\(\int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx\) [60]

Optimal result

Integrand size = 23, antiderivative size = 327 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {2 e^4 (c+d x)^3}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {5 e^4 (c+d x)^5}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{16 b^3 d}-\frac {27 e^4 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b^3 d}-\frac {25 e^4 \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{32 b^3 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{32 b^3 d} \] Output:

-1/2*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) 
)^2+2*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c))-5/2*e^4*(d*x+c)^5/b^2/d/(a+ 
b*arccosh(d*x+c))-1/16*e^4*Chi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b^3/d-27/ 
32*e^4*Chi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b^3/d-25/32*e^4*Chi(5*(a+ 
b*arccosh(d*x+c))/b)*sinh(5*a/b)/b^3/d+1/16*e^4*cosh(a/b)*Shi((a+b*arccosh 
(d*x+c))/b)/b^3/d+27/32*e^4*cosh(3*a/b)*Shi(3*(a+b*arccosh(d*x+c))/b)/b^3/ 
d+25/32*e^4*cosh(5*a/b)*Shi(5*(a+b*arccosh(d*x+c))/b)/b^3/d
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.99 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx=\frac {e^4 \left (-\frac {16 b^2 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^2}+\frac {16 b \left (4 (c+d x)^3-5 (c+d x)^5\right )}{a+b \text {arccosh}(c+d x)}+48 \left (\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+\text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )+25 \left (-2 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{32 b^3 d} \] Input:

Integrate[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^3,x]
 

Output:

(e^4*((-16*b^2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(a + b*Ar 
cCosh[c + d*x])^2 + (16*b*(4*(c + d*x)^3 - 5*(c + d*x)^5))/(a + b*ArcCosh[ 
c + d*x]) + 48*(CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] + CoshInteg 
ral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] - Cosh[a/b]*SinhIntegral[a/b 
 + ArcCosh[c + d*x]] - Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x 
])]) + 25*(-2*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 3*CoshInteg 
ral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcC 
osh[c + d*x])]*Sinh[(5*a)/b] + 2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + 
d*x]] + 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])] + Cosh[(5 
*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c + d*x])])))/(32*b^3*d)
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6411, 27, 6301, 6366, 6302, 25, 5971, 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.

\(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^3} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^3}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^3}d(c+d x)}{d}\)

\(\Big \downarrow \) 6301

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

\(\Big \downarrow \) 6366

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

Int[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^3,x]
 

Output:

(e^4*(-1/2*(Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*(a + b*Ar 
cCosh[c + d*x])^2) - (2*(-((c + d*x)^3/(b*(a + b*ArcCosh[c + d*x]))) + (3* 
(-1/4*(CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b]) - (CoshIntegral 
[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/4 + (Cosh[a/b]*SinhIntegra 
l[(a + b*ArcCosh[c + d*x])/b])/4 + (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*A 
rcCosh[c + d*x]))/b])/4))/b^2))/b + (5*(-((c + d*x)^5/(b*(a + b*ArcCosh[c 
+ d*x]))) + (5*(-1/8*(CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b]) 
- (3*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/16 - (Cos 
hIntegral[(5*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(5*a)/b])/16 + (Cosh[a/b]*S 
inhIntegral[(a + b*ArcCosh[c + d*x])/b])/8 + (3*Cosh[(3*a)/b]*SinhIntegral 
[(3*(a + b*ArcCosh[c + d*x]))/b])/16 + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + 
 b*ArcCosh[c + d*x]))/b])/16))/b^2))/(2*b)))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 5971

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 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 6302

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

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. \(992\) vs. \(2(307)=614\).

Time = 0.26 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.04

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(993\)
default	\(\text {Expression too large to display}\)	\(993\)

Input:

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/64*(-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*(5*b*arccosh(d*x+c)+5*a-b)/b^2/(b^2*arccosh(d* 
x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+25/64*e^4/b^3*exp(5*a/b)*Ei(1,5*arccosh(d 
*x+c)+5*a/b)-3/64*(-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*(3*b*arccosh(d*x+c)+3*a-b 
)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+27/64*e^4/b^3*exp(3* 
a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)-1/32*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d 
*x+c)*e^4*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d 
*x+c)+a^2)+1/32*e^4/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/32/b*e^4*(d*x+ 
c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-1/32/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))-1/32/b^3*e^4*exp 
(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-3/64/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))^2-9/64/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)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x 
+c))-27/64/b^3*e^4*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b)-1/64/b*e^4*(1 
6*(d*x+c)^5-20*(d*x+c)^3+16*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^4+5*d* 
x+5*c-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))/(a+b*arccosh(d*x+c))^2-5/64/b^2*e^4*(16*(d*x+c)^5-20*(d*x+c)^...
 

Fricas [F]

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

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

Sympy [F]

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

integrate((d*e*x+c*e)**4/(a+b*acosh(d*x+c))**3,x)
 

Output:

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

Maxima [F]

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

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")
 

Output:

-1/2*((5*a*d^11*e^4 + b*d^11*e^4)*x^11 + 11*(5*a*c*d^10*e^4 + b*c*d^10*e^4 
)*x^10 + (5*(55*c^2*d^9*e^4 - 3*d^9*e^4)*a + (55*c^2*d^9*e^4 - 3*d^9*e^4)* 
b)*x^9 + 3*(5*(55*c^3*d^8*e^4 - 9*c*d^8*e^4)*a + (55*c^3*d^8*e^4 - 9*c*d^8 
*e^4)*b)*x^8 + 3*(5*(110*c^4*d^7*e^4 - 36*c^2*d^7*e^4 + d^7*e^4)*a + (110* 
c^4*d^7*e^4 - 36*c^2*d^7*e^4 + d^7*e^4)*b)*x^7 + 21*(5*(22*c^5*d^6*e^4 - 1 
2*c^3*d^6*e^4 + c*d^6*e^4)*a + (22*c^5*d^6*e^4 - 12*c^3*d^6*e^4 + c*d^6*e^ 
4)*b)*x^6 + (5*(462*c^6*d^5*e^4 - 378*c^4*d^5*e^4 + 63*c^2*d^5*e^4 - d^5*e 
^4)*a + (462*c^6*d^5*e^4 - 378*c^4*d^5*e^4 + 63*c^2*d^5*e^4 - d^5*e^4)*b)* 
x^5 + (5*(330*c^7*d^4*e^4 - 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 - 5*c*d^4*e^ 
4)*a + (330*c^7*d^4*e^4 - 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 - 5*c*d^4*e^4) 
*b)*x^4 + ((5*a*d^8*e^4 + b*d^8*e^4)*x^8 + 8*(5*a*c*d^7*e^4 + b*c*d^7*e^4) 
*x^7 + (4*(35*c^2*d^6*e^4 - 2*d^6*e^4)*a + (28*c^2*d^6*e^4 - d^6*e^4)*b)*x 
^6 + 2*(4*(35*c^3*d^5*e^4 - 6*c*d^5*e^4)*a + (28*c^3*d^5*e^4 - 3*c*d^5*e^4 
)*b)*x^5 + ((350*c^4*d^4*e^4 - 120*c^2*d^4*e^4 + 3*d^4*e^4)*a + 5*(14*c^4* 
d^4*e^4 - 3*c^2*d^4*e^4)*b)*x^4 + 4*((70*c^5*d^3*e^4 - 40*c^3*d^3*e^4 + 3* 
c*d^3*e^4)*a + (14*c^5*d^3*e^4 - 5*c^3*d^3*e^4)*b)*x^3 + (2*(70*c^6*d^2*e^ 
4 - 60*c^4*d^2*e^4 + 9*c^2*d^2*e^4)*a + (28*c^6*d^2*e^4 - 15*c^4*d^2*e^4)* 
b)*x^2 + (5*c^8*e^4 - 8*c^6*e^4 + 3*c^4*e^4)*a + (c^8*e^4 - c^6*e^4)*b + 2 
*(2*(10*c^7*d*e^4 - 12*c^5*d*e^4 + 3*c^3*d*e^4)*a + (4*c^7*d*e^4 - 3*c^5*d 
*e^4)*b)*x)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + (5*(165*c^8*d^3*e...
 

Giac [F]

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

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^3,x)
 

Output:

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^3, x)
 

Reduce [F]

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

int((d*e*x+c*e)^4/(a+b*acosh(d*x+c))^3,x)
 

Output:

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