Integrand size = 23, antiderivative size = 218 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {16}{75} b^2 e^4 x+\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{75 d}-\frac {8 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{75 d}-\frac {2 b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))^2}{5 d} \] Output:
16/75*b^2*e^4*x+8/225*b^2*e^4*(d*x+c)^3/d+2/125*b^2*e^4*(d*x+c)^5/d-16/75* b*e^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d-8/75*b*e^4*(d *x+c-1)^(1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d-2/25*b*e^4* (d*x+c-1)^(1/2)*(d*x+c)^4*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d+1/5*e^4*( d*x+c)^5*(a+b*arccosh(d*x+c))^2/d
Time = 0.37 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.01 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e^4 \left (240 b^2 (c+d x)+40 b^2 (c+d x)^3+9 \left (25 a^2+2 b^2\right ) (c+d x)^5+30 a b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-8-4 (c+d x)^2-3 (c+d x)^4\right )+30 b \left (15 a (c+d x)^5-8 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-4 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}-3 b \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+225 b^2 (c+d x)^5 \text {arccosh}(c+d x)^2\right )}{1125 d} \] Input:
Integrate[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^2,x]
Output:
(e^4*(240*b^2*(c + d*x) + 40*b^2*(c + d*x)^3 + 9*(25*a^2 + 2*b^2)*(c + d*x )^5 + 30*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-8 - 4*(c + d*x)^2 - 3* (c + d*x)^4) + 30*b*(15*a*(c + d*x)^5 - 8*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 4*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] - 3*b*Sqrt[- 1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + 225*b^2*(c + d*x)^5*ArcCosh[c + d*x]^2))/(1125*d)
Time = 1.58 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6411, 27, 6298, 6354, 15, 6354, 15, 6330, 24}
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.
| \(\displaystyle \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int e^4 (c+d x)^4 (a+b \text {arccosh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int (c+d x)^4 (a+b \text {arccosh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \int \frac {(c+d x)^5 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \left (\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{5} b \int (c+d x)^4d(c+d x)+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \left (\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))-\frac {1}{25} b (c+d x)^5\right )\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{3} b \int (c+d x)^2d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))\right )+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))-\frac {1}{25} b (c+d x)^5\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))-\frac {1}{25} b (c+d x)^5\right )\right )}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \left (\frac {4}{5} \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b \int 1d(c+d x)\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )+\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))-\frac {1}{25} b (c+d x)^5\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4 (a+b \text {arccosh}(c+d x))+\frac {4}{5} \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b (c+d x)\right )-\frac {1}{9} b (c+d x)^3\right )-\frac {1}{25} b (c+d x)^5\right )\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^2,x]
Output:
(e^4*(((c + d*x)^5*(a + b*ArcCosh[c + d*x])^2)/5 - (2*b*(-1/25*(b*(c + d*x )^5) + (Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/5 + (4*(-1/9*(b*(c + d*x)^3) + (Sqrt[-1 + c + d*x]*(c + d*x)^2*Sq rt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/3 + (2*(-(b*(c + d*x)) + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])))/3))/5))/5))/d
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.25 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5}+e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {8 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}+\frac {8 \left (d x +c \right )^{3}}{225}\right )+2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\) | \(218\) |
| default | \(\frac {\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5}+e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {8 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}+\frac {8 \left (d x +c \right )^{3}}{225}\right )+2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\) | \(218\) |
| parts | \(\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{25}-\frac {8 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}+\frac {8 \left (d x +c \right )^{3}}{225}\right )}{d}+\frac {2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\) | \(223\) |
| orering | \(\frac {\left (549 d^{7} x^{7}+3843 c \,d^{6} x^{6}+11529 c^{2} d^{5} x^{5}+19215 c^{3} d^{4} x^{4}+19215 c^{4} d^{3} x^{3}+200 d^{5} x^{5}+11385 c^{5} d^{2} x^{2}+1000 c \,d^{4} x^{4}+3555 c^{6} d x +2000 c^{2} d^{3} x^{3}+405 c^{7}+1680 c^{3} d^{2} x^{2}+360 c^{4} d x +1760 d^{3} x^{3}+60 c^{5}+3360 c \,d^{2} x^{2}+1440 c^{2} d x +240 c^{3}-2880 d x -480 c \right ) \left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}{1125 d \left (d x +c \right )^{6}}-\frac {\left (108 d^{7} x^{7}+756 c \,d^{6} x^{6}+2268 c^{2} d^{5} x^{5}+3780 c^{3} d^{4} x^{4}+3780 c^{4} d^{3} x^{3}+83 d^{5} x^{5}+2205 c^{5} d^{2} x^{2}+415 c \,d^{4} x^{4}+630 c^{6} d x +830 c^{2} d^{3} x^{3}+45 c^{7}+690 c^{3} d^{2} x^{2}+135 c^{4} d x +740 d^{3} x^{3}+15 c^{5}+1380 c \,d^{2} x^{2}+540 c^{2} d x +60 c^{3}-1080 d x -120 c \right ) \left (4 \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} d e +\frac {2 \left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}\right )}{1125 d^{2} \left (d x +c \right )^{5}}+\frac {x \left (9 d^{4} x^{4}+45 c \,d^{3} x^{3}+90 c^{2} d^{2} x^{2}+90 c^{3} d x +45 c^{4}+20 d^{2} x^{2}+60 c d x +60 c^{2}+120\right ) \left (d x +c -1\right ) \left (d x +c +1\right ) \left (12 \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2} d^{2} e^{2}+\frac {16 \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) d^{2} e b}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}+\frac {2 \left (d e x +c e \right )^{4} b^{2} d^{2}}{\left (d x +c -1\right ) \left (d x +c +1\right )}-\frac {\left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b \,d^{2}}{\left (d x +c -1\right )^{\frac {3}{2}} \sqrt {d x +c +1}}-\frac {\left (d e x +c e \right )^{4} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b \,d^{2}}{\sqrt {d x +c -1}\, \left (d x +c +1\right )^{\frac {3}{2}}}\right )}{1125 d^{2} \left (d x +c \right )^{4}}\) | \(714\) |
Input:
int((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/5*e^4*a^2*(d*x+c)^5+e^4*b^2*(1/5*(d*x+c)^5*arccosh(d*x+c)^2-16/75*a rccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-2/25*(d*x+c)^4*arccosh(d*x+c )*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-8/75*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x +c+1)^(1/2)*(d*x+c)^2+16/75*d*x+16/75*c+2/125*(d*x+c)^5+8/225*(d*x+c)^3)+2 *e^4*a*b*(1/5*(d*x+c)^5*arccosh(d*x+c)-1/75*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2 )*(3*(d*x+c)^4+4*(d*x+c)^2+8)))
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (192) = 384\).
Time = 0.10 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.83 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} d^{5} e^{4} x^{5} + 45 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c d^{4} e^{4} x^{4} + 10 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{2} + 4 \, b^{2}\right )} d^{3} e^{4} x^{3} + 30 \, {\left (3 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{3} + 4 \, b^{2} c\right )} d^{2} e^{4} x^{2} + 15 \, {\left (3 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{4} + 8 \, b^{2} c^{2} + 16 \, b^{2}\right )} d e^{4} x + 225 \, {\left (b^{2} d^{5} e^{4} x^{5} + 5 \, b^{2} c d^{4} e^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} e^{4} x^{3} + 10 \, b^{2} c^{3} d^{2} e^{4} x^{2} + 5 \, b^{2} c^{4} d e^{4} x + b^{2} c^{5} e^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 30 \, {\left (15 \, a b d^{5} e^{4} x^{5} + 75 \, a b c d^{4} e^{4} x^{4} + 150 \, a b c^{2} d^{3} e^{4} x^{3} + 150 \, a b c^{3} d^{2} e^{4} x^{2} + 75 \, a b c^{4} d e^{4} x + 15 \, a b c^{5} e^{4} - {\left (3 \, b^{2} d^{4} e^{4} x^{4} + 12 \, b^{2} c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b^{2} c^{2} + 2 \, b^{2}\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b^{2} c^{3} + 2 \, b^{2} c\right )} d e^{4} x + {\left (3 \, b^{2} c^{4} + 4 \, b^{2} c^{2} + 8 \, b^{2}\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 30 \, {\left (3 \, a b d^{4} e^{4} x^{4} + 12 \, a b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, a b c^{2} + 2 \, a b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, a b c^{3} + 2 \, a b c\right )} d e^{4} x + {\left (3 \, a b c^{4} + 4 \, a b c^{2} + 8 \, a b\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{1125 \, d} \] Input:
integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^2,x, algorithm="fricas")
Output:
1/1125*(9*(25*a^2 + 2*b^2)*d^5*e^4*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*e^4*x^4 + 10*(9*(25*a^2 + 2*b^2)*c^2 + 4*b^2)*d^3*e^4*x^3 + 30*(3*(25*a^2 + 2*b^2 )*c^3 + 4*b^2*c)*d^2*e^4*x^2 + 15*(3*(25*a^2 + 2*b^2)*c^4 + 8*b^2*c^2 + 16 *b^2)*d*e^4*x + 225*(b^2*d^5*e^4*x^5 + 5*b^2*c*d^4*e^4*x^4 + 10*b^2*c^2*d^ 3*e^4*x^3 + 10*b^2*c^3*d^2*e^4*x^2 + 5*b^2*c^4*d*e^4*x + b^2*c^5*e^4)*log( d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 30*(15*a*b*d^5*e^4*x^5 + 75*a*b*c*d^4*e^4*x^4 + 150*a*b*c^2*d^3*e^4*x^3 + 150*a*b*c^3*d^2*e^4*x^2 + 75*a*b*c^4*d*e^4*x + 15*a*b*c^5*e^4 - (3*b^2*d^4*e^4*x^4 + 12*b^2*c*d^3*e ^4*x^3 + 2*(9*b^2*c^2 + 2*b^2)*d^2*e^4*x^2 + 4*(3*b^2*c^3 + 2*b^2*c)*d*e^4 *x + (3*b^2*c^4 + 4*b^2*c^2 + 8*b^2)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 ))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 30*(3*a*b*d^4*e^4*x^ 4 + 12*a*b*c*d^3*e^4*x^3 + 2*(9*a*b*c^2 + 2*a*b)*d^2*e^4*x^2 + 4*(3*a*b*c^ 3 + 2*a*b*c)*d*e^4*x + (3*a*b*c^4 + 4*a*b*c^2 + 8*a*b)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=e^{4} \left (\int a^{2} c^{4}\, dx + \int a^{2} d^{4} x^{4}\, dx + \int b^{2} c^{4} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 a^{2} c d^{3} x^{3}\, dx + \int 6 a^{2} c^{2} d^{2} x^{2}\, dx + \int 4 a^{2} c^{3} d x\, dx + \int b^{2} d^{4} x^{4} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 b^{2} c d^{3} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 6 b^{2} c^{2} d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 b^{2} c^{3} d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 8 a b c d^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 12 a b c^{2} d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 8 a b c^{3} d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**4*(a+b*acosh(d*x+c))**2,x)
Output:
e**4*(Integral(a**2*c**4, x) + Integral(a**2*d**4*x**4, x) + Integral(b**2 *c**4*acosh(c + d*x)**2, x) + Integral(2*a*b*c**4*acosh(c + d*x), x) + Int egral(4*a**2*c*d**3*x**3, x) + Integral(6*a**2*c**2*d**2*x**2, x) + Integr al(4*a**2*c**3*d*x, x) + Integral(b**2*d**4*x**4*acosh(c + d*x)**2, x) + I ntegral(2*a*b*d**4*x**4*acosh(c + d*x), x) + Integral(4*b**2*c*d**3*x**3*a cosh(c + d*x)**2, x) + Integral(6*b**2*c**2*d**2*x**2*acosh(c + d*x)**2, x ) + Integral(4*b**2*c**3*d*x*acosh(c + d*x)**2, x) + Integral(8*a*b*c*d**3 *x**3*acosh(c + d*x), x) + Integral(12*a*b*c**2*d**2*x**2*acosh(c + d*x), x) + Integral(8*a*b*c**3*d*x*acosh(c + d*x), x))
\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^2,x, algorithm="maxima")
Output:
1/5*a^2*d^4*e^4*x^5 + a^2*c*d^3*e^4*x^4 + 2*a^2*c^2*d^2*e^4*x^3 + 2*a^2*c^ 3*d*e^4*x^2 + 2*(2*x^2*arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2 *sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a*b*c^3*d*e^4 + 2/3*(6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^2/ d^2 - 15*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/ d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^ 2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a*b*c^2*d^2*e^4 + 1/12*(24*x^4*arccosh(d*x + c) - (6*sqrt(d^2*x^ 2 + 2*c*d*x + c^2 - 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^ 2/d^3 + 105*c^4*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)* d)/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2 *log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 - 105*sq rt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1 )*c/d^5)*d)*a*b*c*d^3*e^4 + 1/300*(120*x^5*arccosh(d*x + c) - (24*sqrt(d^2 *x^2 + 2*c*d*x + c^2 - 1)*x^4/d^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1...
\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*e*x+c*e)^4*(a+b*arccosh(d*x+c))^2,x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^4*(b*arccosh(d*x + c) + a)^2, x)
Timed out. \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((c*e + d*e*x)^4*(a + b*acosh(c + d*x))^2,x)
Output:
int((c*e + d*e*x)^4*(a + b*acosh(c + d*x))^2, x)
\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e^{4} \left (75 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{4}d x \right ) b^{2} d^{5}-16 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b +15 a^{2} d^{5} x^{5}+150 \mathit {acosh} \left (d x +c \right ) a b \,c^{5}+144 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,c^{4}-8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,c^{2}-120 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a b \,c^{5}+75 a^{2} c^{4} d x +150 a^{2} c^{3} d^{2} x^{2}+150 a^{2} c^{2} d^{3} x^{3}+75 a^{2} c \,d^{4} x^{4}+75 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c^{4} d +300 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{3}d x \right ) b^{2} c \,d^{4}+450 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} c^{2} d^{3}+300 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} c^{3} d^{2}+30 \mathit {acosh} \left (d x +c \right ) a b \,d^{5} x^{5}-6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,d^{4} x^{4}-8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,d^{2} x^{2}-150 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a b \,c^{4}+150 \mathit {acosh} \left (d x +c \right ) a b \,c^{4} d x +300 \mathit {acosh} \left (d x +c \right ) a b \,c^{3} d^{2} x^{2}+300 \mathit {acosh} \left (d x +c \right ) a b \,c^{2} d^{3} x^{3}+150 \mathit {acosh} \left (d x +c \right ) a b c \,d^{4} x^{4}-24 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,c^{3} d x -36 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b \,c^{2} d^{2} x^{2}-24 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b c \,d^{3} x^{3}-16 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b c d x \right )}{75 d} \] Input:
int((d*e*x+c*e)^4*(a+b*acosh(d*x+c))^2,x)
Output:
(e**4*(150*acosh(c + d*x)*a*b*c**5 + 150*acosh(c + d*x)*a*b*c**4*d*x + 300 *acosh(c + d*x)*a*b*c**3*d**2*x**2 + 300*acosh(c + d*x)*a*b*c**2*d**3*x**3 + 150*acosh(c + d*x)*a*b*c*d**4*x**4 + 30*acosh(c + d*x)*a*b*d**5*x**5 + 144*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c**4 - 24*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c**3*d*x - 36*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a *b*c**2*d**2*x**2 - 8*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c**2 - 24*s qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c*d**3*x**3 - 16*sqrt(c**2 + 2*c*d *x + d**2*x**2 - 1)*a*b*c*d*x - 6*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b *d**4*x**4 - 8*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*d**2*x**2 - 16*sqr t(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b - 150*sqrt(c + d*x + 1)*sqrt(c + d*x - 1)*a*b*c**4 + 75*int(acosh(c + d*x)**2,x)*b**2*c**4*d + 75*int(acosh(c + d*x)**2*x**4,x)*b**2*d**5 + 300*int(acosh(c + d*x)**2*x**3,x)*b**2*c*d** 4 + 450*int(acosh(c + d*x)**2*x**2,x)*b**2*c**2*d**3 + 300*int(acosh(c + d *x)**2*x,x)*b**2*c**3*d**2 - 120*log(sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a*b*c**5 + 75*a**2*c**4*d*x + 150*a**2*c**3*d**2*x**2 + 150*a** 2*c**2*d**3*x**3 + 75*a**2*c*d**4*x**4 + 15*a**2*d**5*x**5))/(75*d)