3.148 \(\int \frac{(c e+d e x)^4}{(a+b \cosh ^{-1}(c+d x))^4} \, dx\)

Optimal. Leaf size=431 \[ \frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{48 b^4 d}+\frac{27 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^4 d}+\frac{125 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{96 b^4 d}-\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{48 b^4 d}-\frac{27 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^4 d}-\frac{125 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{96 b^4 d}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{25 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{2 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]

[Out]

-(e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3) + (2*e^4*(c + d*x)^
3)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcCosh[c + d*x])^2) + (2*e^4*Sq
rt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b^3*d*(a + b*ArcCosh[c + d*x])) - (25*e^4*Sqrt[-1 + c + d*x]*
(c + d*x)^4*Sqrt[1 + c + d*x])/(6*b^3*d*(a + b*ArcCosh[c + d*x])) + (e^4*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh
[c + d*x])/b])/(48*b^4*d) + (27*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(32*b^4*d) + (
125*e^4*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(96*b^4*d) - (e^4*Sinh[a/b]*SinhIntegral[(
a + b*ArcCosh[c + d*x])/b])/(48*b^4*d) - (27*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(
32*b^4*d) - (125*e^4*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(96*b^4*d)

________________________________________________________________________________________

Rubi [A]  time = 1.10399, antiderivative size = 427, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5866, 12, 5668, 5775, 5666, 3303, 3298, 3301} \[ \frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}+\frac{27 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}+\frac{125 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}-\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}-\frac{27 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}-\frac{125 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{25 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{2 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3) + (2*e^4*(c + d*x)^
3)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcCosh[c + d*x])^2) + (2*e^4*Sq
rt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b^3*d*(a + b*ArcCosh[c + d*x])) - (25*e^4*Sqrt[-1 + c + d*x]*
(c + d*x)^4*Sqrt[1 + c + d*x])/(6*b^3*d*(a + b*ArcCosh[c + d*x])) + (e^4*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[
c + d*x]])/(48*b^4*d) + (27*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcCosh[c + d*x]])/(32*b^4*d) + (125*e^
4*Cosh[(5*a)/b]*CoshIntegral[(5*a)/b + 5*ArcCosh[c + d*x]])/(96*b^4*d) - (e^4*Sinh[a/b]*SinhIntegral[a/b + Arc
Cosh[c + d*x]])/(48*b^4*d) - (27*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c + d*x]])/(32*b^4*d) - (1
25*e^4*Sinh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c + d*x]])/(96*b^4*d)

Rule 5866

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
]

Rule 12

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

Rule 5668

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 5775

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*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), 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, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5666

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

Rule 3303

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 3298

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 3301

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]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^4 x^4}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac{\left (4 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac{\left (5 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b^2 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{6 b^2 d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{25 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 (a+b x)}-\frac{3 \cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^3 d}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 (a+b x)}-\frac{9 \cosh (3 x)}{16 (a+b x)}-\frac{5 \cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{25 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^4 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 b^3 d}+\frac{\left (125 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{96 b^3 d}-\frac{\left (3 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (75 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(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}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{25 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{\left (e^4 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (25 e^4 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 b^3 d}-\frac{\left (3 e^4 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (75 e^4 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^3 d}+\frac{\left (125 e^4 \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{96 b^3 d}+\frac{\left (e^4 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}-\frac{\left (25 e^4 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 b^3 d}+\frac{\left (3 e^4 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}-\frac{\left (75 e^4 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac{\left (125 e^4 \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{96 b^3 d}\\ &=-\frac{e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{25 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^4 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}+\frac{27 e^4 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}+\frac{125 e^4 \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}-\frac{e^4 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}-\frac{27 e^4 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}-\frac{125 e^4 \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}\\ \end{align*}

Mathematica [A]  time = 1.99598, size = 424, normalized size = 0.98 \[ \frac{e^4 \left (-\frac{32 b^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{16 b^2 \left (4 (c+d x)^3-5 (c+d x)^5\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+384 \left (\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-544 \left (3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )+125 \left (10 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+5 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-10 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-5 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )-\frac{16 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (25 (c+d x)^4-12 (c+d x)^2\right )}{a+b \cosh ^{-1}(c+d x)}\right )}{96 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*((-32*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^3 + (16*b^2*(4*(c +
d*x)^3 - 5*(c + d*x)^5))/(a + b*ArcCosh[c + d*x])^2 - (16*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-12*(c + d*x
)^2 + 25*(c + d*x)^4))/(a + b*ArcCosh[c + d*x]) + 384*(Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d*x]] - Sinh[a
/b]*SinhIntegral[a/b + ArcCosh[c + d*x]]) - 544*(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])]) + 125*(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])] - 1
0*Sinh[a/b]*SinhIntegral[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])])))/(96*b^4*d)

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Maple [B]  time = 0.272, size = 1375, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/192*(-16*(d*x+c)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+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*(25*b^2*arccosh(d*x+c)^2+50*a*b*arccosh(d*x+
c)-5*arccosh(d*x+c)*b^2+25*a^2-5*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)-125/192*e^4/b^4*exp(5*a/b)*Ei(1,5*arccosh(d*x+c)+5*a/b)+1/64*(-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*(9*b^2*arccosh(d*x+c)^2+18*a*b*arccosh(d*
x+c)-3*arccosh(d*x+c)*b^2+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*arccos
h(d*x+c)+a^3)-27/64*e^4/b^4*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/96*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x
+c)*e^4*(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)-arccosh(d*x+c)*b^2+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)-1/96*e^4/b^4*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/48*e^4/
b*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^3-1/96*e^4/b^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)
^(1/2))/(a+b*arccosh(d*x+c))^2-1/96*e^4/b^3*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/96*
e^4/b^4*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/32*e^4/b*(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))^3-3/64*e^4/b^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))^2-9/64*e^4/b^3*(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*e^4/b^4*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b)-1/96*e^4/b*(16*(d*x+c)^5-20*(d*x+c)^3+16*(d*x+
c)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+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))^3-5/192*e^4/b^2*(16*(d*x+c)^5-20*(d*x+c)^3+16*(d*x+c)^4*(d*x+c-1)^(1/2)*(d
*x+c+1)^(1/2)+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*arc
cosh(d*x+c))^2-25/192*e^4/b^3*(16*(d*x+c)^5-20*(d*x+c)^3+16*(d*x+c)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+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))-125/192*e
^4/b^4*exp(-5*a/b)*Ei(1,-5*arccosh(d*x+c)-5*a/b))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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^{4} \operatorname{arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname{arcosh}\left (d x + c\right ) + a^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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