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

Optimal. Leaf size=192 \[ -\frac{3 b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e}-\frac{3 b^3 \text{PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac{2 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e}-\frac{3 b^4 \text{PolyLog}\left (5,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e} \]

[Out]

(a + b*ArcCosh[c + d*x])^5/(5*b*d*e) + ((a + b*ArcCosh[c + d*x])^4*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e) - (
2*b*(a + b*ArcCosh[c + d*x])^3*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(d*e) - (3*b^2*(a + b*ArcCosh[c + d*x])^2
*PolyLog[3, -E^(-2*ArcCosh[c + d*x])])/(d*e) - (3*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[4, -E^(-2*ArcCosh[c + d
*x])])/(d*e) - (3*b^4*PolyLog[5, -E^(-2*ArcCosh[c + d*x])])/(2*d*e)

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Rubi [A]  time = 0.265944, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5866, 12, 5660, 3718, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3 b^2 \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e}+\frac{3 b^3 \text{PolyLog}\left (4,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}+\frac{2 b \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e}-\frac{3 b^4 \text{PolyLog}\left (5,-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a + b*ArcCosh[c + d*x])^5/(5*b*d*e) + ((a + b*ArcCosh[c + d*x])^4*Log[1 + E^(2*ArcCosh[c + d*x])])/(d*e) + (
2*b*(a + b*ArcCosh[c + d*x])^3*PolyLog[2, -E^(2*ArcCosh[c + d*x])])/(d*e) - (3*b^2*(a + b*ArcCosh[c + d*x])^2*
PolyLog[3, -E^(2*ArcCosh[c + d*x])])/(d*e) + (3*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[4, -E^(2*ArcCosh[c + d*x]
)])/(d*e) - (3*b^4*PolyLog[5, -E^(2*ArcCosh[c + d*x])])/(2*d*e)

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 5660

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

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4}{c e+d e x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^4}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^4 \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^4}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{(4 b) \operatorname{Subst}\left (\int (a+b x)^3 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \text{Li}_4\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text{Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac{3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac{3 b^4 \text{Li}_5\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ \end{align*}

Mathematica [A]  time = 0.762255, size = 308, normalized size = 1.6 \[ \frac{-3 b^2 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2-3 a b^3 \text{PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )-2 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3-3 b^4 \cosh ^{-1}(c+d x) \text{PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )-\frac{3}{2} b^4 \text{PolyLog}\left (5,-e^{-2 \cosh ^{-1}(c+d x)}\right )+2 a^2 b^2 \cosh ^{-1}(c+d x)^3+6 a^2 b^2 \cosh ^{-1}(c+d x)^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )+2 a^3 b \cosh ^{-1}(c+d x)^2+4 a^3 b \cosh ^{-1}(c+d x) \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )+a^4 \log (c+d x)+a b^3 \cosh ^{-1}(c+d x)^4+4 a b^3 \cosh ^{-1}(c+d x)^3 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )+\frac{1}{5} b^4 \cosh ^{-1}(c+d x)^5+b^4 \cosh ^{-1}(c+d x)^4 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )}{d e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a^3*b*ArcCosh[c + d*x]^2 + 2*a^2*b^2*ArcCosh[c + d*x]^3 + a*b^3*ArcCosh[c + d*x]^4 + (b^4*ArcCosh[c + d*x]^
5)/5 + 4*a^3*b*ArcCosh[c + d*x]*Log[1 + E^(-2*ArcCosh[c + d*x])] + 6*a^2*b^2*ArcCosh[c + d*x]^2*Log[1 + E^(-2*
ArcCosh[c + d*x])] + 4*a*b^3*ArcCosh[c + d*x]^3*Log[1 + E^(-2*ArcCosh[c + d*x])] + b^4*ArcCosh[c + d*x]^4*Log[
1 + E^(-2*ArcCosh[c + d*x])] + a^4*Log[c + d*x] - 2*b*(a + b*ArcCosh[c + d*x])^3*PolyLog[2, -E^(-2*ArcCosh[c +
d*x])] - 3*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[3, -E^(-2*ArcCosh[c + d*x])] - 3*a*b^3*PolyLog[4, -E^(-2*Ar
cCosh[c + d*x])] - 3*b^4*ArcCosh[c + d*x]*PolyLog[4, -E^(-2*ArcCosh[c + d*x])] - (3*b^4*PolyLog[5, -E^(-2*ArcC
osh[c + d*x])])/2)/(d*e)

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Maple [B]  time = 0.033, size = 727, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*a^4/e*ln(d*x+c)-1/5/d*b^4/e*arccosh(d*x+c)^5+1/d*b^4/e*arccosh(d*x+c)^4*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1
)^(1/2))^2+1)+2/d*b^4/e*arccosh(d*x+c)^3*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3/d*b^4/e*arcco
sh(d*x+c)^2*polylog(3,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/d*b^4/e*arccosh(d*x+c)*polylog(4,-(d*x+c+(
d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3/2/d*b^4/e*polylog(5,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-1/d*a*b^3
/e*arccosh(d*x+c)^4+4/d*a*b^3/e*arccosh(d*x+c)^3*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2+1)+6/d*a*b^3/e*a
rccosh(d*x+c)^2*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-6/d*a*b^3/e*arccosh(d*x+c)*polylog(3,-(d
*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/d*a*b^3/e*polylog(4,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-2/d
*a^2*b^2/e*arccosh(d*x+c)^3+6/d*a^2*b^2/e*arccosh(d*x+c)^2*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2+1)+6/d
*a^2*b^2/e*arccosh(d*x+c)*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3/d*a^2*b^2/e*polylog(3,-(d*x+
c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-2/d*a^3*b/e*arccosh(d*x+c)^2+4/d*a^3*b/e*arccosh(d*x+c)*ln((d*x+c+(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2))^2+1)+2/d*a^3*b/e*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((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)/(d*e*x + c*e), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{4}}{c + d x}\, dx + \int \frac{b^{4} \operatorname{acosh}^{4}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac{4 a b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac{6 a^{2} b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac{4 a^{3} b \operatorname{acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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