\(\int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx\) [150]

Optimal result

Integrand size = 23, antiderivative size = 352 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d} \]

[Out]

1/3*e^2*(d*x+c)/b^2/d/(a+b*arccosh(d*x+c))^2-1/2*e^2*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))^2+1/24*e^2*Chi((a+b*
arccosh(d*x+c))/b)*cosh(a/b)/b^4/d+9/8*e^2*Chi(3*(a+b*arccosh(d*x+c))/b)*cosh(3*a/b)/b^4/d-1/24*e^2*Shi((a+b*a
rccosh(d*x+c))/b)*sinh(a/b)/b^4/d-9/8*e^2*Shi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b^4/d-1/3*e^2*(d*x+c)^2*(d
*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^3+1/3*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*ar
ccosh(d*x+c))-3/2*e^2*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+c))

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5996, 12, 5886, 5951, 5885, 3384, 3379, 3382, 5880, 5953} \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b d (a+b \text {arccosh}(c+d x))^3} \]

[In]

Int[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^4,x]

[Out]

-1/3*(e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x])^3) + (e^2*(c + d*x))
/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcCosh[c + d*x])^2) + (e^2*Sqrt[-1
+ c + d*x]*Sqrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) - (3*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[
1 + c + d*x])/(2*b^3*d*(a + b*ArcCosh[c + d*x])) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(2
4*b^4*d) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(8*b^4*d) - (e^2*Sinh[a/b]*SinhI
ntegral[(a + b*ArcCosh[c + d*x])/b])/(24*b^4*d) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]
))/b])/(8*b^4*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 5880

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c
*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[
-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5885

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[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] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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 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 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

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^2 x^2}{(a+b \text {arccosh}(x))^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{(a+b \text {arccosh}(x))^4} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}-\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))^3} \, dx,x,c+d x\right )}{3 b d}+\frac {e^2 \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))^3} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 \text {Subst}\left (\int \frac {1}{(a+b \text {arccosh}(x))^2} \, dx,x,c+d x\right )}{3 b^2 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^2}{(a+b \text {arccosh}(x))^2} \, dx,x,c+d x\right )}{2 b^2 d} \\ & = -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{2 b^4 d}-\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} (a+b \text {arccosh}(x))} \, dx,x,c+d x\right )}{3 b^3 d} \\ & = -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {e^2 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{3 b^4 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{8 b^4 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{8 b^4 d} \\ & = -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {\left (e^2 \cosh \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 )}{3 b^4 d}+\frac {\left (3 e^2 \cosh \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 )}{8 b^4 d}+\frac {\left (9 e^2 \cosh \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 )}{8 b^4 d}+\frac {\left (e^2 \sinh \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 )}{3 b^4 d}-\frac {\left (3 e^2 \sinh \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 )}{8 b^4 d}-\frac {\left (9 e^2 \sinh \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 )}{8 b^4 d} \\ & = -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.77 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\frac {e^2 \left (-\frac {8 b^3 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^3}+\frac {4 b^2 \left (2 (c+d x)-3 (c+d x)^3\right )}{(a+b \text {arccosh}(c+d x))^2}-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-2+9 (c+d x)^2\right )}{a+b \text {arccosh}(c+d x)}-80 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+80 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+27 \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{24 b^4 d} \]

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(e^2*((-8*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^3 + (4*b^2*(2*(c + d*
x) - 3*(c + d*x)^3))/(a + b*ArcCosh[c + d*x])^2 - (4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-2 + 9*(c + d*x)^
2))/(a + b*ArcCosh[c + d*x]) - 80*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d*x]] + 80*Sinh[a/b]*SinhIntegral[a
/b + ArcCosh[c + d*x]] + 27*(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])])))/(24*b^4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(776\) vs. \(2(322)=644\).

Time = 0.65 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.21

method	result	size
derivativedivides	\(\frac {\frac {\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^{2} \left (9 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arccosh}\left (d x +c \right )-3 b^{2} \operatorname {arccosh}\left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \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 )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \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 )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \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 )}{16 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\)	\(777\)
default	\(\frac {\frac {\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^{2} \left (9 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arccosh}\left (d x +c \right )-3 b^{2} \operatorname {arccosh}\left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \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 )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \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 )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \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 )}{16 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\)	\(777\)

[In]

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/48*(-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^2*(9*b^2*arccosh(d*x+c)^2+18*a*b*arccosh(d*x+c)-3*b^2*arccosh(d*x+c)+9*a^2-3*a*b+2*b^2)/b^3/(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)-9/16*e^2/b^4*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/
b)+1/48*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^2*(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)-b^2*arccosh(d*
x+c)+a^2-a*b+2*b^2)/b^3/(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)-1/48*e^2/b^
4*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/24/b*e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^3-
1/48/b^2*e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-1/48/b^3*e^2*(d*x+c+(d*x+c-1)^(1/2
)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/48/b^4*e^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/24/b*e^2*(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))
^3-1/16/b^2*e^2*(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))^2-3/16/b^3*e^2*(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))-9/16/b^4*e^2*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))

Fricas [F]

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

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^
2*arccosh(d*x + c)^2 + 4*a^3*b*arccosh(d*x + c) + a^4), x)

Sympy [F]

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

[In]

integrate((d*e*x+c*e)**2/(a+b*acosh(d*x+c))**4,x)

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

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

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int((c*e + d*e*x)^2/(a + b*acosh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^2/(a + b*acosh(c + d*x))^4, x)