Integrand size = 18, antiderivative size = 337 \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \sqrt {-1+c x} \sqrt {1+c x}}{315 c^9}-\frac {4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{945 c^9}-\frac {2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) (-1+c x)^{5/2} (1+c x)^{5/2}}{525 c^9}-\frac {4 b e^3 \left (9 c^2 d+7 e\right ) (-1+c x)^{7/2} (1+c x)^{7/2}}{441 c^9}-\frac {b e^4 (-1+c x)^{9/2} (1+c x)^{9/2}}{81 c^9}+d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x)) \] Output:
-1/315*b*(315*c^8*d^4+420*c^6*d^3*e+378*c^4*d^2*e^2+180*c^2*d*e^3+35*e^4)* (c*x-1)^(1/2)*(c*x+1)^(1/2)/c^9-4/945*b*e*(105*c^6*d^3+189*c^4*d^2*e+135*c ^2*d*e^2+35*e^3)*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^9-2/525*b*e^2*(63*c^4*d^2+9 0*c^2*d*e+35*e^2)*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^9-4/441*b*e^3*(9*c^2*d+7*e )*(c*x-1)^(7/2)*(c*x+1)^(7/2)/c^9-1/81*b*e^4*(c*x-1)^(9/2)*(c*x+1)^(9/2)/c ^9+d^4*x*(a+b*arccosh(c*x))+4/3*d^3*e*x^3*(a+b*arccosh(c*x))+6/5*d^2*e^2*x ^5*(a+b*arccosh(c*x))+4/7*d*e^3*x^7*(a+b*arccosh(c*x))+1/9*e^4*x^9*(a+b*ar ccosh(c*x))
Time = 0.22 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.79 \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^4+320 c^2 e^3 \left (81 d+7 e x^2\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )+8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )\right )}{c^9}+315 b x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right ) \text {arccosh}(c x)}{99225} \] Input:
Integrate[(d + e*x^2)^4*(a + b*ArcCosh[c*x]),x]
Output:
(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e ^4*x^8) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^4 + 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) + 8*c^6*e*(110 25*d^3 + 3969*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3*e*x^2 + 23814*d^2*e^2*x^4 + 8100*d*e^3*x^6 + 1225*e^4*x^8)))/c^ 9 + 315*b*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 3 5*e^4*x^8)*ArcCosh[c*x])/99225
Time = 0.94 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6323, 27, 2113, 2331, 2389, 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.
| \(\displaystyle \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6323 |
| \(\displaystyle -b c \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {35 \left (c^2 x^2-1\right )^{7/2} e^4}{c^8}+\frac {20 \left (9 d c^2+7 e\right ) \left (c^2 x^2-1\right )^{5/2} e^3}{c^8}+\frac {6 \left (63 d^2 c^4+90 d e c^2+35 e^2\right ) \left (c^2 x^2-1\right )^{3/2} e^2}{c^8}+\frac {4 \left (105 d^3 c^6+189 d^2 e c^4+135 d e^2 c^2+35 e^3\right ) \sqrt {c^2 x^2-1} e}{c^8}+\frac {315 d^4 c^8+420 d^3 e c^6+378 d^2 e^2 c^4+180 d e^3 c^2+35 e^4}{c^8 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^4 x (a+b \text {arccosh}(c x))+\frac {4}{3} d^3 e x^3 (a+b \text {arccosh}(c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \text {arccosh}(c x))+\frac {4}{7} d e^3 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^4 x^9 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {40 e^3 \left (c^2 x^2-1\right )^{7/2} \left (9 c^2 d+7 e\right )}{7 c^{10}}+\frac {70 e^4 \left (c^2 x^2-1\right )^{9/2}}{9 c^{10}}+\frac {12 e^2 \left (c^2 x^2-1\right )^{5/2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{5 c^{10}}+\frac {8 e \left (c^2 x^2-1\right )^{3/2} \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{3 c^{10}}+\frac {2 \sqrt {c^2 x^2-1} \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right )}{c^{10}}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[(d + e*x^2)^4*(a + b*ArcCosh[c*x]),x]
Output:
-1/630*(b*c*Sqrt[-1 + c^2*x^2]*((2*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4* d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*Sqrt[-1 + c^2*x^2])/c^10 + (8*e*(105*c^6 *d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(-1 + c^2*x^2)^(3/2))/(3*c^ 10) + (12*e^2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(-1 + c^2*x^2)^(5/2))/(5* c^10) + (40*e^3*(9*c^2*d + 7*e)*(-1 + c^2*x^2)^(7/2))/(7*c^10) + (70*e^4*( -1 + c^2*x^2)^(9/2))/(9*c^10)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^4*x*(a + b*ArcCosh[c*x]) + (4*d^3*e*x^3*(a + b*ArcCosh[c*x]))/3 + (6*d^2*e^2*x^5* (a + b*ArcCosh[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcCosh[c*x]))/7 + (e^4*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[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ m]/(a*c + b*d*x^2)^FracPart[m]) Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a *d, 0] && EqQ[m, n] && !IntegerQ[m]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(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}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Time = 0.15 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.90
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{4} x^{9}+\frac {4}{7} d \,e^{3} x^{7}+\frac {6}{5} d^{2} e^{2} x^{5}+\frac {4}{3} d^{3} e \,x^{3}+d^{4} x \right )+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{4} x^{9}}{9}+\frac {4 c \,\operatorname {arccosh}\left (c x \right ) d \,e^{3} x^{7}}{7}+\frac {6 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e^{2} x^{5}}{5}+\frac {4 c \,\operatorname {arccosh}\left (c x \right ) d^{3} e \,x^{3}}{3}+\operatorname {arccosh}\left (c x \right ) d^{4} c x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 e^{4} x^{8} c^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right )}{99225 c^{8}}\right )}{c}\) | \(302\) |
| derivativedivides | \(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d^{4} c^{9} x +\frac {4 \,\operatorname {arccosh}\left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \,\operatorname {arccosh}\left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 e^{4} x^{8} c^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right )}{99225}\right )}{c^{8}}}{c}\) | \(331\) |
| default | \(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d^{4} c^{9} x +\frac {4 \,\operatorname {arccosh}\left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \,\operatorname {arccosh}\left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 e^{4} x^{8} c^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right )}{99225}\right )}{c^{8}}}{c}\) | \(331\) |
| orering | \(\frac {x \left (20825 c^{10} e^{5} x^{10}+132525 c^{10} d \,e^{4} x^{8}+366282 c^{10} d^{2} e^{3} x^{6}+1400 c^{8} e^{5} x^{8}+604170 c^{10} d^{3} e^{2} x^{4}+12960 c^{8} d \,e^{4} x^{6}+1025325 c^{10} d^{4} e \,x^{2}+63504 c^{8} d^{2} e^{3} x^{4}+2240 c^{6} e^{5} x^{6}+99225 c^{10} d^{5}+352800 c^{8} d^{3} e^{2} x^{2}+25920 c^{6} d \,e^{4} x^{4}-793800 c^{8} d^{4} e +254016 c^{6} d^{2} e^{3} x^{2}+4480 c^{4} e^{5} x^{4}-705600 c^{6} d^{3} e^{2}+103680 c^{4} d \,e^{4} x^{2}-508032 c^{4} d^{2} e^{3}+17920 c^{2} e^{5} x^{2}-207360 c^{2} d \,e^{4}-35840 e^{5}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{99225 \left (e \,x^{2}+d \right ) c^{10}}-\frac {\left (1225 e^{4} x^{8} c^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (8 \left (e \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e x +\frac {\left (e \,x^{2}+d \right )^{4} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{99225 c^{10} \left (e \,x^{2}+d \right )^{4}}\) | \(499\) |
Input:
int((e*x^2+d)^4*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/9*e^4*x^9+4/7*d*e^3*x^7+6/5*d^2*e^2*x^5+4/3*d^3*e*x^3+d^4*x)+b/c*(1/9 *c*arccosh(c*x)*e^4*x^9+4/7*c*arccosh(c*x)*d*e^3*x^7+6/5*c*arccosh(c*x)*d^ 2*e^2*x^5+4/3*c*arccosh(c*x)*d^3*e*x^3+arccosh(c*x)*d^4*c*x-1/99225/c^8*(c *x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*e^4*x^8+8100*c^8*d*e^3*x^6+23814*c^8*d ^2*e^2*x^4+1400*c^6*e^4*x^6+44100*c^8*d^3*e*x^2+9720*c^6*d*e^3*x^4+99225*c ^8*d^4+31752*c^6*d^2*e^2*x^2+1680*c^4*e^4*x^4+88200*c^6*d^3*e+12960*c^4*d* e^3*x^2+63504*c^4*d^2*e^2+2240*c^2*e^4*x^2+25920*c^2*d*e^3+4480*e^4))
Time = 0.12 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \, {\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} + 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} + 25920 \, b c^{2} d e^{3} + 100 \, {\left (81 \, b c^{8} d e^{3} + 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \, {\left (3969 \, b c^{8} d^{2} e^{2} + 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \, {\left (11025 \, b c^{8} d^{3} e + 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} + 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \] Input:
integrate((e*x^2+d)^4*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
1/99225*(11025*a*c^9*e^4*x^9 + 56700*a*c^9*d*e^3*x^7 + 119070*a*c^9*d^2*e^ 2*x^5 + 132300*a*c^9*d^3*e*x^3 + 99225*a*c^9*d^4*x + 315*(35*b*c^9*e^4*x^9 + 180*b*c^9*d*e^3*x^7 + 378*b*c^9*d^2*e^2*x^5 + 420*b*c^9*d^3*e*x^3 + 315 *b*c^9*d^4*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*c^8*e^4*x^8 + 99225*b *c^8*d^4 + 88200*b*c^6*d^3*e + 63504*b*c^4*d^2*e^2 + 25920*b*c^2*d*e^3 + 1 00*(81*b*c^8*d*e^3 + 14*b*c^6*e^4)*x^6 + 4480*b*e^4 + 6*(3969*b*c^8*d^2*e^ 2 + 1620*b*c^6*d*e^3 + 280*b*c^4*e^4)*x^4 + 4*(11025*b*c^8*d^3*e + 7938*b* c^6*d^2*e^2 + 3240*b*c^4*d*e^3 + 560*b*c^2*e^4)*x^2)*sqrt(c^2*x^2 - 1))/c^ 9
\[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{4}\, dx \] Input:
integrate((e*x**2+d)**4*(a+b*acosh(c*x)),x)
Output:
Integral((a + b*acosh(c*x))*(d + e*x**2)**4, x)
Time = 0.03 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.23 \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} e + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} e^{2} + \frac {4}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e^{3} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{4} + a d^{4} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{4}}{c} \] Input:
integrate((e*x^2+d)^4*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
1/9*a*e^4*x^9 + 4/7*a*d*e^3*x^7 + 6/5*a*d^2*e^2*x^5 + 4/3*a*d^3*e*x^3 + 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))*b*d^3*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)*b*d^2*e^2 + 4/ 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^3 + 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^4 + a*d^4*x + (c*x*arc cosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^4/c
Exception generated. \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x^2+d)^4*(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. \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^4 \,d x \] Input:
int((a + b*acosh(c*x))*(d + e*x^2)^4,x)
Output:
int((a + b*acosh(c*x))*(d + e*x^2)^4, x)
Time = 0.21 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.43 \[ \int \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x)) \, dx=\frac {99225 a \,c^{9} d^{4} x +11025 a \,c^{9} e^{4} x^{9}+56700 \mathit {acosh} \left (c x \right ) b \,c^{9} d \,e^{3} x^{7}-44100 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{3} e \,x^{2}-23814 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{2} e^{2} x^{4}-8100 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d \,e^{3} x^{6}-31752 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{2} e^{2} x^{2}-9720 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d \,e^{3} x^{4}-12960 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d \,e^{3} x^{2}+132300 \mathit {acosh} \left (c x \right ) b \,c^{9} d^{3} e \,x^{3}+119070 \mathit {acosh} \left (c x \right ) b \,c^{9} d^{2} e^{2} x^{5}+99225 \mathit {acosh} \left (c x \right ) b \,c^{9} d^{4} x +11025 \mathit {acosh} \left (c x \right ) b \,c^{9} e^{4} x^{9}-1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} e^{4} x^{8}-88200 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{3} e -1400 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e^{4} x^{6}-63504 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d^{2} e^{2}-1680 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{4} x^{4}-25920 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d \,e^{3}-2240 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{4} x^{2}-99225 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{8} d^{4}+132300 a \,c^{9} d^{3} e \,x^{3}+119070 a \,c^{9} d^{2} e^{2} x^{5}+56700 a \,c^{9} d \,e^{3} x^{7}-4480 \sqrt {c^{2} x^{2}-1}\, b \,e^{4}}{99225 c^{9}} \] Input:
int((e*x^2+d)^4*(a+b*acosh(c*x)),x)
Output:
(99225*acosh(c*x)*b*c**9*d**4*x + 132300*acosh(c*x)*b*c**9*d**3*e*x**3 + 1 19070*acosh(c*x)*b*c**9*d**2*e**2*x**5 + 56700*acosh(c*x)*b*c**9*d*e**3*x* *7 + 11025*acosh(c*x)*b*c**9*e**4*x**9 - 44100*sqrt(c**2*x**2 - 1)*b*c**8* d**3*e*x**2 - 23814*sqrt(c**2*x**2 - 1)*b*c**8*d**2*e**2*x**4 - 8100*sqrt( c**2*x**2 - 1)*b*c**8*d*e**3*x**6 - 1225*sqrt(c**2*x**2 - 1)*b*c**8*e**4*x **8 - 88200*sqrt(c**2*x**2 - 1)*b*c**6*d**3*e - 31752*sqrt(c**2*x**2 - 1)* b*c**6*d**2*e**2*x**2 - 9720*sqrt(c**2*x**2 - 1)*b*c**6*d*e**3*x**4 - 1400 *sqrt(c**2*x**2 - 1)*b*c**6*e**4*x**6 - 63504*sqrt(c**2*x**2 - 1)*b*c**4*d **2*e**2 - 12960*sqrt(c**2*x**2 - 1)*b*c**4*d*e**3*x**2 - 1680*sqrt(c**2*x **2 - 1)*b*c**4*e**4*x**4 - 25920*sqrt(c**2*x**2 - 1)*b*c**2*d*e**3 - 2240 *sqrt(c**2*x**2 - 1)*b*c**2*e**4*x**2 - 4480*sqrt(c**2*x**2 - 1)*b*e**4 - 99225*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c**8*d**4 + 99225*a*c**9*d**4*x + 1323 00*a*c**9*d**3*e*x**3 + 119070*a*c**9*d**2*e**2*x**5 + 56700*a*c**9*d*e**3 *x**7 + 11025*a*c**9*e**4*x**9)/(99225*c**9)