Integrand size = 21, antiderivative size = 261 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{315 c^9}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{945 c^9}-\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) (-1+c x)^{5/2} (1+c x)^{5/2}}{525 c^9}-\frac {2 b e \left (9 c^2 d+14 e\right ) (-1+c x)^{7/2} (1+c x)^{7/2}}{441 c^9}-\frac {b e^2 (-1+c x)^{9/2} (1+c x)^{9/2}}{81 c^9}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x)) \] Output:
-1/315*b*(63*c^4*d^2+90*c^2*d*e+35*e^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^9-2/ 945*b*(63*c^4*d^2+135*c^2*d*e+70*e^2)*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^9-1/52 5*b*(21*c^4*d^2+90*c^2*d*e+70*e^2)*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^9-2/441*b *e*(9*c^2*d+14*e)*(c*x-1)^(7/2)*(c*x+1)^(7/2)/c^9-1/81*b*e^2*(c*x-1)^(9/2) *(c*x+1)^(9/2)/c^9+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))
Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.74 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^2+160 c^2 e \left (81 d+14 e x^2\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )\right )}{c^9}+315 b x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right ) \text {arccosh}(c x)}{99225} \] Input:
Integrate[x^4*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]
Output:
(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
Time = 0.64 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6373, 27, 1905, 1578, 1195, 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.
| \(\displaystyle \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {35 e^2 \left (c^2 x^2-1\right )^{7/2}}{c^8}+\frac {10 e \left (9 d c^2+14 e\right ) \left (c^2 x^2-1\right )^{5/2}}{c^8}+\frac {3 \left (21 d^2 c^4+90 d e c^2+70 e^2\right ) \left (c^2 x^2-1\right )^{3/2}}{c^8}+\frac {2 \left (63 d^2 c^4+135 d e c^2+70 e^2\right ) \sqrt {c^2 x^2-1}}{c^8}+\frac {63 d^2 c^4+90 d e c^2+35 e^2}{c^8 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {20 e \left (c^2 x^2-1\right )^{7/2} \left (9 c^2 d+14 e\right )}{7 c^{10}}+\frac {70 e^2 \left (c^2 x^2-1\right )^{9/2}}{9 c^{10}}+\frac {6 \left (c^2 x^2-1\right )^{5/2} \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{5 c^{10}}+\frac {4 \left (c^2 x^2-1\right )^{3/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{3 c^{10}}+\frac {2 \sqrt {c^2 x^2-1} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{c^{10}}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[x^4*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]
Output:
-1/630*(b*c*Sqrt[-1 + c^2*x^2]*((2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*Sqrt [-1 + c^2*x^2])/c^10 + (4*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(-1 + c^2*x^ 2)^(3/2))/(3*c^10) + (6*(21*c^4*d^2 + 90*c^2*d*e + 70*e^2)*(-1 + c^2*x^2)^ (5/2))/(5*c^10) + (20*e*(9*c^2*d + 14*e)*(-1 + c^2*x^2)^(7/2))/(7*c^10) + (70*e^2*(-1 + c^2*x^2)^(9/2))/(9*c^10)))/(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
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[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] && Int egerQ[(m - 1)/2]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
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]))
Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.82
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{2} x^{9}+\frac {2}{7} d e \,x^{7}+\frac {1}{5} d^{2} x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e^{2} x^{9}}{9}+\frac {2 c^{5} \operatorname {arccosh}\left (c x \right ) d e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d^{2}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 e^{2} x^{8} c^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 x^{6} c^{6} e^{2}+4860 x^{4} c^{6} d e +5292 x^{2} c^{6} d^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 c^{2} e^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225 c^{4}}\right )}{c^{5}}\) | \(214\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} d^{2} c^{9} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{9} x^{5}}{5}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 e^{2} x^{8} c^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 x^{6} c^{6} e^{2}+4860 x^{4} c^{6} d e +5292 x^{2} c^{6} d^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 c^{2} e^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225}\right )}{c^{4}}}{c^{5}}\) | \(227\) |
| default | \(\frac {\frac {a \left (\frac {1}{5} d^{2} c^{9} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{9} x^{5}}{5}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 e^{2} x^{8} c^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 x^{6} c^{6} e^{2}+4860 x^{4} c^{6} d e +5292 x^{2} c^{6} d^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 c^{2} e^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225}\right )}{c^{4}}}{c^{5}}\) | \(227\) |
| orering | \(\frac {\left (20825 c^{10} e^{3} x^{12}+76675 c^{10} d \,e^{2} x^{10}+96147 c^{10} d^{2} e \,x^{8}+1400 c^{8} e^{3} x^{10}+35721 c^{10} d^{3} x^{6}+7180 c^{8} d \,e^{2} x^{8}+13824 c^{8} d^{2} e \,x^{6}+2240 x^{8} e^{3} c^{6}+5292 c^{8} d^{3} x^{4}+14080 x^{6} e^{2} c^{6} d +48816 x^{4} e \,c^{6} d^{2}+4480 x^{6} e^{3} c^{4}+21168 c^{6} d^{3} x^{2}+54080 x^{4} e^{2} c^{4} d -58752 x^{2} e \,c^{4} d^{2}+17920 x^{4} e^{3} c^{2}-42336 c^{4} d^{3}-94720 x^{2} e^{2} c^{2} d -51840 c^{2} d^{2} e -35840 x^{2} e^{3}-17920 d \,e^{2}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{99225 x \left (e \,x^{2}+d \right ) c^{10}}-\frac {\left (1225 e^{2} x^{8} c^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 x^{6} c^{6} e^{2}+4860 x^{4} c^{6} d e +5292 x^{2} c^{6} d^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 c^{2} e^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (4 x^{3} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+4 x^{5} \left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e +\frac {x^{4} \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{99225 c^{10} \left (e \,x^{2}+d \right )^{2} x^{4}}\) | \(472\) |
Input:
int(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
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*arccosh(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+64 80*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))
Time = 0.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.89 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{2} x^{9} + 28350 \, a c^{9} d e x^{7} + 19845 \, a c^{9} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} e^{2} x^{9} + 90 \, b c^{9} d e x^{7} + 63 \, b c^{9} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{2} x^{8} + 10584 \, b c^{4} d^{2} + 50 \, {\left (81 \, b c^{8} d e + 28 \, b c^{6} e^{2}\right )} x^{6} + 12960 \, b c^{2} d e + 3 \, {\left (1323 \, b c^{8} d^{2} + 1620 \, b c^{6} d e + 560 \, b c^{4} e^{2}\right )} x^{4} + 4480 \, b e^{2} + 4 \, {\left (1323 \, b c^{6} d^{2} + 1620 \, b c^{4} d e + 560 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \] Input:
integrate(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
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 + sq rt(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
\[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \] Input:
integrate(x**4*(e*x**2+d)**2*(a+b*acosh(c*x)),x)
Output:
Integral(x**4*(a + b*acosh(c*x))*(d + e*x**2)**2, x)
Time = 0.04 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.17 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{2} \] Input:
integrate(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
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*sqrt(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
Exception generated. \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*(e*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \] Input:
int(x^4*(a + b*acosh(c*x))*(d + e*x^2)^2,x)
Output:
int(x^4*(a + b*acosh(c*x))*(d + e*x^2)^2, x)
Time = 0.19 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.29 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {19845 \mathit {acosh} \left (c x \right ) b \,c^{9} d^{2} x^{5}+28350 \mathit {acosh} \left (c x \right ) b \,c^{9} d e \,x^{7}+11025 \mathit {acosh} \left (c x \right ) b \,c^{9} e^{2} x^{9}-3969 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{2} x^{4}-4050 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d e \,x^{6}-1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} e^{2} x^{8}-5292 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{2} x^{2}-4860 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d e \,x^{4}-1400 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e^{2} x^{6}-10584 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d^{2}-6480 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d e \,x^{2}-1680 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{2} x^{4}-12960 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d e -2240 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{2} x^{2}-4480 \sqrt {c^{2} x^{2}-1}\, b \,e^{2}+19845 a \,c^{9} d^{2} x^{5}+28350 a \,c^{9} d e \,x^{7}+11025 a \,c^{9} e^{2} x^{9}}{99225 c^{9}} \] Input:
int(x^4*(e*x^2+d)^2*(a+b*acosh(c*x)),x)
Output:
(19845*acosh(c*x)*b*c**9*d**2*x**5 + 28350*acosh(c*x)*b*c**9*d*e*x**7 + 11 025*acosh(c*x)*b*c**9*e**2*x**9 - 3969*sqrt(c**2*x**2 - 1)*b*c**8*d**2*x** 4 - 4050*sqrt(c**2*x**2 - 1)*b*c**8*d*e*x**6 - 1225*sqrt(c**2*x**2 - 1)*b* c**8*e**2*x**8 - 5292*sqrt(c**2*x**2 - 1)*b*c**6*d**2*x**2 - 4860*sqrt(c** 2*x**2 - 1)*b*c**6*d*e*x**4 - 1400*sqrt(c**2*x**2 - 1)*b*c**6*e**2*x**6 - 10584*sqrt(c**2*x**2 - 1)*b*c**4*d**2 - 6480*sqrt(c**2*x**2 - 1)*b*c**4*d* e*x**2 - 1680*sqrt(c**2*x**2 - 1)*b*c**4*e**2*x**4 - 12960*sqrt(c**2*x**2 - 1)*b*c**2*d*e - 2240*sqrt(c**2*x**2 - 1)*b*c**2*e**2*x**2 - 4480*sqrt(c* *2*x**2 - 1)*b*e**2 + 19845*a*c**9*d**2*x**5 + 28350*a*c**9*d*e*x**7 + 110 25*a*c**9*e**2*x**9)/(99225*c**9)