 

Output:

int((x^4*(a + b*acosh(c*x)))/(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3}+16 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e^{4} x^{2}+8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{5} x^{4}-3 a \,d^{2} e x -5 a d \,e^{2} x^{3}}{8 d \,e^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x^4*(a+b*acosh(c*x))/(e*x^2+d)^3,x)
 

Output:

(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 6*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/ 
(sqrt(e)*sqrt(d)))*a*e**2*x**4 + 8*int((acosh(c*x)*x**4)/(d**3 + 3*d**2*e* 
x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3 + 16*int((acosh(c*x)*x**4 
)/(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**2 + 
 8*int((acosh(c*x)*x**4)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6 
),x)*b*d*e**5*x**4 - 3*a*d**2*e*x - 5*a*d*e**2*x**3)/(8*d*e**3*(d**2 + 2*d 
*e*x**2 + e**2*x**4))