\(\int x^4 (d+e x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [470]

Optimal result

Integrand size = 21, antiderivative size = 319 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x)) \]

[Out]

1/5*d^2*x^5*(a+b*arccosh(c*x))+2/7*d*e*x^7*(a+b*arccosh(c*x))+1/9*e^2*x^9*(a+b*arccosh(c*x))+1/315*b*(63*c^4*d
^2+90*c^2*d*e+35*e^2)*(-c^2*x^2+1)/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/945*b*(63*c^4*d^2+135*c^2*d*e+70*e^2)*(-c
^2*x^2+1)^2/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/525*b*(21*c^4*d^2+90*c^2*d*e+70*e^2)*(-c^2*x^2+1)^3/c^9/(c*x-1)^
(1/2)/(c*x+1)^(1/2)-2/441*b*e*(9*c^2*d+14*e)*(-c^2*x^2+1)^4/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/81*b*e^2*(-c^2*x
^2+1)^5/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5958, 12, 534, 1265, 911, 1167} \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {2 b e \left (1-c^2 x^2\right )^4 \left (9 c^2 d+14 e\right )}{441 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right )^3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{525 c^9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b \left (1-c^2 x^2\right )^2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{945 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{315 c^9 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(b*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*(63*c^4*d^2
+ 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(21*c^4*d^2 + 90*c^2*d*e
+ 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*e*(9*c^2*d + 14*e)*(1 - c^2*x^2)^4)/
(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*
x^5*(a + b*ArcCosh[c*x]))/5 + (2*d*e*x^7*(a + b*ArcCosh[c*x]))/7 + (e^2*x^9*(a + b*ArcCosh[c*x]))/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*
a2 + b1*b2*x^n)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 5958

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]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[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] && LeQ[m + p, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-(b c) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{315 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {1}{315} (b c) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2 \left (\frac {63 c^4 d^2+90 c^2 d e+35 e^2}{c^4}-\frac {\left (-90 c^2 d e-70 e^2\right ) x^2}{c^4}+\frac {35 e^2 x^4}{c^4}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{315 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {63 c^4 d^2+90 c^2 d e+35 e^2}{c^8}+\frac {2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) x^2}{c^8}+\frac {3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) x^4}{c^8}+\frac {10 e \left (9 c^2 d+14 e\right ) x^6}{c^8}+\frac {35 e^2 x^8}{c^8}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{315 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.60 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^2+160 c^2 e \left (81 d+14 e x^2\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )\right )}{c^9}+315 b x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right ) \text {arccosh}(c x)}{99225} \]

[In]

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

[Out]

(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^2 + 160*c^2*e*(81*d +
14*e*x^2) + 24*c^4*(441*d^2 + 270*d*e*x^2 + 70*e^2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) +
c^8*(3969*d^2*x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcCosh[
c*x])/99225

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.67

method	result	size
parts	\(a \left (\frac {1}{9} e^{2} x^{9}+\frac {2}{7} d e \,x^{7}+\frac {1}{5} d^{2} x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e^{2} x^{9}}{9}+\frac {2 c^{5} \operatorname {arccosh}\left (c x \right ) d e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d^{2}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{2} x^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 c^{6} e^{2} x^{6}+4860 c^{6} d e \,x^{4}+5292 c^{6} d^{2} x^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 e^{2} c^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225 c^{4}}\right )}{c^{5}}\)	\(214\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{5} c^{9} d^{2} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} x^{5}}{5}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{2} x^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 c^{6} e^{2} x^{6}+4860 c^{6} d e \,x^{4}+5292 c^{6} d^{2} x^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 e^{2} c^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225}\right )}{c^{4}}}{c^{5}}\)	\(227\)
default	\(\frac {\frac {a \left (\frac {1}{5} c^{9} d^{2} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} x^{5}}{5}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{2} x^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 c^{6} e^{2} x^{6}+4860 c^{6} d e \,x^{4}+5292 c^{6} d^{2} x^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 e^{2} c^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225}\right )}{c^{4}}}{c^{5}}\)	\(227\)

[In]

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

[Out]

a*(1/9*e^2*x^9+2/7*d*e*x^7+1/5*d^2*x^5)+b/c^5*(1/9*c^5*arccosh(c*x)*e^2*x^9+2/7*c^5*arccosh(c*x)*d*e*x^7+1/5*a
rccosh(c*x)*c^5*x^5*d^2-1/99225/c^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*e^2*x^8+4050*c^8*d*e*x^6+3969*c^8*d^
2*x^4+1400*c^6*e^2*x^6+4860*c^6*d*e*x^4+5292*c^6*d^2*x^2+1680*c^4*e^2*x^4+6480*c^4*d*e*x^2+10584*c^4*d^2+2240*
c^2*e^2*x^2+12960*c^2*d*e+4480*e^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.72 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{2} x^{9} + 28350 \, a c^{9} d e x^{7} + 19845 \, a c^{9} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} e^{2} x^{9} + 90 \, b c^{9} d e x^{7} + 63 \, b c^{9} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{2} x^{8} + 10584 \, b c^{4} d^{2} + 50 \, {\left (81 \, b c^{8} d e + 28 \, b c^{6} e^{2}\right )} x^{6} + 12960 \, b c^{2} d e + 3 \, {\left (1323 \, b c^{8} d^{2} + 1620 \, b c^{6} d e + 560 \, b c^{4} e^{2}\right )} x^{4} + 4480 \, b e^{2} + 4 \, {\left (1323 \, b c^{6} d^{2} + 1620 \, b c^{4} d e + 560 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \]

[In]

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

[Out]

1/99225*(11025*a*c^9*e^2*x^9 + 28350*a*c^9*d*e*x^7 + 19845*a*c^9*d^2*x^5 + 315*(35*b*c^9*e^2*x^9 + 90*b*c^9*d*
e*x^7 + 63*b*c^9*d^2*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*c^8*e^2*x^8 + 10584*b*c^4*d^2 + 50*(81*b*c^8*
d*e + 28*b*c^6*e^2)*x^6 + 12960*b*c^2*d*e + 3*(1323*b*c^8*d^2 + 1620*b*c^6*d*e + 560*b*c^4*e^2)*x^4 + 4480*b*e
^2 + 4*(1323*b*c^6*d^2 + 1620*b*c^4*d*e + 560*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^9

Sympy [F]

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

[In]

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

[Out]

Integral(x**4*(a + b*acosh(c*x))*(d + e*x**2)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.96 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{2} \]

[In]

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

[Out]

1/9*a*e^2*x^9 + 2/7*a*d*e*x^7 + 1/5*a*d^2*x^5 + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*s
qrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*d^2 + 2/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 -
1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*d*e +
1/2835*(315*x^9*arccosh(c*x) - (35*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqrt(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x^2
- 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c^10)*c)*b*e^2

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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