\(\int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx\) [142]

Optimal result

Integrand size = 23, antiderivative size = 327 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{16 b^3 d}-\frac {27 e^4 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b^3 d}-\frac {25 e^4 \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d} \]

[Out]

2*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))-5/2*e^4*(d*x+c)^5/b^2/d/(a+b*arccosh(d*x+c))+1/16*e^4*cosh(a/b)*Shi
((a+b*arccosh(d*x+c))/b)/b^3/d+27/32*e^4*cosh(3*a/b)*Shi(3*(a+b*arccosh(d*x+c))/b)/b^3/d+25/32*e^4*cosh(5*a/b)
*Shi(5*(a+b*arccosh(d*x+c))/b)/b^3/d-1/16*e^4*Chi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b^3/d-27/32*e^4*Chi(3*(a+b
*arccosh(d*x+c))/b)*sinh(3*a/b)/b^3/d-25/32*e^4*Chi(5*(a+b*arccosh(d*x+c))/b)*sinh(5*a/b)/b^3/d-1/2*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))^2

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12, 5886, 5951, 5887, 5556, 3384, 3379, 3382} \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{16 b^3 d}-\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}-\frac {25 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{2 b d (a+b \text {arccosh}(c+d x))^2} \]

[In]

Int[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

-1/2*(e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x])^2) + (2*e^4*(c + d*x
)^3)/(b^2*d*(a + b*ArcCosh[c + d*x])) - (5*e^4*(c + d*x)^5)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e^4*CoshInte
gral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(16*b^3*d) - (27*e^4*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*
Sinh[(3*a)/b])/(32*b^3*d) - (25*e^4*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(5*a)/b])/(32*b^3*d) + (
e^4*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(16*b^3*d) + (27*e^4*Cosh[(3*a)/b]*SinhIntegral[(3*(a
+ b*ArcCosh[c + d*x]))/b])/(32*b^3*d) + (25*e^4*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(3
2*b^3*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5886

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] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCosh[c
*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5951

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rule 5996

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^4 x^4}{(a+b \text {arccosh}(x))^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{(a+b \text {arccosh}(x))^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}-\frac {\left (2 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))^2} \, dx,x,c+d x\right )}{b d}+\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))^2} \, dx,x,c+d x\right )}{2 b d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {\left (6 e^4\right ) \text {Subst}\left (\int \frac {x^2}{a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {x^4}{a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{2 b^2 d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}+\frac {\left (6 e^4\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^3 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}+\frac {\left (6 e^4\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b^3 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{32 b^3 d}+\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d}+\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d}-\frac {\left (25 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{16 b^3 d}-\frac {\left (75 e^4\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{32 b^3 d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {\left (3 e^4 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d}+\frac {\left (25 e^4 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{16 b^3 d}-\frac {\left (3 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d}+\frac {\left (75 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{32 b^3 d}+\frac {\left (25 e^4 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{32 b^3 d}+\frac {\left (3 e^4 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d}-\frac {\left (25 e^4 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{16 b^3 d}+\frac {\left (3 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^3 d}-\frac {\left (75 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{32 b^3 d}-\frac {\left (25 e^4 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{32 b^3 d} \\ & = -\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{16 b^3 d}-\frac {27 e^4 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b^3 d}-\frac {25 e^4 \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.99 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=\frac {e^4 \left (-\frac {16 b^2 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^2}+\frac {16 b \left (4 (c+d x)^3-5 (c+d x)^5\right )}{a+b \text {arccosh}(c+d x)}+48 \left (\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+\text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )+25 \left (-2 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{32 b^3 d} \]

[In]

Integrate[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(e^4*((-16*b^2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^2 + (16*b*(4*(c + d*
x)^3 - 5*(c + d*x)^5))/(a + b*ArcCosh[c + d*x]) + 48*(CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] + CoshInt
egral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] - Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] - Cosh[(3*a)/
b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])]) + 25*(-2*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 3*CoshI
ntegral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcCosh[c + d*x])]*Sinh[(5*a)/b] + 2
*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])] + C
osh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c + d*x])])))/(32*b^3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(992\) vs. \(2(307)=614\).

Time = 1.25 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.04

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(993\)
default	\(\text {Expression too large to display}\)	\(993\)

[In]

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/64*(-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)^5-20*(d*x+c)^3+5*d*x+5*c)*e^4*(5*b*arccosh(d*x+c)+5*a-b)/b^2/(b^2*arccosh(
d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+25/64*e^4/b^3*exp(5*a/b)*Ei(1,5*arccosh(d*x+c)+5*a/b)-3/64*(-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*(3*b*arccosh(d*x+c)+3
*a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+27/64*e^4/b^3*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b
)-1/32*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^4*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arc
cosh(d*x+c)+a^2)+1/32*e^4/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/32/b*e^4*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1
/2))/(a+b*arccosh(d*x+c))^2-1/32/b^2*e^4*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/32/b^3
*e^4*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-3/64/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))^2-9/64/b^2*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))-27/64/b^3*e^4*exp(-3*
a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b)-1/64/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))^2-5/64/b^2*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))-25/64/b^3*e^4*exp
(-5*a/b)*Ei(1,-5*arccosh(d*x+c)-5*a/b))

Fricas [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

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^3*arccosh(d*x + c)^3
+ 3*a*b^2*arccosh(d*x + c)^2 + 3*a^2*b*arccosh(d*x + c) + a^3), x)

Sympy [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**4/(a+b*acosh(d*x+c))**3,x)

[Out]

e**4*(Integral(c**4/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)**3), x)
+ Integral(d**4*x**4/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)**3),
x) + Integral(4*c*d**3*x**3/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)
**3), x) + Integral(6*c**2*d**2*x**2/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh
(c + d*x)**3), x) + Integral(4*c**3*d*x/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*ac
osh(c + d*x)**3), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*((5*a*d^11*e^4 + b*d^11*e^4)*x^11 + 11*(5*a*c*d^10*e^4 + b*c*d^10*e^4)*x^10 + (5*(55*c^2*d^9*e^4 - 3*d^9*
e^4)*a + (55*c^2*d^9*e^4 - 3*d^9*e^4)*b)*x^9 + 3*(5*(55*c^3*d^8*e^4 - 9*c*d^8*e^4)*a + (55*c^3*d^8*e^4 - 9*c*d
^8*e^4)*b)*x^8 + 3*(5*(110*c^4*d^7*e^4 - 36*c^2*d^7*e^4 + d^7*e^4)*a + (110*c^4*d^7*e^4 - 36*c^2*d^7*e^4 + d^7
*e^4)*b)*x^7 + 21*(5*(22*c^5*d^6*e^4 - 12*c^3*d^6*e^4 + c*d^6*e^4)*a + (22*c^5*d^6*e^4 - 12*c^3*d^6*e^4 + c*d^
6*e^4)*b)*x^6 + (5*(462*c^6*d^5*e^4 - 378*c^4*d^5*e^4 + 63*c^2*d^5*e^4 - d^5*e^4)*a + (462*c^6*d^5*e^4 - 378*c
^4*d^5*e^4 + 63*c^2*d^5*e^4 - d^5*e^4)*b)*x^5 + (5*(330*c^7*d^4*e^4 - 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 - 5*c*
d^4*e^4)*a + (330*c^7*d^4*e^4 - 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 - 5*c*d^4*e^4)*b)*x^4 + ((5*a*d^8*e^4 + b*d^
8*e^4)*x^8 + 8*(5*a*c*d^7*e^4 + b*c*d^7*e^4)*x^7 + (4*(35*c^2*d^6*e^4 - 2*d^6*e^4)*a + (28*c^2*d^6*e^4 - d^6*e
^4)*b)*x^6 + 2*(4*(35*c^3*d^5*e^4 - 6*c*d^5*e^4)*a + (28*c^3*d^5*e^4 - 3*c*d^5*e^4)*b)*x^5 + ((350*c^4*d^4*e^4
- 120*c^2*d^4*e^4 + 3*d^4*e^4)*a + 5*(14*c^4*d^4*e^4 - 3*c^2*d^4*e^4)*b)*x^4 + 4*((70*c^5*d^3*e^4 - 40*c^3*d^
3*e^4 + 3*c*d^3*e^4)*a + (14*c^5*d^3*e^4 - 5*c^3*d^3*e^4)*b)*x^3 + (2*(70*c^6*d^2*e^4 - 60*c^4*d^2*e^4 + 9*c^2
*d^2*e^4)*a + (28*c^6*d^2*e^4 - 15*c^4*d^2*e^4)*b)*x^2 + (5*c^8*e^4 - 8*c^6*e^4 + 3*c^4*e^4)*a + (c^8*e^4 - c^
6*e^4)*b + 2*(2*(10*c^7*d*e^4 - 12*c^5*d*e^4 + 3*c^3*d*e^4)*a + (4*c^7*d*e^4 - 3*c^5*d*e^4)*b)*x)*(d*x + c + 1
)^(3/2)*(d*x + c - 1)^(3/2) + (5*(165*c^8*d^3*e^4 - 252*c^6*d^3*e^4 + 105*c^4*d^3*e^4 - 10*c^2*d^3*e^4)*a + (1
65*c^8*d^3*e^4 - 252*c^6*d^3*e^4 + 105*c^4*d^3*e^4 - 10*c^2*d^3*e^4)*b)*x^3 + (3*(5*a*d^9*e^4 + b*d^9*e^4)*x^9
+ 27*(5*a*c*d^8*e^4 + b*c*d^8*e^4)*x^8 + ((540*c^2*d^7*e^4 - 31*d^7*e^4)*a + (108*c^2*d^7*e^4 - 5*d^7*e^4)*b)
*x^7 + 7*((180*c^3*d^6*e^4 - 31*c*d^6*e^4)*a + (36*c^3*d^6*e^4 - 5*c*d^6*e^4)*b)*x^6 + ((1890*c^4*d^5*e^4 - 65
1*c^2*d^5*e^4 + 20*d^5*e^4)*a + (378*c^4*d^5*e^4 - 105*c^2*d^5*e^4 + 2*d^5*e^4)*b)*x^5 + (5*(378*c^5*d^4*e^4 -
217*c^3*d^4*e^4 + 20*c*d^4*e^4)*a + (378*c^5*d^4*e^4 - 175*c^3*d^4*e^4 + 10*c*d^4*e^4)*b)*x^4 + ((1260*c^6*d^
3*e^4 - 1085*c^4*d^3*e^4 + 200*c^2*d^3*e^4 - 4*d^3*e^4)*a + (252*c^6*d^3*e^4 - 175*c^4*d^3*e^4 + 20*c^2*d^3*e^
4)*b)*x^3 + ((540*c^7*d^2*e^4 - 651*c^5*d^2*e^4 + 200*c^3*d^2*e^4 - 12*c*d^2*e^4)*a + (108*c^7*d^2*e^4 - 105*c
^5*d^2*e^4 + 20*c^3*d^2*e^4)*b)*x^2 + (15*c^9*e^4 - 31*c^7*e^4 + 20*c^5*e^4 - 4*c^3*e^4)*a + (3*c^9*e^4 - 5*c^
7*e^4 + 2*c^5*e^4)*b + ((135*c^8*d*e^4 - 217*c^6*d*e^4 + 100*c^4*d*e^4 - 12*c^2*d*e^4)*a + (27*c^8*d*e^4 - 35*
c^6*d*e^4 + 10*c^4*d*e^4)*b)*x)*(d*x + c + 1)*(d*x + c - 1) + (5*(55*c^9*d^2*e^4 - 108*c^7*d^2*e^4 + 63*c^5*d^
2*e^4 - 10*c^3*d^2*e^4)*a + (55*c^9*d^2*e^4 - 108*c^7*d^2*e^4 + 63*c^5*d^2*e^4 - 10*c^3*d^2*e^4)*b)*x^2 + (3*(
5*a*d^10*e^4 + b*d^10*e^4)*x^10 + 30*(5*a*c*d^9*e^4 + b*c*d^9*e^4)*x^9 + ((675*c^2*d^8*e^4 - 38*d^8*e^4)*a + (
135*c^2*d^8*e^4 - 7*d^8*e^4)*b)*x^8 + 8*((225*c^3*d^7*e^4 - 38*c*d^7*e^4)*a + (45*c^3*d^7*e^4 - 7*c*d^7*e^4)*b
)*x^7 + (2*(1575*c^4*d^6*e^4 - 532*c^2*d^6*e^4 + 16*d^6*e^4)*a + (630*c^4*d^6*e^4 - 196*c^2*d^6*e^4 + 5*d^6*e^
4)*b)*x^6 + 2*(2*(945*c^5*d^5*e^4 - 532*c^3*d^5*e^4 + 48*c*d^5*e^4)*a + (378*c^5*d^5*e^4 - 196*c^3*d^5*e^4 + 1
5*c*d^5*e^4)*b)*x^5 + ((3150*c^6*d^4*e^4 - 2660*c^4*d^4*e^4 + 480*c^2*d^4*e^4 - 9*d^4*e^4)*a + (630*c^6*d^4*e^
4 - 490*c^4*d^4*e^4 + 75*c^2*d^4*e^4 - d^4*e^4)*b)*x^4 + 4*((450*c^7*d^3*e^4 - 532*c^5*d^3*e^4 + 160*c^3*d^3*e
^4 - 9*c*d^3*e^4)*a + (90*c^7*d^3*e^4 - 98*c^5*d^3*e^4 + 25*c^3*d^3*e^4 - c*d^3*e^4)*b)*x^3 + ((675*c^8*d^2*e^
4 - 1064*c^6*d^2*e^4 + 480*c^4*d^2*e^4 - 54*c^2*d^2*e^4)*a + (135*c^8*d^2*e^4 - 196*c^6*d^2*e^4 + 75*c^4*d^2*e
^4 - 6*c^2*d^2*e^4)*b)*x^2 + (15*c^10*e^4 - 38*c^8*e^4 + 32*c^6*e^4 - 9*c^4*e^4)*a + (3*c^10*e^4 - 7*c^8*e^4 +
5*c^6*e^4 - c^4*e^4)*b + 2*((75*c^9*d*e^4 - 152*c^7*d*e^4 + 96*c^5*d*e^4 - 18*c^3*d*e^4)*a + (15*c^9*d*e^4 -
28*c^7*d*e^4 + 15*c^5*d*e^4 - 2*c^3*d*e^4)*b)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 5*(c^11*e^4 - 3*c^9*e^4
+ 3*c^7*e^4 - c^5*e^4)*a + (c^11*e^4 - 3*c^9*e^4 + 3*c^7*e^4 - c^5*e^4)*b + (5*(11*c^10*d*e^4 - 27*c^8*d*e^4
+ 21*c^6*d*e^4 - 5*c^4*d*e^4)*a + (11*c^10*d*e^4 - 27*c^8*d*e^4 + 21*c^6*d*e^4 - 5*c^4*d*e^4)*b)*x + (5*b*d^11
*e^4*x^11 + 55*b*c*d^10*e^4*x^10 + 5*(55*c^2*d^9*e^4 - 3*d^9*e^4)*b*x^9 + 15*(55*c^3*d^8*e^4 - 9*c*d^8*e^4)*b*
x^8 + 15*(110*c^4*d^7*e^4 - 36*c^2*d^7*e^4 + d^7*e^4)*b*x^7 + 105*(22*c^5*d^6*e^4 - 12*c^3*d^6*e^4 + c*d^6*e^4
)*b*x^6 + 5*(462*c^6*d^5*e^4 - 378*c^4*d^5*e^4 + 63*c^2*d^5*e^4 - d^5*e^4)*b*x^5 + 5*(330*c^7*d^4*e^4 - 378*c^
5*d^4*e^4 + 105*c^3*d^4*e^4 - 5*c*d^4*e^4)*b*x^4 + 5*(165*c^8*d^3*e^4 - 252*c^6*d^3*e^4 + 105*c^4*d^3*e^4 - 10
*c^2*d^3*e^4)*b*x^3 + (5*b*d^8*e^4*x^8 + 40*b*c*d^7*e^4*x^7 + 4*(35*c^2*d^6*e^4 - 2*d^6*e^4)*b*x^6 + 8*(35*c^3
*d^5*e^4 - 6*c*d^5*e^4)*b*x^5 + (350*c^4*d^4*e^4 - 120*c^2*d^4*e^4 + 3*d^4*e^4)*b*x^4 + 4*(70*c^5*d^3*e^4 - 40
*c^3*d^3*e^4 + 3*c*d^3*e^4)*b*x^3 + 2*(70*c^6*d^2*e^4 - 60*c^4*d^2*e^4 + 9*c^2*d^2*e^4)*b*x^2 + 4*(10*c^7*d*e^
4 - 12*c^5*d*e^4 + 3*c^3*d*e^4)*b*x + (5*c^8*e^4 - 8*c^6*e^4 + 3*c^4*e^4)*b)*(d*x + c + 1)^(3/2)*(d*x + c - 1)
^(3/2) + 5*(55*c^9*d^2*e^4 - 108*c^7*d^2*e^4 + 63*c^5*d^2*e^4 - 10*c^3*d^2*e^4)*b*x^2 + (15*b*d^9*e^4*x^9 + 13
5*b*c*d^8*e^4*x^8 + (540*c^2*d^7*e^4 - 31*d^7*e^4)*b*x^7 + 7*(180*c^3*d^6*e^4 - 31*c*d^6*e^4)*b*x^6 + (1890*c^
4*d^5*e^4 - 651*c^2*d^5*e^4 + 20*d^5*e^4)*b*x^5 + 5*(378*c^5*d^4*e^4 - 217*c^3*d^4*e^4 + 20*c*d^4*e^4)*b*x^4 +
(1260*c^6*d^3*e^4 - 1085*c^4*d^3*e^4 + 200*c^2*d^3*e^4 - 4*d^3*e^4)*b*x^3 + (540*c^7*d^2*e^4 - 651*c^5*d^2*e^
4 + 200*c^3*d^2*e^4 - 12*c*d^2*e^4)*b*x^2 + (135*c^8*d*e^4 - 217*c^6*d*e^4 + 100*c^4*d*e^4 - 12*c^2*d*e^4)*b*x
+ (15*c^9*e^4 - 31*c^7*e^4 + 20*c^5*e^4 - 4*c^3*e^4)*b)*(d*x + c + 1)*(d*x + c - 1) + 5*(11*c^10*d*e^4 - 27*c
^8*d*e^4 + 21*c^6*d*e^4 - 5*c^4*d*e^4)*b*x + (15*b*d^10*e^4*x^10 + 150*b*c*d^9*e^4*x^9 + (675*c^2*d^8*e^4 - 38
*d^8*e^4)*b*x^8 + 8*(225*c^3*d^7*e^4 - 38*c*d^7*e^4)*b*x^7 + 2*(1575*c^4*d^6*e^4 - 532*c^2*d^6*e^4 + 16*d^6*e^
4)*b*x^6 + 4*(945*c^5*d^5*e^4 - 532*c^3*d^5*e^4 + 48*c*d^5*e^4)*b*x^5 + (3150*c^6*d^4*e^4 - 2660*c^4*d^4*e^4 +
480*c^2*d^4*e^4 - 9*d^4*e^4)*b*x^4 + 4*(450*c^7*d^3*e^4 - 532*c^5*d^3*e^4 + 160*c^3*d^3*e^4 - 9*c*d^3*e^4)*b*
x^3 + (675*c^8*d^2*e^4 - 1064*c^6*d^2*e^4 + 480*c^4*d^2*e^4 - 54*c^2*d^2*e^4)*b*x^2 + 2*(75*c^9*d*e^4 - 152*c^
7*d*e^4 + 96*c^5*d*e^4 - 18*c^3*d*e^4)*b*x + (15*c^10*e^4 - 38*c^8*e^4 + 32*c^6*e^4 - 9*c^4*e^4)*b)*sqrt(d*x +
c + 1)*sqrt(d*x + c - 1) + 5*(c^11*e^4 - 3*c^9*e^4 + 3*c^7*e^4 - c^5*e^4)*b)*log(d*x + sqrt(d*x + c + 1)*sqrt
(d*x + c - 1) + c))/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 3*(5*c^2*d^5 - d^5)*a^2*b^2*x^4 + 4*(5*c^3*d^4 -
3*c*d^4)*a^2*b^2*x^3 + 3*(5*c^4*d^3 - 6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 6*(c^5*d^2 - 2*c^3*d^2 + c*d^2)*a^2*b^2*x
+ (c^6*d - 3*c^4*d + 3*c^2*d - d)*a^2*b^2 + (a^2*b^2*d^4*x^3 + 3*a^2*b^2*c*d^3*x^2 + 3*a^2*b^2*c^2*d^2*x + a^
2*b^2*c^3*d)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 3*(a^2*b^2*d^5*x^4 + 4*a^2*b^2*c*d^4*x^3 + (6*c^2*d^3 -
d^3)*a^2*b^2*x^2 + 2*(2*c^3*d^2 - c*d^2)*a^2*b^2*x + (c^4*d - c^2*d)*a^2*b^2)*(d*x + c + 1)*(d*x + c - 1) + (
b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 3*(5*c^2*d^5 - d^5)*b^4*x^4 + 4*(5*c^3*d^4 - 3*c*d^4)*b^4*x^3 + 3*(5*c^4*d^3 -
6*c^2*d^3 + d^3)*b^4*x^2 + 6*(c^5*d^2 - 2*c^3*d^2 + c*d^2)*b^4*x + (c^6*d - 3*c^4*d + 3*c^2*d - d)*b^4 + (b^4
*d^4*x^3 + 3*b^4*c*d^3*x^2 + 3*b^4*c^2*d^2*x + b^4*c^3*d)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 3*(b^4*d^5
*x^4 + 4*b^4*c*d^4*x^3 + (6*c^2*d^3 - d^3)*b^4*x^2 + 2*(2*c^3*d^2 - c*d^2)*b^4*x + (c^4*d - c^2*d)*b^4)*(d*x +
c + 1)*(d*x + c - 1) + 3*(b^4*d^6*x^5 + 5*b^4*c*d^5*x^4 + 2*(5*c^2*d^4 - d^4)*b^4*x^3 + 2*(5*c^3*d^3 - 3*c*d^
3)*b^4*x^2 + (5*c^4*d^2 - 6*c^2*d^2 + d^2)*b^4*x + (c^5*d - 2*c^3*d + c*d)*b^4)*sqrt(d*x + c + 1)*sqrt(d*x + c
- 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 + 3*(a^2*b^2*d^6*x^5 + 5*a^2*b^2*c*d^5*x^4 + 2*(5*
c^2*d^4 - d^4)*a^2*b^2*x^3 + 2*(5*c^3*d^3 - 3*c*d^3)*a^2*b^2*x^2 + (5*c^4*d^2 - 6*c^2*d^2 + d^2)*a^2*b^2*x + (
c^5*d - 2*c^3*d + c*d)*a^2*b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 3
*(5*c^2*d^5 - d^5)*a*b^3*x^4 + 4*(5*c^3*d^4 - 3*c*d^4)*a*b^3*x^3 + 3*(5*c^4*d^3 - 6*c^2*d^3 + d^3)*a*b^3*x^2 +
6*(c^5*d^2 - 2*c^3*d^2 + c*d^2)*a*b^3*x + (c^6*d - 3*c^4*d + 3*c^2*d - d)*a*b^3 + (a*b^3*d^4*x^3 + 3*a*b^3*c*
d^3*x^2 + 3*a*b^3*c^2*d^2*x + a*b^3*c^3*d)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 3*(a*b^3*d^5*x^4 + 4*a*b^
3*c*d^4*x^3 + (6*c^2*d^3 - d^3)*a*b^3*x^2 + 2*(2*c^3*d^2 - c*d^2)*a*b^3*x + (c^4*d - c^2*d)*a*b^3)*(d*x + c +
1)*(d*x + c - 1) + 3*(a*b^3*d^6*x^5 + 5*a*b^3*c*d^5*x^4 + 2*(5*c^2*d^4 - d^4)*a*b^3*x^3 + 2*(5*c^3*d^3 - 3*c*d
^3)*a*b^3*x^2 + (5*c^4*d^2 - 6*c^2*d^2 + d^2)*a*b^3*x + (c^5*d - 2*c^3*d + c*d)*a*b^3)*sqrt(d*x + c + 1)*sqrt(
d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)) + integrate(1/2*(25*d^12*e^4*x^12 + 300*c*d^
11*e^4*x^11 + 25*c^12*e^4 - 100*c^10*e^4 + 150*c^8*e^4 + 50*(33*c^2*d^10*e^4 - 2*d^10*e^4)*x^10 - 100*c^6*e^4
+ 500*(11*c^3*d^9*e^4 - 2*c*d^9*e^4)*x^9 + 75*(165*c^4*d^8*e^4 - 60*c^2*d^8*e^4 + 2*d^8*e^4)*x^8 + 25*c^4*e^4
+ 600*(33*c^5*d^7*e^4 - 20*c^3*d^7*e^4 + 2*c*d^7*e^4)*x^7 + 100*(231*c^6*d^6*e^4 - 210*c^4*d^6*e^4 + 42*c^2*d^
6*e^4 - d^6*e^4)*x^6 + 600*(33*c^7*d^5*e^4 - 42*c^5*d^5*e^4 + 14*c^3*d^5*e^4 - c*d^5*e^4)*x^5 + (25*d^8*e^4*x^
8 + 200*c*d^7*e^4*x^7 + 25*c^8*e^4 - 24*c^6*e^4 + 3*c^4*e^4 + 4*(175*c^2*d^6*e^4 - 6*d^6*e^4)*x^6 + 8*(175*c^3
*d^5*e^4 - 18*c*d^5*e^4)*x^5 + (1750*c^4*d^4*e^4 - 360*c^2*d^4*e^4 + 3*d^4*e^4)*x^4 + 4*(350*c^5*d^3*e^4 - 120
*c^3*d^3*e^4 + 3*c*d^3*e^4)*x^3 + 2*(350*c^6*d^2*e^4 - 180*c^4*d^2*e^4 + 9*c^2*d^2*e^4)*x^2 + 4*(50*c^7*d*e^4
- 36*c^5*d*e^4 + 3*c^3*d*e^4)*x)*(d*x + c + 1)^2*(d*x + c - 1)^2 + 25*(495*c^8*d^4*e^4 - 840*c^6*d^4*e^4 + 420
*c^4*d^4*e^4 - 60*c^2*d^4*e^4 + d^4*e^4)*x^4 + (100*d^9*e^4*x^9 + 900*c*d^8*e^4*x^8 + 100*c^9*e^4 - 172*c^7*e^
4 + 87*c^5*e^4 + 4*(900*c^2*d^7*e^4 - 43*d^7*e^4)*x^7 - 12*c^3*e^4 + 28*(300*c^3*d^6*e^4 - 43*c*d^6*e^4)*x^6 +
3*(4200*c^4*d^5*e^4 - 1204*c^2*d^5*e^4 + 29*d^5*e^4)*x^5 + 5*(2520*c^5*d^4*e^4 - 1204*c^3*d^4*e^4 + 87*c*d^4*
e^4)*x^4 + 2*(4200*c^6*d^3*e^4 - 3010*c^4*d^3*e^4 + 435*c^2*d^3*e^4 - 6*d^3*e^4)*x^3 + 6*(600*c^7*d^2*e^4 - 60
2*c^5*d^2*e^4 + 145*c^3*d^2*e^4 - 6*c*d^2*e^4)*x^2 + (900*c^8*d*e^4 - 1204*c^6*d*e^4 + 435*c^4*d*e^4 - 36*c^2*
d*e^4)*x)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 100*(55*c^9*d^3*e^4 - 120*c^7*d^3*e^4 + 84*c^5*d^3*e^4 - 2
0*c^3*d^3*e^4 + c*d^3*e^4)*x^3 + 3*(50*d^10*e^4*x^10 + 500*c*d^9*e^4*x^9 + 50*c^10*e^4 - 124*c^8*e^4 + 105*c^6
*e^4 + 2*(1125*c^2*d^8*e^4 - 62*d^8*e^4)*x^8 - 35*c^4*e^4 + 16*(375*c^3*d^7*e^4 - 62*c*d^7*e^4)*x^7 + 7*(1500*
c^4*d^6*e^4 - 496*c^2*d^6*e^4 + 15*d^6*e^4)*x^6 + 4*c^2*e^4 + 14*(900*c^5*d^5*e^4 - 496*c^3*d^5*e^4 + 45*c*d^5
*e^4)*x^5 + 35*(300*c^6*d^4*e^4 - 248*c^4*d^4*e^4 + 45*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(1500*c^7*d^3*e^4 - 1736
*c^5*d^3*e^4 + 525*c^3*d^3*e^4 - 35*c*d^3*e^4)*x^3 + (2250*c^8*d^2*e^4 - 3472*c^6*d^2*e^4 + 1575*c^4*d^2*e^4 -
210*c^2*d^2*e^4 + 4*d^2*e^4)*x^2 + 2*(250*c^9*d*e^4 - 496*c^7*d*e^4 + 315*c^5*d*e^4 - 70*c^3*d*e^4 + 4*c*d*e^
4)*x)*(d*x + c + 1)*(d*x + c - 1) + 150*(11*c^10*d^2*e^4 - 30*c^8*d^2*e^4 + 28*c^6*d^2*e^4 - 10*c^4*d^2*e^4 +
c^2*d^2*e^4)*x^2 + (100*d^11*e^4*x^11 + 1100*c*d^10*e^4*x^10 + 100*c^11*e^4 - 324*c^9*e^4 + 381*c^7*e^4 + 4*(1
375*c^2*d^9*e^4 - 81*d^9*e^4)*x^9 - 193*c^5*e^4 + 12*(1375*c^3*d^8*e^4 - 243*c*d^8*e^4)*x^8 + 3*(11000*c^4*d^7
*e^4 - 3888*c^2*d^7*e^4 + 127*d^7*e^4)*x^7 + 36*c^3*e^4 + 21*(2200*c^5*d^6*e^4 - 1296*c^3*d^6*e^4 + 127*c*d^6*
e^4)*x^6 + (46200*c^6*d^5*e^4 - 40824*c^4*d^5*e^4 + 8001*c^2*d^5*e^4 - 193*d^5*e^4)*x^5 + (33000*c^7*d^4*e^4 -
40824*c^5*d^4*e^4 + 13335*c^3*d^4*e^4 - 965*c*d^4*e^4)*x^4 + (16500*c^8*d^3*e^4 - 27216*c^6*d^3*e^4 + 13335*c
^4*d^3*e^4 - 1930*c^2*d^3*e^4 + 36*d^3*e^4)*x^3 + (5500*c^9*d^2*e^4 - 11664*c^7*d^2*e^4 + 8001*c^5*d^2*e^4 - 1
930*c^3*d^2*e^4 + 108*c*d^2*e^4)*x^2 + (1100*c^10*d*e^4 - 2916*c^8*d*e^4 + 2667*c^6*d*e^4 - 965*c^4*d*e^4 + 10
8*c^2*d*e^4)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 100*(3*c^11*d*e^4 - 10*c^9*d*e^4 + 12*c^7*d*e^4 - 6*c^5*
d*e^4 + c^3*d*e^4)*x)/(a*b^2*d^8*x^8 + 8*a*b^2*c*d^7*x^7 + 4*(7*c^2*d^6 - d^6)*a*b^2*x^6 + 8*(7*c^3*d^5 - 3*c*
d^5)*a*b^2*x^5 + 2*(35*c^4*d^4 - 30*c^2*d^4 + 3*d^4)*a*b^2*x^4 + 8*(7*c^5*d^3 - 10*c^3*d^3 + 3*c*d^3)*a*b^2*x^
3 + 4*(7*c^6*d^2 - 15*c^4*d^2 + 9*c^2*d^2 - d^2)*a*b^2*x^2 + (a*b^2*d^4*x^4 + 4*a*b^2*c*d^3*x^3 + 6*a*b^2*c^2*
d^2*x^2 + 4*a*b^2*c^3*d*x + a*b^2*c^4)*(d*x + c + 1)^2*(d*x + c - 1)^2 + 8*(c^7*d - 3*c^5*d + 3*c^3*d - c*d)*a
*b^2*x + 4*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + (10*c^2*d^3 - d^3)*a*b^2*x^3 + (10*c^3*d^2 - 3*c*d^2)*a*b^2*x^
2 + (5*c^4*d - 3*c^2*d)*a*b^2*x + (c^5 - c^3)*a*b^2)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + (c^8 - 4*c^6 +
6*c^4 - 4*c^2 + 1)*a*b^2 + 6*(a*b^2*d^6*x^6 + 6*a*b^2*c*d^5*x^5 + (15*c^2*d^4 - 2*d^4)*a*b^2*x^4 + 4*(5*c^3*d^
3 - 2*c*d^3)*a*b^2*x^3 + (15*c^4*d^2 - 12*c^2*d^2 + d^2)*a*b^2*x^2 + 2*(3*c^5*d - 4*c^3*d + c*d)*a*b^2*x + (c^
6 - 2*c^4 + c^2)*a*b^2)*(d*x + c + 1)*(d*x + c - 1) + 4*(a*b^2*d^7*x^7 + 7*a*b^2*c*d^6*x^6 + 3*(7*c^2*d^5 - d^
5)*a*b^2*x^5 + 5*(7*c^3*d^4 - 3*c*d^4)*a*b^2*x^4 + (35*c^4*d^3 - 30*c^2*d^3 + 3*d^3)*a*b^2*x^3 + 3*(7*c^5*d^2
- 10*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 + (7*c^6*d - 15*c^4*d + 9*c^2*d - d)*a*b^2*x + (c^7 - 3*c^5 + 3*c^3 - c)*a*b
^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^3*d^8*x^8 + 8*b^3*c*d^7*x^7 + 4*(7*c^2*d^6 - d^6)*b^3*x^6 + 8*(7*
c^3*d^5 - 3*c*d^5)*b^3*x^5 + 2*(35*c^4*d^4 - 30*c^2*d^4 + 3*d^4)*b^3*x^4 + 8*(7*c^5*d^3 - 10*c^3*d^3 + 3*c*d^3
)*b^3*x^3 + 4*(7*c^6*d^2 - 15*c^4*d^2 + 9*c^2*d^2 - d^2)*b^3*x^2 + (b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*
d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*(d*x + c + 1)^2*(d*x + c - 1)^2 + 8*(c^7*d - 3*c^5*d + 3*c^3*d - c*d)*b^3*x
+ 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + (10*c^2*d^3 - d^3)*b^3*x^3 + (10*c^3*d^2 - 3*c*d^2)*b^3*x^2 + (5*c^4*d -
3*c^2*d)*b^3*x + (c^5 - c^3)*b^3)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + (c^8 - 4*c^6 + 6*c^4 - 4*c^2 + 1)
*b^3 + 6*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + (15*c^2*d^4 - 2*d^4)*b^3*x^4 + 4*(5*c^3*d^3 - 2*c*d^3)*b^3*x^3 + (15
*c^4*d^2 - 12*c^2*d^2 + d^2)*b^3*x^2 + 2*(3*c^5*d - 4*c^3*d + c*d)*b^3*x + (c^6 - 2*c^4 + c^2)*b^3)*(d*x + c +
1)*(d*x + c - 1) + 4*(b^3*d^7*x^7 + 7*b^3*c*d^6*x^6 + 3*(7*c^2*d^5 - d^5)*b^3*x^5 + 5*(7*c^3*d^4 - 3*c*d^4)*b
^3*x^4 + (35*c^4*d^3 - 30*c^2*d^3 + 3*d^3)*b^3*x^3 + 3*(7*c^5*d^2 - 10*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (7*c^6*d -
15*c^4*d + 9*c^2*d - d)*b^3*x + (c^7 - 3*c^5 + 3*c^3 - c)*b^3)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x +
sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)), x)

Giac [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^3, x)