3.27 \(\int \frac{(d+e x)^3}{a+b \cosh ^{-1}(c x)} \, dx\)

Optimal. Leaf size=394 \[ -\frac{3 d^2 e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac{3 d e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac{3 d e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}-\frac{e^3 \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}-\frac{e^3 \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}+\frac{3 d^2 e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac{3 d e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{3 d e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{e^3 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}+\frac{e^3 \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}-\frac{d^3 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{d^3 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]

[Out]

-((d^3*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(b*c)) - (3*d*e^2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b
])/(4*b*c^3) - (3*d^2*e*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(2*b*c^2) - (e^3*CoshIntegral[(2
*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(4*b*c^4) - (3*d*e^2*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/
b])/(4*b*c^3) - (e^3*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]]*Sinh[(4*a)/b])/(8*b*c^4) + (d^3*Cosh[a/b]*SinhInte
gral[a/b + ArcCosh[c*x]])/(b*c) + (3*d*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(4*b*c^3) + (3*d^2*e*Co
sh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(2*b*c^2) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*Ar
cCosh[c*x]])/(4*b*c^4) + (3*d*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(4*b*c^3) + (e^3*Cosh[
(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(8*b*c^4)

________________________________________________________________________________________

Rubi [A]  time = 1.17025, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5806, 6742, 3303, 3298, 3301, 5448, 12} \[ -\frac{3 d^2 e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac{3 d e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac{3 d e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}-\frac{e^3 \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}-\frac{e^3 \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}+\frac{3 d^2 e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac{3 d e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{3 d e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{e^3 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}+\frac{e^3 \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}-\frac{d^3 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac{d^3 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^3*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(b*c)) - (3*d*e^2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b
])/(4*b*c^3) - (3*d^2*e*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(2*b*c^2) - (e^3*CoshIntegral[(2
*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(4*b*c^4) - (3*d*e^2*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/
b])/(4*b*c^3) - (e^3*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]]*Sinh[(4*a)/b])/(8*b*c^4) + (d^3*Cosh[a/b]*SinhInte
gral[a/b + ArcCosh[c*x]])/(b*c) + (3*d*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(4*b*c^3) + (3*d^2*e*Co
sh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(2*b*c^2) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*Ar
cCosh[c*x]])/(4*b*c^4) + (3*d*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(4*b*c^3) + (e^3*Cosh[
(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(8*b*c^4)

Rule 5806

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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]

Rule 5448

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 12

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

Rubi steps

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

Mathematica [A]  time = 0.567684, size = 287, normalized size = 0.73 \[ \frac{-2 c d \sinh \left (\frac{a}{b}\right ) \left (4 c^2 d^2+3 e^2\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-2 e \sinh \left (\frac{2 a}{b}\right ) \left (6 c^2 d^2+e^2\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+12 c^2 d^2 e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+8 c^3 d^3 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-6 c d e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-e^3 \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+6 c d e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+6 c d e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 e^3 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+e^3 \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )}{8 b c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*d*(4*c^2*d^2 + 3*e^2)*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 2*e*(6*c^2*d^2 + e^2)*CoshIntegral[2*
(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - 6*c*d*e^2*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - e^3*CoshI
ntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + 8*c^3*d^3*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 6*c*d*e
^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 12*c^2*d^2*e*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])
] + 2*e^3*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 6*c*d*e^2*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + A
rcCosh[c*x])] + e^3*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])])/(8*b*c^4)

________________________________________________________________________________________

Maple [A]  time = 0.141, size = 394, normalized size = 1. \begin{align*}{\frac{1}{c} \left ( -{\frac{{e}^{3}}{16\,{c}^{3}b}{{\rm e}^{-4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-4\,{\rm arccosh} \left (cx\right )-4\,{\frac{a}{b}} \right ) }+{\frac{{e}^{3}}{16\,{c}^{3}b}{{\rm e}^{4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,4\,{\rm arccosh} \left (cx\right )+4\,{\frac{a}{b}} \right ) }+{\frac{3\,e{d}^{2}}{4\,bc}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (cx\right )+2\,{\frac{a}{b}} \right ) }+{\frac{{e}^{3}}{8\,{c}^{3}b}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (cx\right )+2\,{\frac{a}{b}} \right ) }-{\frac{3\,e{d}^{2}}{4\,bc}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (cx\right )-2\,{\frac{a}{b}} \right ) }-{\frac{{e}^{3}}{8\,{c}^{3}b}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (cx\right )-2\,{\frac{a}{b}} \right ) }-{\frac{3\,d{e}^{2}}{8\,{c}^{2}b}{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ) }+{\frac{{d}^{3}}{2\,b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }+{\frac{3\,d{e}^{2}}{8\,{c}^{2}b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }-{\frac{{d}^{3}}{2\,b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }-{\frac{3\,d{e}^{2}}{8\,{c}^{2}b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }+{\frac{3\,d{e}^{2}}{8\,{c}^{2}b}{{\rm e}^{3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-1/16/c^3*e^3/b*exp(-4*a/b)*Ei(1,-4*arccosh(c*x)-4*a/b)+1/16/c^3*e^3/b*exp(4*a/b)*Ei(1,4*arccosh(c*x)+4*a
/b)+3/4/c*e/b*exp(2*a/b)*Ei(1,2*arccosh(c*x)+2*a/b)*d^2+1/8/c^3*e^3/b*exp(2*a/b)*Ei(1,2*arccosh(c*x)+2*a/b)-3/
4/c*e/b*exp(-2*a/b)*Ei(1,-2*arccosh(c*x)-2*a/b)*d^2-1/8/c^3*e^3/b*exp(-2*a/b)*Ei(1,-2*arccosh(c*x)-2*a/b)-3/8/
c^2*d*e^2/b*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b)+1/2*d^3/b*exp(a/b)*Ei(1,arccosh(c*x)+a/b)+3/8/c^2*d/b*exp(
a/b)*Ei(1,arccosh(c*x)+a/b)*e^2-1/2*d^3/b*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)-3/8/c^2*d/b*exp(-a/b)*Ei(1,-arccos
h(c*x)-a/b)*e^2+3/8/c^2*d*e^2/b*exp(3*a/b)*Ei(1,3*arccosh(c*x)+3*a/b))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3}}{a + b \operatorname{acosh}{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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