Integrand size = 14, antiderivative size = 312 \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=-\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \sqrt {-1+a x} \sqrt {1+a x}}{315 a^9}-\frac {4 d \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right ) (-1+a x)^{3/2} (1+a x)^{3/2}}{945 a^9}-\frac {2 d^2 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right ) (-1+a x)^{5/2} (1+a x)^{5/2}}{525 a^9}-\frac {4 d^3 \left (9 a^2 c+7 d\right ) (-1+a x)^{7/2} (1+a x)^{7/2}}{441 a^9}-\frac {d^4 (-1+a x)^{9/2} (1+a x)^{9/2}}{81 a^9}+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x) \] Output:
-1/315*(315*a^8*c^4+420*a^6*c^3*d+378*a^4*c^2*d^2+180*a^2*c*d^3+35*d^4)*(a *x-1)^(1/2)*(a*x+1)^(1/2)/a^9-4/945*d*(105*a^6*c^3+189*a^4*c^2*d+135*a^2*c *d^2+35*d^3)*(a*x-1)^(3/2)*(a*x+1)^(3/2)/a^9-2/525*d^2*(63*a^4*c^2+90*a^2* c*d+35*d^2)*(a*x-1)^(5/2)*(a*x+1)^(5/2)/a^9-4/441*d^3*(9*a^2*c+7*d)*(a*x-1 )^(7/2)*(a*x+1)^(7/2)/a^9-1/81*d^4*(a*x-1)^(9/2)*(a*x+1)^(9/2)/a^9+c^4*x*a rccosh(a*x)+4/3*c^3*d*x^3*arccosh(a*x)+6/5*c^2*d^2*x^5*arccosh(a*x)+4/7*c* d^3*x^7*arccosh(a*x)+1/9*d^4*x^9*arccosh(a*x)
Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.69 \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (4480 d^4+320 a^2 d^3 \left (81 c+7 d x^2\right )+48 a^4 d^2 \left (1323 c^2+270 c d x^2+35 d^2 x^4\right )+8 a^6 d \left (11025 c^3+3969 c^2 d x^2+1215 c d^2 x^4+175 d^3 x^6\right )+a^8 \left (99225 c^4+44100 c^3 d x^2+23814 c^2 d^2 x^4+8100 c d^3 x^6+1225 d^4 x^8\right )\right )}{99225 a^9}+\frac {1}{315} x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \text {arccosh}(a x) \] Input:
Integrate[(c + d*x^2)^4*ArcCosh[a*x],x]
Output:
-1/99225*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(4480*d^4 + 320*a^2*d^3*(81*c + 7*d *x^2) + 48*a^4*d^2*(1323*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 8*a^6*d*(11025* c^3 + 3969*c^2*d*x^2 + 1215*c*d^2*x^4 + 175*d^3*x^6) + a^8*(99225*c^4 + 44 100*c^3*d*x^2 + 23814*c^2*d^2*x^4 + 8100*c*d^3*x^6 + 1225*d^4*x^8)))/a^9 + (x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^ 8)*ArcCosh[a*x])/315
Time = 1.39 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, 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.
| \(\displaystyle \int \text {arccosh}(a x) \left (c+d x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 6323 |
| \(\displaystyle -a \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{315 \sqrt {a x-1} \sqrt {a x+1}}dx+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} a \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{\sqrt {a x-1} \sqrt {a x+1}}dx+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x)\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {x \left (35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4\right )}{\sqrt {a^2 x^2-1}}dx}{315 \sqrt {a x-1} \sqrt {a x+1}}+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x)\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {35 d^4 x^8+180 c d^3 x^6+378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4}{\sqrt {a^2 x^2-1}}dx^2}{630 \sqrt {a x-1} \sqrt {a x+1}}+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x)\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \left (\frac {35 \left (a^2 x^2-1\right )^{7/2} d^4}{a^8}+\frac {20 \left (9 c a^2+7 d\right ) \left (a^2 x^2-1\right )^{5/2} d^3}{a^8}+\frac {6 \left (63 c^2 a^4+90 c d a^2+35 d^2\right ) \left (a^2 x^2-1\right )^{3/2} d^2}{a^8}+\frac {4 \left (105 c^3 a^6+189 c^2 d a^4+135 c d^2 a^2+35 d^3\right ) \sqrt {a^2 x^2-1} d}{a^8}+\frac {315 c^4 a^8+420 c^3 d a^6+378 c^2 d^2 a^4+180 c d^3 a^2+35 d^4}{a^8 \sqrt {a^2 x^2-1}}\right )dx^2}{630 \sqrt {a x-1} \sqrt {a x+1}}+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (\frac {40 d^3 \left (a^2 x^2-1\right )^{7/2} \left (9 a^2 c+7 d\right )}{7 a^{10}}+\frac {70 d^4 \left (a^2 x^2-1\right )^{9/2}}{9 a^{10}}+\frac {12 d^2 \left (a^2 x^2-1\right )^{5/2} \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{5 a^{10}}+\frac {8 d \left (a^2 x^2-1\right )^{3/2} \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right )}{3 a^{10}}+\frac {2 \sqrt {a^2 x^2-1} \left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right )}{a^{10}}\right )}{630 \sqrt {a x-1} \sqrt {a x+1}}+c^4 x \text {arccosh}(a x)+\frac {4}{3} c^3 d x^3 \text {arccosh}(a x)+\frac {6}{5} c^2 d^2 x^5 \text {arccosh}(a x)+\frac {4}{7} c d^3 x^7 \text {arccosh}(a x)+\frac {1}{9} d^4 x^9 \text {arccosh}(a x)\) |
Input:
Int[(c + d*x^2)^4*ArcCosh[a*x],x]
Output:
-1/630*(a*Sqrt[-1 + a^2*x^2]*((2*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^ 2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Sqrt[-1 + a^2*x^2])/a^10 + (8*d*(105*a^6*c ^3 + 189*a^4*c^2*d + 135*a^2*c*d^2 + 35*d^3)*(-1 + a^2*x^2)^(3/2))/(3*a^10 ) + (12*d^2*(63*a^4*c^2 + 90*a^2*c*d + 35*d^2)*(-1 + a^2*x^2)^(5/2))/(5*a^ 10) + (40*d^3*(9*a^2*c + 7*d)*(-1 + a^2*x^2)^(7/2))/(7*a^10) + (70*d^4*(-1 + a^2*x^2)^(9/2))/(9*a^10)))/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^4*x*ArcCo sh[a*x] + (4*c^3*d*x^3*ArcCosh[a*x])/3 + (6*c^2*d^2*x^5*ArcCosh[a*x])/5 + (4*c*d^3*x^7*ArcCosh[a*x])/7 + (d^4*x^9*ArcCosh[a*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 = 0.14 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.79
| method | result | size |
| parts | \(\frac {d^{4} x^{9} \operatorname {arccosh}\left (a x \right )}{9}+\frac {4 c \,d^{3} x^{7} \operatorname {arccosh}\left (a x \right )}{7}+\frac {6 c^{2} d^{2} x^{5} \operatorname {arccosh}\left (a x \right )}{5}+\frac {4 c^{3} d \,x^{3} \operatorname {arccosh}\left (a x \right )}{3}+c^{4} x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (1225 a^{8} d^{4} x^{8}+8100 a^{8} c \,d^{3} x^{6}+23814 a^{8} c^{2} d^{2} x^{4}+1400 a^{6} d^{4} x^{6}+44100 a^{8} c^{3} d \,x^{2}+9720 a^{6} c \,d^{3} x^{4}+99225 a^{8} c^{4}+31752 a^{6} c^{2} d^{2} x^{2}+1680 a^{4} d^{4} x^{4}+88200 a^{6} c^{3} d +12960 a^{4} c \,d^{3} x^{2}+63504 a^{4} c^{2} d^{2}+2240 a^{2} d^{4} x^{2}+25920 a^{2} c \,d^{3}+4480 d^{4}\right )}{99225 a^{9}}\) | \(246\) |
| derivativedivides | \(\frac {\operatorname {arccosh}\left (a x \right ) c^{4} a x +\frac {4 a \,\operatorname {arccosh}\left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccosh}\left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccosh}\left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{4} x^{9}}{9}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (1225 a^{8} d^{4} x^{8}+8100 a^{8} c \,d^{3} x^{6}+23814 a^{8} c^{2} d^{2} x^{4}+1400 a^{6} d^{4} x^{6}+44100 a^{8} c^{3} d \,x^{2}+9720 a^{6} c \,d^{3} x^{4}+99225 a^{8} c^{4}+31752 a^{6} c^{2} d^{2} x^{2}+1680 a^{4} d^{4} x^{4}+88200 a^{6} c^{3} d +12960 a^{4} c \,d^{3} x^{2}+63504 a^{4} c^{2} d^{2}+2240 a^{2} d^{4} x^{2}+25920 a^{2} c \,d^{3}+4480 d^{4}\right )}{99225 a^{8}}}{a}\) | \(255\) |
| default | \(\frac {\operatorname {arccosh}\left (a x \right ) c^{4} a x +\frac {4 a \,\operatorname {arccosh}\left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccosh}\left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccosh}\left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{4} x^{9}}{9}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (1225 a^{8} d^{4} x^{8}+8100 a^{8} c \,d^{3} x^{6}+23814 a^{8} c^{2} d^{2} x^{4}+1400 a^{6} d^{4} x^{6}+44100 a^{8} c^{3} d \,x^{2}+9720 a^{6} c \,d^{3} x^{4}+99225 a^{8} c^{4}+31752 a^{6} c^{2} d^{2} x^{2}+1680 a^{4} d^{4} x^{4}+88200 a^{6} c^{3} d +12960 a^{4} c \,d^{3} x^{2}+63504 a^{4} c^{2} d^{2}+2240 a^{2} d^{4} x^{2}+25920 a^{2} c \,d^{3}+4480 d^{4}\right )}{99225 a^{8}}}{a}\) | \(255\) |
| orering | \(\frac {x \left (20825 a^{10} d^{5} x^{10}+132525 a^{10} c \,d^{4} x^{8}+366282 a^{10} c^{2} d^{3} x^{6}+1400 a^{8} d^{5} x^{8}+604170 a^{10} c^{3} d^{2} x^{4}+12960 a^{8} c \,d^{4} x^{6}+1025325 a^{10} c^{4} d \,x^{2}+63504 a^{8} c^{2} d^{3} x^{4}+2240 a^{6} d^{5} x^{6}+99225 a^{10} c^{5}+352800 a^{8} c^{3} d^{2} x^{2}+25920 a^{6} c \,d^{4} x^{4}-793800 a^{8} c^{4} d +254016 a^{6} c^{2} d^{3} x^{2}+4480 a^{4} d^{5} x^{4}-705600 a^{6} c^{3} d^{2}+103680 a^{4} c \,d^{4} x^{2}-508032 a^{4} c^{2} d^{3}+17920 a^{2} d^{5} x^{2}-207360 a^{2} c \,d^{4}-35840 d^{5}\right ) \operatorname {arccosh}\left (a x \right )}{99225 \left (d \,x^{2}+c \right ) a^{10}}-\frac {\left (1225 a^{8} d^{4} x^{8}+8100 a^{8} c \,d^{3} x^{6}+23814 a^{8} c^{2} d^{2} x^{4}+1400 a^{6} d^{4} x^{6}+44100 a^{8} c^{3} d \,x^{2}+9720 a^{6} c \,d^{3} x^{4}+99225 a^{8} c^{4}+31752 a^{6} c^{2} d^{2} x^{2}+1680 a^{4} d^{4} x^{4}+88200 a^{6} c^{3} d +12960 a^{4} c \,d^{3} x^{2}+63504 a^{4} c^{2} d^{2}+2240 a^{2} d^{4} x^{2}+25920 a^{2} c \,d^{3}+4480 d^{4}\right ) \left (a x -1\right ) \left (a x +1\right ) \left (8 \left (d \,x^{2}+c \right )^{3} \operatorname {arccosh}\left (a x \right ) d x +\frac {\left (d \,x^{2}+c \right )^{4} a}{\sqrt {a x -1}\, \sqrt {a x +1}}\right )}{99225 a^{10} \left (d \,x^{2}+c \right )^{4}}\) | \(490\) |
Input:
int((d*x^2+c)^4*arccosh(a*x),x,method=_RETURNVERBOSE)
Output:
1/9*d^4*x^9*arccosh(a*x)+4/7*c*d^3*x^7*arccosh(a*x)+6/5*c^2*d^2*x^5*arccos h(a*x)+4/3*c^3*d*x^3*arccosh(a*x)+c^4*x*arccosh(a*x)-1/99225/a^9*(a*x-1)^( 1/2)*(a*x+1)^(1/2)*(1225*a^8*d^4*x^8+8100*a^8*c*d^3*x^6+23814*a^8*c^2*d^2* x^4+1400*a^6*d^4*x^6+44100*a^8*c^3*d*x^2+9720*a^6*c*d^3*x^4+99225*a^8*c^4+ 31752*a^6*c^2*d^2*x^2+1680*a^4*d^4*x^4+88200*a^6*c^3*d+12960*a^4*c*d^3*x^2 +63504*a^4*c^2*d^2+2240*a^2*d^4*x^2+25920*a^2*c*d^3+4480*d^4)
Time = 0.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.80 \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=\frac {315 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (1225 \, a^{8} d^{4} x^{8} + 99225 \, a^{8} c^{4} + 88200 \, a^{6} c^{3} d + 63504 \, a^{4} c^{2} d^{2} + 100 \, {\left (81 \, a^{8} c d^{3} + 14 \, a^{6} d^{4}\right )} x^{6} + 25920 \, a^{2} c d^{3} + 6 \, {\left (3969 \, a^{8} c^{2} d^{2} + 1620 \, a^{6} c d^{3} + 280 \, a^{4} d^{4}\right )} x^{4} + 4480 \, d^{4} + 4 \, {\left (11025 \, a^{8} c^{3} d + 7938 \, a^{6} c^{2} d^{2} + 3240 \, a^{4} c d^{3} + 560 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{99225 \, a^{9}} \] Input:
integrate((d*x^2+c)^4*arccosh(a*x),x, algorithm="fricas")
Output:
1/99225*(315*(35*a^9*d^4*x^9 + 180*a^9*c*d^3*x^7 + 378*a^9*c^2*d^2*x^5 + 4 20*a^9*c^3*d*x^3 + 315*a^9*c^4*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (1225*a^8 *d^4*x^8 + 99225*a^8*c^4 + 88200*a^6*c^3*d + 63504*a^4*c^2*d^2 + 100*(81*a ^8*c*d^3 + 14*a^6*d^4)*x^6 + 25920*a^2*c*d^3 + 6*(3969*a^8*c^2*d^2 + 1620* a^6*c*d^3 + 280*a^4*d^4)*x^4 + 4480*d^4 + 4*(11025*a^8*c^3*d + 7938*a^6*c^ 2*d^2 + 3240*a^4*c*d^3 + 560*a^2*d^4)*x^2)*sqrt(a^2*x^2 - 1))/a^9
\[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=\int \left (c + d x^{2}\right )^{4} \operatorname {acosh}{\left (a x \right )}\, dx \] Input:
integrate((d*x**2+c)**4*acosh(a*x),x)
Output:
Integral((c + d*x**2)**4*acosh(a*x), x)
Time = 0.04 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.23 \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=-\frac {1}{99225} \, {\left (\frac {1225 \, \sqrt {a^{2} x^{2} - 1} d^{4} x^{8}}{a^{2}} + \frac {8100 \, \sqrt {a^{2} x^{2} - 1} c d^{3} x^{6}}{a^{2}} + \frac {23814 \, \sqrt {a^{2} x^{2} - 1} c^{2} d^{2} x^{4}}{a^{2}} + \frac {1400 \, \sqrt {a^{2} x^{2} - 1} d^{4} x^{6}}{a^{4}} + \frac {44100 \, \sqrt {a^{2} x^{2} - 1} c^{3} d x^{2}}{a^{2}} + \frac {9720 \, \sqrt {a^{2} x^{2} - 1} c d^{3} x^{4}}{a^{4}} + \frac {99225 \, \sqrt {a^{2} x^{2} - 1} c^{4}}{a^{2}} + \frac {31752 \, \sqrt {a^{2} x^{2} - 1} c^{2} d^{2} x^{2}}{a^{4}} + \frac {1680 \, \sqrt {a^{2} x^{2} - 1} d^{4} x^{4}}{a^{6}} + \frac {88200 \, \sqrt {a^{2} x^{2} - 1} c^{3} d}{a^{4}} + \frac {12960 \, \sqrt {a^{2} x^{2} - 1} c d^{3} x^{2}}{a^{6}} + \frac {63504 \, \sqrt {a^{2} x^{2} - 1} c^{2} d^{2}}{a^{6}} + \frac {2240 \, \sqrt {a^{2} x^{2} - 1} d^{4} x^{2}}{a^{8}} + \frac {25920 \, \sqrt {a^{2} x^{2} - 1} c d^{3}}{a^{8}} + \frac {4480 \, \sqrt {a^{2} x^{2} - 1} d^{4}}{a^{10}}\right )} a + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname {arcosh}\left (a x\right ) \] Input:
integrate((d*x^2+c)^4*arccosh(a*x),x, algorithm="maxima")
Output:
-1/99225*(1225*sqrt(a^2*x^2 - 1)*d^4*x^8/a^2 + 8100*sqrt(a^2*x^2 - 1)*c*d^ 3*x^6/a^2 + 23814*sqrt(a^2*x^2 - 1)*c^2*d^2*x^4/a^2 + 1400*sqrt(a^2*x^2 - 1)*d^4*x^6/a^4 + 44100*sqrt(a^2*x^2 - 1)*c^3*d*x^2/a^2 + 9720*sqrt(a^2*x^2 - 1)*c*d^3*x^4/a^4 + 99225*sqrt(a^2*x^2 - 1)*c^4/a^2 + 31752*sqrt(a^2*x^2 - 1)*c^2*d^2*x^2/a^4 + 1680*sqrt(a^2*x^2 - 1)*d^4*x^4/a^6 + 88200*sqrt(a^ 2*x^2 - 1)*c^3*d/a^4 + 12960*sqrt(a^2*x^2 - 1)*c*d^3*x^2/a^6 + 63504*sqrt( a^2*x^2 - 1)*c^2*d^2/a^6 + 2240*sqrt(a^2*x^2 - 1)*d^4*x^2/a^8 + 25920*sqrt (a^2*x^2 - 1)*c*d^3/a^8 + 4480*sqrt(a^2*x^2 - 1)*d^4/a^10)*a + 1/315*(35*d ^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315*c^4*x)*arcc osh(a*x)
Time = 0.13 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.01 \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=\frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \sqrt {a^{2} x^{2} - 1}}{315 \, a^{9}} - \frac {44100 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{6} c^{3} d + 23814 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} a^{4} c^{2} d^{2} + 79380 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{4} c^{2} d^{2} + 8100 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {7}{2}} a^{2} c d^{3} + 34020 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} a^{2} c d^{3} + 1225 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {9}{2}} d^{4} + 56700 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d^{3} + 6300 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {7}{2}} d^{4} + 13230 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{4} + 14700 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{4}}{99225 \, a^{9}} \] Input:
integrate((d*x^2+c)^4*arccosh(a*x),x, algorithm="giac")
Output:
1/315*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315* c^4*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/315*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*sqrt(a^2*x^2 - 1)/a^9 - 1/99225 *(44100*(a^2*x^2 - 1)^(3/2)*a^6*c^3*d + 23814*(a^2*x^2 - 1)^(5/2)*a^4*c^2* d^2 + 79380*(a^2*x^2 - 1)^(3/2)*a^4*c^2*d^2 + 8100*(a^2*x^2 - 1)^(7/2)*a^2 *c*d^3 + 34020*(a^2*x^2 - 1)^(5/2)*a^2*c*d^3 + 1225*(a^2*x^2 - 1)^(9/2)*d^ 4 + 56700*(a^2*x^2 - 1)^(3/2)*a^2*c*d^3 + 6300*(a^2*x^2 - 1)^(7/2)*d^4 + 1 3230*(a^2*x^2 - 1)^(5/2)*d^4 + 14700*(a^2*x^2 - 1)^(3/2)*d^4)/a^9
Timed out. \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=\int \mathrm {acosh}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^4 \,d x \] Input:
int(acosh(a*x)*(c + d*x^2)^4,x)
Output:
int(acosh(a*x)*(c + d*x^2)^4, x)
Time = 0.19 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.28 \[ \int \left (c+d x^2\right )^4 \text {arccosh}(a x) \, dx=\frac {99225 \mathit {acosh} \left (a x \right ) a^{9} c^{4} x +132300 \mathit {acosh} \left (a x \right ) a^{9} c^{3} d \,x^{3}+119070 \mathit {acosh} \left (a x \right ) a^{9} c^{2} d^{2} x^{5}+56700 \mathit {acosh} \left (a x \right ) a^{9} c \,d^{3} x^{7}+11025 \mathit {acosh} \left (a x \right ) a^{9} d^{4} x^{9}-44100 \sqrt {a^{2} x^{2}-1}\, a^{8} c^{3} d \,x^{2}-23814 \sqrt {a^{2} x^{2}-1}\, a^{8} c^{2} d^{2} x^{4}-8100 \sqrt {a^{2} x^{2}-1}\, a^{8} c \,d^{3} x^{6}-1225 \sqrt {a^{2} x^{2}-1}\, a^{8} d^{4} x^{8}-88200 \sqrt {a^{2} x^{2}-1}\, a^{6} c^{3} d -31752 \sqrt {a^{2} x^{2}-1}\, a^{6} c^{2} d^{2} x^{2}-9720 \sqrt {a^{2} x^{2}-1}\, a^{6} c \,d^{3} x^{4}-1400 \sqrt {a^{2} x^{2}-1}\, a^{6} d^{4} x^{6}-63504 \sqrt {a^{2} x^{2}-1}\, a^{4} c^{2} d^{2}-12960 \sqrt {a^{2} x^{2}-1}\, a^{4} c \,d^{3} x^{2}-1680 \sqrt {a^{2} x^{2}-1}\, a^{4} d^{4} x^{4}-25920 \sqrt {a^{2} x^{2}-1}\, a^{2} c \,d^{3}-2240 \sqrt {a^{2} x^{2}-1}\, a^{2} d^{4} x^{2}-4480 \sqrt {a^{2} x^{2}-1}\, d^{4}-99225 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{8} c^{4}}{99225 a^{9}} \] Input:
int((d*x^2+c)^4*acosh(a*x),x)
Output:
(99225*acosh(a*x)*a**9*c**4*x + 132300*acosh(a*x)*a**9*c**3*d*x**3 + 11907 0*acosh(a*x)*a**9*c**2*d**2*x**5 + 56700*acosh(a*x)*a**9*c*d**3*x**7 + 110 25*acosh(a*x)*a**9*d**4*x**9 - 44100*sqrt(a**2*x**2 - 1)*a**8*c**3*d*x**2 - 23814*sqrt(a**2*x**2 - 1)*a**8*c**2*d**2*x**4 - 8100*sqrt(a**2*x**2 - 1) *a**8*c*d**3*x**6 - 1225*sqrt(a**2*x**2 - 1)*a**8*d**4*x**8 - 88200*sqrt(a **2*x**2 - 1)*a**6*c**3*d - 31752*sqrt(a**2*x**2 - 1)*a**6*c**2*d**2*x**2 - 9720*sqrt(a**2*x**2 - 1)*a**6*c*d**3*x**4 - 1400*sqrt(a**2*x**2 - 1)*a** 6*d**4*x**6 - 63504*sqrt(a**2*x**2 - 1)*a**4*c**2*d**2 - 12960*sqrt(a**2*x **2 - 1)*a**4*c*d**3*x**2 - 1680*sqrt(a**2*x**2 - 1)*a**4*d**4*x**4 - 2592 0*sqrt(a**2*x**2 - 1)*a**2*c*d**3 - 2240*sqrt(a**2*x**2 - 1)*a**2*d**4*x** 2 - 4480*sqrt(a**2*x**2 - 1)*d**4 - 99225*sqrt(a*x + 1)*sqrt(a*x - 1)*a**8 *c**4)/(99225*a**9)