Integrand size = 20, antiderivative size = 359 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d^2 x+\frac {8 b^2 d e x}{9 c^2}+\frac {16 b^2 e^2 x}{75 c^4}+\frac {4}{27} b^2 d e x^3+\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {8 b d e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3}-\frac {16 b e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{75 c^5}-\frac {4 b d e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c}-\frac {8 b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}+d^2 x (a+b \text {arccosh}(c x))^2+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))^2+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))^2 \]
2*b^2*d^2*x+8/9*b^2*d*e*x/c^2+16/75*b^2*e^2*x/c^4+4/27*b^2*d*e*x^3+8/225*b ^2*e^2*x^3/c^2+2/125*b^2*e^2*x^5+d^2*x*(a+b*arccosh(c*x))^2+2/3*d*e*x^3*(a +b*arccosh(c*x))^2+1/5*e^2*x^5*(a+b*arccosh(c*x))^2-2*b*d^2*(a+b*arccosh(c *x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-8/9*b*d*e*(a+b*arccosh(c*x))*(c*x-1)^(1 /2)*(c*x+1)^(1/2)/c^3-16/75*b*e^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1) ^(1/2)/c^5-4/9*b*d*e*x^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c- 8/75*b*e^2*x^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-2/25*b*e ^2*x^4*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt {-1+c x} \sqrt {1+c x} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )+2 b^2 c x \left (360 e^2+60 c^2 e \left (25 d+e x^2\right )+c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )\right )-30 b \left (-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )\right ) \text {arccosh}(c x)+225 b^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {arccosh}(c x)^2}{3375 c^5} \]
(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - 30*a*b*Sqrt[-1 + c*x]*S qrt[1 + c*x]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^ 2 + 9*e^2*x^4)) + 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375 *d^2 + 250*d*e*x^2 + 27*e^2*x^4)) - 30*b*(-15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(24*e^2 + 4*c^2*e*(25*d + 3 *e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcCosh[c*x] + 225*b^2* c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x]^2)/(3375*c^5)
Time = 1.47 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6324, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle \int \left (d^2 (a+b \text {arccosh}(c x))^2+2 d e x^2 (a+b \text {arccosh}(c x))^2+e^2 x^4 (a+b \text {arccosh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {16 b e^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{75 c^5}-\frac {8 b d e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c^3}-\frac {8 b e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{75 c^3}+d^2 x (a+b \text {arccosh}(c x))^2-\frac {2 b d^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))^2-\frac {4 b d e x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c}+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))^2-\frac {2 b e^2 x^4 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c}+\frac {16 b^2 e^2 x}{75 c^4}+\frac {8 b^2 d e x}{9 c^2}+\frac {8 b^2 e^2 x^3}{225 c^2}+2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5\) |
2*b^2*d^2*x + (8*b^2*d*e*x)/(9*c^2) + (16*b^2*e^2*x)/(75*c^4) + (4*b^2*d*e *x^3)/27 + (8*b^2*e^2*x^3)/(225*c^2) + (2*b^2*e^2*x^5)/125 - (2*b*d^2*Sqrt [-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (8*b*d*e*Sqrt[-1 + c*x] *Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (16*b*e^2*Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(a + b*ArcCosh[c*x]))/(75*c^5) - (4*b*d*e*x^2*Sqrt[-1 + c*x]*S qrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c) - (8*b*e^2*x^2*Sqrt[-1 + c*x]*Sqr t[1 + c*x]*(a + b*ArcCosh[c*x]))/(75*c^3) - (2*b*e^2*x^4*Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c) + d^2*x*(a + b*ArcCosh[c*x])^2 + (2*d*e*x^3*(a + b*ArcCosh[c*x])^2)/3 + (e^2*x^5*(a + b*ArcCosh[c*x])^2)/5
3.6.26.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Time = 0.75 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {2 c^{2} d e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}+\frac {e^{2} \left (225 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-120 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+18 c^{5} x^{5}-240 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+40 c^{3} x^{3}+240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 e^{2} c^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225}\right )}{c^{4}}}{c}\) | \(402\) |
| default | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {2 c^{2} d e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}+\frac {e^{2} \left (225 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-120 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+18 c^{5} x^{5}-240 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+40 c^{3} x^{3}+240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 e^{2} c^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225}\right )}{c^{4}}}{c}\) | \(402\) |
| parts | \(a^{2} \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b^{2} \left (675 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x^{5} e^{2}+2250 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x^{3} d e +3375 \operatorname {arccosh}\left (c x \right )^{2} c^{5} x \,d^{2}-270 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{4} x^{4} e^{2}-1500 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{4} x^{2} d e -6750 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{4} d^{2}-360 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2} e^{2}-3000 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} d e +54 e^{2} c^{5} x^{5}+500 d \,c^{5} e \,x^{3}+6750 d^{2} c^{5} x -720 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}+120 c^{3} x^{3} e^{2}+3000 c^{3} x d e +720 c x \,e^{2}\right )}{3375 c^{5}}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \,\operatorname {arccosh}\left (c x \right ) d e \,x^{3}}{3}+\operatorname {arccosh}\left (c x \right ) c x \,d^{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 e^{2} c^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225 c^{4}}\right )}{c}\) | \(415\) |
1/c*(a^2/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+b^2/c^4*(c^4*d^2* (arccosh(c*x)^2*x*c-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c*x)+2/27 *c^2*d*e*(9*arccosh(c*x)^2*x^3*c^3-6*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1 /2)*c^2*x^2-12*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c^3*x^3+12*c*x)+ 1/1125*e^2*(225*arccosh(c*x)^2*c^5*x^5-90*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arcc osh(c*x)*c^4*x^4-120*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2+18*c ^5*x^5-240*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+40*c^3*x^3+240*c*x))+2 *a*b/c^4*(arccosh(c*x)*d^2*c^5*x+2/3*arccosh(c*x)*d*c^5*e*x^3+1/5*arccosh( c*x)*e^2*c^5*x^5-1/225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(9*c^4*e^2*x^4+50*c^4*d *e*x^2+225*c^4*d^2+12*c^2*e^2*x^2+100*c^2*d*e+24*e^2)))
Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.06 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \, {\left (25 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{5} d e + 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \, {\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 15 \, {\left (225 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} + 200 \, b^{2} c^{3} d e + 48 \, b^{2} c e^{2}\right )} x + 30 \, {\left (45 \, a b c^{5} e^{2} x^{5} + 150 \, a b c^{5} d e x^{3} + 225 \, a b c^{5} d^{2} x - {\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} + 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \, {\left (25 \, b^{2} c^{4} d e + 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 30 \, {\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} + 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \, {\left (25 \, a b c^{4} d e + 6 \, a b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{3375 \, c^{5}} \]
1/3375*(27*(25*a^2 + 2*b^2)*c^5*e^2*x^5 + 10*(25*(9*a^2 + 2*b^2)*c^5*d*e + 12*b^2*c^3*e^2)*x^3 + 225*(3*b^2*c^5*e^2*x^5 + 10*b^2*c^5*d*e*x^3 + 15*b^ 2*c^5*d^2*x)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 15*(225*(a^2 + 2*b^2)*c^5*d^ 2 + 200*b^2*c^3*d*e + 48*b^2*c*e^2)*x + 30*(45*a*b*c^5*e^2*x^5 + 150*a*b*c ^5*d*e*x^3 + 225*a*b*c^5*d^2*x - (9*b^2*c^4*e^2*x^4 + 225*b^2*c^4*d^2 + 10 0*b^2*c^2*d*e + 24*b^2*e^2 + 2*(25*b^2*c^4*d*e + 6*b^2*c^2*e^2)*x^2)*sqrt( c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 30*(9*a*b*c^4*e^2*x^4 + 225*a *b*c^4*d^2 + 100*a*b*c^2*d*e + 24*a*b*e^2 + 2*(25*a*b*c^4*d*e + 6*a*b*c^2* e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^5
\[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}\, dx \]
Time = 0.21 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.19 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \operatorname {arcosh}\left (c x\right )^{2} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b d e - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} a b e^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c \operatorname {arcosh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d^{2}}{c} \]
1/5*b^2*e^2*x^5*arccosh(c*x)^2 + 1/5*a^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arccosh (c*x)^2 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*arccosh(c*x)^2 + 4/9*(3*x^3*arccosh( c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a*b*d*e - 4/27*(3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)*arccosh(c* x) - (c^2*x^3 + 6*x)/c^2)*b^2*d*e + 2/75*(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)*a*b*e^2 - 2/1125*(15*(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*arccosh(c*x) - (9*c^4*x^5 + 20*c^2 *x^3 + 120*x)/c^4)*b^2*e^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x) /c) + a^2*d^2*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^2/c
Exception generated. \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]