Integrand size = 23, antiderivative size = 263 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 b^2 d}+\frac {5 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 b^2 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 b^2 d} \]
1/8*e^4*Chi((a+b*arccosh(d*x+c))/b)*cosh(a/b)/b^2/d+9/16*e^4*Chi(3*(a+b*ar ccosh(d*x+c))/b)*cosh(3*a/b)/b^2/d+5/16*e^4*Chi(5*(a+b*arccosh(d*x+c))/b)* cosh(5*a/b)/b^2/d-1/8*e^4*Shi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b^2/d-9/16 *e^4*Shi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b^2/d-5/16*e^4*Shi(5*(a+b*a rccosh(d*x+c))/b)*sinh(5*a/b)/b^2/d-e^4*(d*x+c)^4*(d*x+c-1)^(1/2)*(d*x+c+1 )^(1/2)/b/d/(a+b*arccosh(d*x+c))
Time = 2.07 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.11 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {e^4 \left (-\frac {16 b (c+d x)^4 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{a+b \text {arccosh}(c+d x)}-16 \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )+5 \left (10 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-10 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-5 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{16 b^2 d} \]
(e^4*((-16*b*(c + d*x)^4*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)) /(a + b*ArcCosh[c + d*x]) - 16*(3*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d*x]] + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c + d*x])] - 3*Sinh[a /b]*SinhIntegral[a/b + ArcCosh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a /b + ArcCosh[c + d*x])]) + 5*(10*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d*x]] + 5*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c + d*x])] + Cosh[(5 *a)/b]*CoshIntegral[5*(a/b + ArcCosh[c + d*x])] - 10*Sinh[a/b]*SinhIntegra l[a/b + ArcCosh[c + d*x]] - 5*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[ c + d*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c + d*x])])))/(16 *b^2*d)
Time = 0.54 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6411, 27, 6300, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {e^4 \left (-\frac {\int \left (-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 (a+b \text {arccosh}(c+d x))}-\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (-\frac {-\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {9}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {5}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {9}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {5}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
(e^4*(-((Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*(a + b*ArcCo sh[c + d*x]))) - (-1/8*(Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b] ) - (9*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/16 - (5 *Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/16 + (Sinh[a/ b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/8 + (9*Sinh[(3*a)/b]*SinhInte gral[(3*(a + b*ArcCosh[c + d*x]))/b])/16 + (5*Sinh[(5*a)/b]*SinhIntegral[( 5*(a + b*ArcCosh[c + d*x]))/b])/16)/b^2))/d
3.2.36.3.1 Defintions of rubi rules used
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_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(664\) vs. \(2(247)=494\).
Time = 1.43 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.53
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-16 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}+12 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+16 \left (d x +c \right )^{5}-20 \left (d x +c \right )^{3}+5 d x +5 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {5 e^{4} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (d x +c \right )+\frac {5 a}{b}\right )}{32 b^{2}}+\frac {3 \left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{4} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{4}}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {e^{4} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{2}}-\frac {3 e^{4} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {e^{4} \left (16 \left (d x +c \right )^{5}-20 \left (d x +c \right )^{3}+16 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}+5 d x +5 c -12 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {5 e^{4} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (d x +c \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{d}\) | \(665\) |
| default | \(\frac {\frac {\left (-16 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}+12 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+16 \left (d x +c \right )^{5}-20 \left (d x +c \right )^{3}+5 d x +5 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {5 e^{4} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (d x +c \right )+\frac {5 a}{b}\right )}{32 b^{2}}+\frac {3 \left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{4}}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{4} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{4}}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {e^{4} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{2}}-\frac {3 e^{4} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {e^{4} \left (16 \left (d x +c \right )^{5}-20 \left (d x +c \right )^{3}+16 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{4}+5 d x +5 c -12 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{32 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {5 e^{4} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (d x +c \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{d}\) | \(665\) |
1/d*(1/32*(-16*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^4+12*(d*x+c)^2*(d*x +c-1)^(1/2)*(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+16*(d*x+c)^5-2 0*(d*x+c)^3+5*d*x+5*c)*e^4/b/(a+b*arccosh(d*x+c))-5/32*e^4/b^2*exp(5*a/b)* Ei(1,5*arccosh(d*x+c)+5*a/b)+3/32*(-4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^ (1/2)+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c)*e^4/b/(a+b*ar ccosh(d*x+c))-9/32*e^4/b^2*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/16*(- (d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^4/b/(a+b*arccosh(d*x+c))-1/16*e^4 /b^2*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/16/b*e^4*(d*x+c+(d*x+c-1)^(1/2)*( d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/16/b^2*e^4*exp(-a/b)*Ei(1,-arccosh( d*x+c)-a/b)-3/32/b*e^4*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c)^2*(d*x+c-1)^(1/2)* (d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-9/32 /b^2*e^4*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b)-1/32/b*e^4*(16*(d*x+c)^ 5-20*(d*x+c)^3+16*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^4+5*d*x+5*c-12*( d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/ (a+b*arccosh(d*x+c))-5/32/b^2*e^4*exp(-5*a/b)*Ei(1,-5*arccosh(d*x+c)-5*a/b ))
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
integral((d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4* x + c^4*e^4)/(b^2*arccosh(d*x + c)^2 + 2*a*b*arccosh(d*x + c) + a^2), x)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d*x)**2) , x) + Integral(d**4*x**4/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d* x)**2), x) + Integral(4*c*d**3*x**3/(a**2 + 2*a*b*acosh(c + d*x) + b**2*ac osh(c + d*x)**2), x) + Integral(6*c**2*d**2*x**2/(a**2 + 2*a*b*acosh(c + d *x) + b**2*acosh(c + d*x)**2), x) + Integral(4*c**3*d*x/(a**2 + 2*a*b*acos h(c + d*x) + b**2*acosh(c + d*x)**2), x))
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
-(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4 - c^5*e^4 + (21*c^2*d^5*e^4 - d^ 5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 - c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 - 2*c^2* d^3*e^4)*x^3 + (21*c^5*d^2*e^4 - 10*c^3*d^2*e^4)*x^2 + (d^6*e^4*x^6 + 6*c* d^5*e^4*x^5 + c^6*e^4 - c^4*e^4 + (15*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(5*c^ 3*d^3*e^4 - c*d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^4 - 2*c^2*d^2*e^4)*x^2 + 2*(3* c^5*d*e^4 - 2*c^3*d*e^4)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (7*c^6*d *e^4 - 5*c^4*d*e^4)*x)/(a*b*d^3*x^2 + 2*a*b*c*d^2*x + (c^2*d - d)*a*b + (a *b*d^2*x + a*b*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^3*x^2 + 2 *b^2*c*d^2*x + (c^2*d - d)*b^2 + (b^2*d^2*x + b^2*c*d)*sqrt(d*x + c + 1)*s qrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)) + in tegrate((5*d^8*e^4*x^8 + 40*c*d^7*e^4*x^7 + 5*c^8*e^4 - 10*c^6*e^4 + 5*c^4 *e^4 + 10*(14*c^2*d^6*e^4 - d^6*e^4)*x^6 + 20*(14*c^3*d^5*e^4 - 3*c*d^5*e^ 4)*x^5 + 5*(70*c^4*d^4*e^4 - 30*c^2*d^4*e^4 + d^4*e^4)*x^4 + 20*(14*c^5*d^ 3*e^4 - 10*c^3*d^3*e^4 + c*d^3*e^4)*x^3 + (5*d^6*e^4*x^6 + 30*c*d^5*e^4*x^ 5 + 5*c^6*e^4 - 3*c^4*e^4 + 3*(25*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(25*c^3*d ^3*e^4 - 3*c*d^3*e^4)*x^3 + 3*(25*c^4*d^2*e^4 - 6*c^2*d^2*e^4)*x^2 + 6*(5* c^5*d*e^4 - 2*c^3*d*e^4)*x)*(d*x + c + 1)*(d*x + c - 1) + 10*(14*c^6*d^2*e ^4 - 15*c^4*d^2*e^4 + 3*c^2*d^2*e^4)*x^2 + (10*d^7*e^4*x^7 + 70*c*d^6*e^4* x^6 + 10*c^7*e^4 - 13*c^5*e^4 + 4*c^3*e^4 + (210*c^2*d^5*e^4 - 13*d^5*e^4) *x^5 + 5*(70*c^3*d^4*e^4 - 13*c*d^4*e^4)*x^4 + 2*(175*c^4*d^3*e^4 - 65*...
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \]