\(\int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^4} \, dx\) [66]

Optimal result

Integrand size = 23, antiderivative size = 431 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {2 e^4 (c+d x)^3}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b^3 d (a+b \text {arccosh}(c+d x))}-\frac {25 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{6 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{48 b^4 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^4 d}+\frac {125 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{96 b^4 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{48 b^4 d}-\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^4 d}-\frac {125 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{96 b^4 d} \] Output:

-1/3*e^4*(d*x+c-1)^(1/2)*(d*x+c)^4*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c) 
)^3+2/3*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))^2-5/6*e^4*(d*x+c)^5/b^2/d 
/(a+b*arccosh(d*x+c))^2+2*e^4*(d*x+c-1)^(1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)/b^ 
3/d/(a+b*arccosh(d*x+c))-25/6*e^4*(d*x+c-1)^(1/2)*(d*x+c)^4*(d*x+c+1)^(1/2 
)/b^3/d/(a+b*arccosh(d*x+c))+1/48*e^4*cosh(a/b)*Chi((a+b*arccosh(d*x+c))/b 
)/b^4/d+27/32*e^4*cosh(3*a/b)*Chi(3*(a+b*arccosh(d*x+c))/b)/b^4/d+125/96*e 
^4*cosh(5*a/b)*Chi(5*(a+b*arccosh(d*x+c))/b)/b^4/d-1/48*e^4*sinh(a/b)*Shi( 
(a+b*arccosh(d*x+c))/b)/b^4/d-27/32*e^4*sinh(3*a/b)*Shi(3*(a+b*arccosh(d*x 
+c))/b)/b^4/d-125/96*e^4*sinh(5*a/b)*Shi(5*(a+b*arccosh(d*x+c))/b)/b^4/d
 

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.98 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^4} \, dx=\frac {e^4 \left (-\frac {32 b^3 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^3}+\frac {16 b^2 \left (4 (c+d x)^3-5 (c+d x)^5\right )}{(a+b \text {arccosh}(c+d x))^2}-\frac {16 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-12 (c+d x)^2+25 (c+d x)^4\right )}{a+b \text {arccosh}(c+d x)}+384 \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-544 \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )+125 \left (10 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-10 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-5 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{96 b^4 d} \] Input:

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

Output:

(e^4*((-32*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(a + b*Ar 
cCosh[c + d*x])^3 + (16*b^2*(4*(c + d*x)^3 - 5*(c + d*x)^5))/(a + b*ArcCos 
h[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]*CoshIntegra 
l[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])] - 10*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)
 

Rubi [A] (verified)

Time = 2.34 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6411, 27, 6301, 6366, 6300, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\)

\(\Big \downarrow \) 6301

\(\displaystyle \frac {e^4 \left (-\frac {4 \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}+\frac {5 \int \frac {(c+d x)^5}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\)

\(\Big \downarrow \) 6366

\(\displaystyle \frac {e^4 \left (-\frac {4 \left (\frac {3 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {5 \left (\frac {5 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^5}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\)

\(\Big \downarrow \) 6300

\(\displaystyle \frac {e^4 \left (-\frac {4 \left (\frac {3 \left (-\frac {\int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {5 \left (\frac {5 \left (-\frac {\int \left (-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 (a+b \text {arccosh}(c+d x))}-\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^5}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (-\frac {4 \left (\frac {3 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {5 \left (\frac {5 \left (-\frac {-\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {9}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {5}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {9}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {5}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^5}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\)

Input:

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

Output:

(e^4*(-1/3*(Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*(a + b*Ar 
cCosh[c + d*x])^3) - (4*(-1/2*(c + d*x)^3/(b*(a + b*ArcCosh[c + d*x])^2) + 
 (3*(-((Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*(a + b*ArcCos 
h[c + d*x]))) - (-1/4*(Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]) 
 - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/4 + (Sin 
h[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/4 + (3*Sinh[(3*a)/b]*Sinh 
Integral[(3*(a + b*ArcCosh[c + d*x]))/b])/4)/b^2))/(2*b)))/(3*b) + (5*(-1/ 
2*(c + d*x)^5/(b*(a + b*ArcCosh[c + d*x])^2) + (5*(-((Sqrt[-1 + c + d*x]*( 
c + d*x)^4*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) - (-1/8*(Cosh[ 
a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]) - (9*Cosh[(3*a)/b]*CoshInte 
gral[(3*(a + b*ArcCosh[c + d*x]))/b])/16 - (5*Cosh[(5*a)/b]*CoshIntegral[( 
5*(a + b*ArcCosh[c + d*x]))/b])/16 + (Sinh[a/b]*SinhIntegral[(a + b*ArcCos 
h[c + d*x])/b])/8 + (9*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d* 
x]))/b])/16 + (5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b 
])/16)/b^2))/(2*b)))/(3*b)))/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6300

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

rule 6301

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] + (-Simp[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] + Simp[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 6366

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] - Simp[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, e 
1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
 

rule 6411

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1374\) vs. \(2(399)=798\).

Time = 0.27 (sec) , antiderivative size = 1375, normalized size of antiderivative = 3.19

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1375\)
default	\(\text {Expression too large to display}\)	\(1375\)

Input:

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

Output:

1/d*(1/192*(-16*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^4+12*(d*x+c-1)^(1/ 
2)*(d*x+c+1)^(1/2)*(d*x+c)^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*b^2*arccosh(d*x+c)+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-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x 
+c)^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*ar 
ccosh(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)-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)-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)-1/96*e^4/b^4*exp(a/b)*Ei(1,a 
rccosh(d*x+c)+a/b)-1/48/b*e^4*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b 
*arccosh(d*x+c))^3-1/96/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))^2-1/96/b^3*e^4*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/ 
(a+b*arccosh(d*x+c))-1/96/b^4*e^4*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/32 
/b*e^4*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2- 
(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^3-3/64/b^2*e^4*(4*(d 
*x+c)^3-3*d*x-3*c+4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2-(d*x+c-1)...
 

Fricas [F]

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

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

Output:

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)
 

Sympy [F]

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

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

Output:

e**4*(Integral(c**4/(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) + Int 
egral(d**4*x**4/(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) + Integra 
l(4*c*d**3*x**3/(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) + Integra 
l(6*c**2*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) + Inte 
gral(4*c**3*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. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^4} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^4} \, dx=e^{4} \left (\left (\int \frac {x^{4}}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) d^{4}+4 \left (\int \frac {x^{3}}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c \,d^{3}+6 \left (\int \frac {x^{2}}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c^{2} d^{2}+4 \left (\int \frac {x}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c^{3} d +\left (\int \frac {1}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c^{4}\right ) \] Input:

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

Output:

e**4*(int(x**4/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*ac 
osh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*d**4 + 4*in 
t(x**3/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + 
d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c*d**3 + 6*int(x**2 
/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)** 
2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c**2*d**2 + 4*int(x/(acos 
h(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2 
*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c**3*d + int(1/(acosh(c + d*x)* 
*4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2*b**2 + 4*a 
cosh(c + d*x)*a**3*b + a**4),x)*c**4)