Integrand size = 20, antiderivative size = 609 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d^2 e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {32 b e^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{49 c}+d^3 x (a+b \text {arccosh}(c x))^2+d^2 e x^3 (a+b \text {arccosh}(c x))^2+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2 \]
2*b^2*d^3*x+4/3*b^2*d^2*e*x/c^2+16/25*b^2*d*e^2*x/c^4+32/245*b^2*e^3*x/c^6 +2/9*b^2*d^2*e*x^3+8/75*b^2*d*e^2*x^3/c^2+16/735*b^2*e^3*x^3/c^4+6/125*b^2 *d*e^2*x^5+12/1225*b^2*e^3*x^5/c^2+2/343*b^2*e^3*x^7+d^3*x*(a+b*arccosh(c* x))^2+d^2*e*x^3*(a+b*arccosh(c*x))^2+3/5*d*e^2*x^5*(a+b*arccosh(c*x))^2+1/ 7*e^3*x^7*(a+b*arccosh(c*x))^2-2*b*d^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c *x+1)^(1/2)/c-4/3*b*d^2*e*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c ^3-16/25*b*d*e^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-32/245 *b*e^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^7-2/3*b*d^2*e*x^2* (a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-8/25*b*d*e^2*x^2*(a+b*arc cosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-16/245*b*e^3*x^2*(a+b*arccosh(c *x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-6/25*b*d*e^2*x^4*(a+b*arccosh(c*x))*( c*x-1)^(1/2)*(c*x+1)^(1/2)/c-12/245*b*e^3*x^4*(a+b*arccosh(c*x))*(c*x-1)^( 1/2)*(c*x+1)^(1/2)/c^3-2/49*b*e^3*x^6*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c* x+1)^(1/2)/c
Time = 0.37 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.74 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {11025 a^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-210 a b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )+2 b^2 c x \left (25200 e^3+840 c^2 e^2 \left (147 d+5 e x^2\right )+210 c^4 e \left (1225 d^2+98 d e x^2+9 e^2 x^4\right )+c^6 \left (385875 d^3+42875 d^2 e x^2+9261 d e^2 x^4+1125 e^3 x^6\right )\right )-210 b \left (-105 a c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )\right ) \text {arccosh}(c x)+11025 b^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \text {arccosh}(c x)^2}{385875 c^7} \]
(11025*a^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - 210* a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e *x^2 + 441*d*e^2*x^4 + 75*e^3*x^6)) + 2*b^2*c*x*(25200*e^3 + 840*c^2*e^2*( 147*d + 5*e*x^2) + 210*c^4*e*(1225*d^2 + 98*d*e*x^2 + 9*e^2*x^4) + c^6*(38 5875*d^3 + 42875*d^2*e*x^2 + 9261*d*e^2*x^4 + 1125*e^3*x^6)) - 210*b*(-105 *a*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225* d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e ^2*x^4 + 75*e^3*x^6)))*ArcCosh[c*x] + 11025*b^2*c^7*x*(35*d^3 + 35*d^2*e*x ^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcCosh[c*x]^2)/(385875*c^7)
Time = 2.39 (sec) , antiderivative size = 609, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle \int \left (d^3 (a+b \text {arccosh}(c x))^2+3 d^2 e x^2 (a+b \text {arccosh}(c x))^2+3 d e^2 x^4 (a+b \text {arccosh}(c x))^2+e^3 x^6 (a+b \text {arccosh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {32 b e^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{245 c^7}-\frac {16 b d e^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c^5}-\frac {16 b e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{245 c^5}-\frac {4 b d^2 e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {8 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c^3}-\frac {12 b e^3 x^4 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{245 c^3}+d^3 x (a+b \text {arccosh}(c x))^2-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}+d^2 e x^3 (a+b \text {arccosh}(c x))^2-\frac {2 b d^2 e x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c}+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))^2-\frac {6 b d e^2 x^4 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c}+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))^2-\frac {2 b e^3 x^6 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{49 c}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {12 b^2 e^3 x^5}{1225 c^2}+2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7\) |
2*b^2*d^3*x + (4*b^2*d^2*e*x)/(3*c^2) + (16*b^2*d*e^2*x)/(25*c^4) + (32*b^ 2*e^3*x)/(245*c^6) + (2*b^2*d^2*e*x^3)/9 + (8*b^2*d*e^2*x^3)/(75*c^2) + (1 6*b^2*e^3*x^3)/(735*c^4) + (6*b^2*d*e^2*x^5)/125 + (12*b^2*e^3*x^5)/(1225* c^2) + (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b* ArcCosh[c*x]))/c - (4*b*d^2*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[ c*x]))/(3*c^3) - (16*b*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c *x]))/(25*c^5) - (32*b*e^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x ]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh [c*x]))/(3*c) - (8*b*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh [c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCo sh[c*x]))/(245*c^5) - (6*b*d*e^2*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*A rcCosh[c*x]))/(25*c) - (12*b*e^3*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*A rcCosh[c*x]))/(245*c^3) - (2*b*e^3*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b *ArcCosh[c*x]))/(49*c) + d^3*x*(a + b*ArcCosh[c*x])^2 + d^2*e*x^3*(a + b*A rcCosh[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcCosh[c*x])^2)/5 + (e^3*x^7*(a + b* ArcCosh[c*x])^2)/7
3.6.25.3.1 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 = 0.81 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \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 {c^{4} d^{2} 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 )}{9}+\frac {c^{2} d \,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 )}{375}+\frac {e^{3} \left (3675 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{7}-1050 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}-1260 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+150 c^{7} x^{7}-1680 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+252 c^{5} x^{5}-3360 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+560 c^{3} x^{3}+3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(632\) |
| default | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \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 {c^{4} d^{2} 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 )}{9}+\frac {c^{2} d \,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 )}{375}+\frac {e^{3} \left (3675 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{7}-1050 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}-1260 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+150 c^{7} x^{7}-1680 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+252 c^{5} x^{5}-3360 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+560 c^{3} x^{3}+3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(632\) |
| parts | \(a^{2} \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b^{2} \left (-15750 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{6} e^{3}-92610 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{4} d \,e^{2}-257250 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} x^{2} d^{2} e -771750 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{6} d^{3}+55125 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{7} e^{3}+231525 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{5} d \,e^{2}+385875 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x^{3} d^{2} e +385875 \operatorname {arccosh}\left (c x \right )^{2} c^{7} x \,d^{3}-18900 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4} e^{3}-123480 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{2} d \,e^{2}-514500 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{4} d^{2} e +2250 e^{3} c^{7} x^{7}+18522 d \,c^{7} e^{2} x^{5}+85750 d^{2} c^{7} e \,x^{3}+771750 d^{3} c^{7} x -25200 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{2} x^{2} e^{3}-246960 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) c^{2} d \,e^{2}+3780 c^{5} x^{5} e^{3}+41160 c^{5} x^{3} d \,e^{2}+514500 c^{5} x \,d^{2} e -50400 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right ) e^{3}+8400 c^{3} x^{3} e^{3}+246960 c^{3} x d \,e^{2}+50400 c x \,e^{3}\right )}{385875 c^{7}}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{5}}{5}+c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{3}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675 c^{6}}\right )}{c}\) | \(677\) |
1/c*(a^2/c^6*(d^3*c^7*x+d^2*c^7*e*x^3+3/5*d*c^7*e^2*x^5+1/7*e^3*c^7*x^7)+b ^2/c^6*(c^6*d^3*(arccosh(c*x)^2*x*c-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^( 1/2)+2*c*x)+1/9*c^4*d^2*e*(9*arccosh(c*x)^2*x^3*c^3-6*(c*x+1)^(1/2)*arccos h(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)+1/375*c^2*d*e^2*(225*arccosh(c*x)^2*c^5*x^5-90*(c*x-1)^(1 /2)*(c*x+1)^(1/2)*arccosh(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)+1/25725*e^3*(3675*arccosh(c*x)^2*c^7*x^7-1050*arccosh( c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^6*x^6-1260*(c*x-1)^(1/2)*(c*x+1)^(1/2)* arccosh(c*x)*c^4*x^4+150*c^7*x^7-1680*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^( 1/2)*c^2*x^2+252*c^5*x^5-3360*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+560 *c^3*x^3+3360*c*x))+2*a*b/c^6*(arccosh(c*x)*d^3*c^7*x+arccosh(c*x)*d^2*c^7 *e*x^3+3/5*arccosh(c*x)*d*c^7*e^2*x^5+1/7*arccosh(c*x)*e^3*c^7*x^7-1/3675* (c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*e^3*x^6+441*c^6*d*e^2*x^4+1225*c^6*d^2 *e*x^2+90*c^4*e^3*x^4+3675*c^6*d^3+588*c^4*d*e^2*x^2+2450*c^4*d^2*e+120*c^ 2*e^3*x^2+1176*c^2*d*e^2+240*e^3)))
Time = 0.27 (sec) , antiderivative size = 586, normalized size of antiderivative = 0.96 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \, {\left (49 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d e^{2} + 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \, {\left (1225 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} e + 1176 \, b^{2} c^{5} d e^{2} + 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \, {\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 105 \, {\left (3675 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} + 4900 \, b^{2} c^{5} d^{2} e + 2352 \, b^{2} c^{3} d e^{2} + 480 \, b^{2} c e^{3}\right )} x + 210 \, {\left (525 \, a b c^{7} e^{3} x^{7} + 2205 \, a b c^{7} d e^{2} x^{5} + 3675 \, a b c^{7} d^{2} e x^{3} + 3675 \, a b c^{7} d^{3} x - {\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} + 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} + 240 \, b^{2} e^{3} + 9 \, {\left (49 \, b^{2} c^{6} d e^{2} + 10 \, b^{2} c^{4} e^{3}\right )} x^{4} + {\left (1225 \, b^{2} c^{6} d^{2} e + 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 210 \, {\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} + 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} + 240 \, a b e^{3} + 9 \, {\left (49 \, a b c^{6} d e^{2} + 10 \, a b c^{4} e^{3}\right )} x^{4} + {\left (1225 \, a b c^{6} d^{2} e + 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{385875 \, c^{7}} \]
1/385875*(1125*(49*a^2 + 2*b^2)*c^7*e^3*x^7 + 189*(49*(25*a^2 + 2*b^2)*c^7 *d*e^2 + 20*b^2*c^5*e^3)*x^5 + 35*(1225*(9*a^2 + 2*b^2)*c^7*d^2*e + 1176*b ^2*c^5*d*e^2 + 240*b^2*c^3*e^3)*x^3 + 11025*(5*b^2*c^7*e^3*x^7 + 21*b^2*c^ 7*d*e^2*x^5 + 35*b^2*c^7*d^2*e*x^3 + 35*b^2*c^7*d^3*x)*log(c*x + sqrt(c^2* x^2 - 1))^2 + 105*(3675*(a^2 + 2*b^2)*c^7*d^3 + 4900*b^2*c^5*d^2*e + 2352* b^2*c^3*d*e^2 + 480*b^2*c*e^3)*x + 210*(525*a*b*c^7*e^3*x^7 + 2205*a*b*c^7 *d*e^2*x^5 + 3675*a*b*c^7*d^2*e*x^3 + 3675*a*b*c^7*d^3*x - (75*b^2*c^6*e^3 *x^6 + 3675*b^2*c^6*d^3 + 2450*b^2*c^4*d^2*e + 1176*b^2*c^2*d*e^2 + 240*b^ 2*e^3 + 9*(49*b^2*c^6*d*e^2 + 10*b^2*c^4*e^3)*x^4 + (1225*b^2*c^6*d^2*e + 588*b^2*c^4*d*e^2 + 120*b^2*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))*log(c*x + sqr t(c^2*x^2 - 1)) - 210*(75*a*b*c^6*e^3*x^6 + 3675*a*b*c^6*d^3 + 2450*a*b*c^ 4*d^2*e + 1176*a*b*c^2*d*e^2 + 240*a*b*e^3 + 9*(49*a*b*c^6*d*e^2 + 10*a*b* c^4*e^3)*x^4 + (1225*a*b*c^6*d^2*e + 588*a*b*c^4*d*e^2 + 120*a*b*c^2*e^3)* x^2)*sqrt(c^2*x^2 - 1))/c^7
\[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{3}\, dx \]
Time = 0.22 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{7} \, b^{2} e^{3} x^{7} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} e^{3} x^{7} + \frac {3}{5} \, b^{2} d e^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + \frac {3}{5} \, a^{2} d e^{2} x^{5} + b^{2} d^{2} e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + a^{2} d^{2} e x^{3} + b^{2} d^{3} x \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{3} \, {\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^{2} e - \frac {2}{9} \, {\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^{2} e + \frac {2}{25} \, {\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 d e^{2} - \frac {2}{375} \, {\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} d e^{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 )} a b e^{3} - \frac {2}{25725} \, {\left (105 \, {\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 \operatorname {arcosh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} e^{3} + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d^{3}}{c} \]
1/7*b^2*e^3*x^7*arccosh(c*x)^2 + 1/7*a^2*e^3*x^7 + 3/5*b^2*d*e^2*x^5*arcco sh(c*x)^2 + 3/5*a^2*d*e^2*x^5 + b^2*d^2*e*x^3*arccosh(c*x)^2 + a^2*d^2*e*x ^3 + b^2*d^3*x*arccosh(c*x)^2 + 2/3*(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^2*e - 2/9*(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^2*e + 2/25*(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*d*e^2 - 2/3 75*(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*d*e^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)*a*b*e^3 - 2/25725*(105*(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*arccosh(c*x) - (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^ 6)*b^2*e^3 + 2*b^2*d^3*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d^3*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^3/c
Exception generated. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^3 \,d x \]