\(\int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^2} \, dx\) [55]

Optimal result

Integrand size = 23, antiderivative size = 195 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^2} \, dx=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 b^2 d} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 2.08 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.18 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {e^3 \left (-\frac {2 b (c+d x)^3 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{a+b \text {arccosh}(c+d x)}+4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+3 \log (a+b \text {arccosh}(c+d x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-3 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\log (a+b \text {arccosh}(c+d x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{2 b^2 d} \] Input:

Integrate[(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^2,x]
 

Output:

(e^3*((-2*b*(c + d*x)^3*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))/ 
(a + b*ArcCosh[c + d*x]) + 4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c 
 + d*x])] + Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c + d*x])] + 3*Log 
[a + b*ArcCosh[c + d*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c 
 + d*x])] - 3*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c + d*x])] + Lo 
g[a + b*ArcCosh[c + d*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c 
+ d*x])]) - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c + d*x])]))/(2*b^ 
2*d)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6411, 27, 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.

\(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^2} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 6300

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

\(\Big \downarrow \) 2009

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

Input:

Int[(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^2,x]
 

Output:

(e^3*(-((Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(b*(a + b*ArcCo 
sh[c + d*x]))) - (-1/2*(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d 
*x]))/b]) - (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/2 
 + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/2 + (Sinh[ 
(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/2)/b^2))/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 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. \(417\) vs. \(2(183)=366\).

Time = 0.20 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.14

method	result	size
derivativedivides	\(\frac {\frac {\left (-8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3}}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{4 b^{2}}+\frac {\left (-2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3}}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{4 b^{2}}}{d}\)	\(418\)
default	\(\frac {\frac {\left (-8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3}}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{4 b^{2}}+\frac {\left (-2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3}}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{4 b^{2}}}{d}\)	\(418\)

Input:

int((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d*e*x+c*e)**3/(a+b*acosh(d*x+c))**2,x)
 

Output:

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

Maxima [F]

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

integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^2,x, algorithm="maxima")
 

Output:

-(d^6*e^3*x^6 + 6*c*d^5*e^3*x^5 + c^6*e^3 - c^4*e^3 + (15*c^2*d^4*e^3 - d^ 
4*e^3)*x^4 + 4*(5*c^3*d^3*e^3 - c*d^3*e^3)*x^3 + 3*(5*c^4*d^2*e^3 - 2*c^2* 
d^2*e^3)*x^2 + (d^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + c^5*e^3 - c^3*e^3 + (10*c^ 
2*d^3*e^3 - d^3*e^3)*x^3 + (10*c^3*d^2*e^3 - 3*c*d^2*e^3)*x^2 + (5*c^4*d*e 
^3 - 3*c^2*d*e^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(3*c^5*d*e^3 
- 2*c^3*d*e^3)*x)/(a*b*d^3*x^2 + 2*a*b*c*d^2*x + (c^2*d - d)*a*b + (a*b*d^ 
2*x + a*b*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^3*x^2 + 2*b^2* 
c*d^2*x + (c^2*d - d)*b^2 + (b^2*d^2*x + b^2*c*d)*sqrt(d*x + c + 1)*sqrt(d 
*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)) + integra 
te((4*d^7*e^3*x^7 + 28*c*d^6*e^3*x^6 + 4*c^7*e^3 - 8*c^5*e^3 + 4*c^3*e^3 + 
 4*(21*c^2*d^5*e^3 - 2*d^5*e^3)*x^5 + 20*(7*c^3*d^4*e^3 - 2*c*d^4*e^3)*x^4 
 + 4*(35*c^4*d^3*e^3 - 20*c^2*d^3*e^3 + d^3*e^3)*x^3 + 2*(2*d^5*e^3*x^5 + 
10*c*d^4*e^3*x^4 + 2*c^5*e^3 - c^3*e^3 + (20*c^2*d^3*e^3 - d^3*e^3)*x^3 + 
(20*c^3*d^2*e^3 - 3*c*d^2*e^3)*x^2 + (10*c^4*d*e^3 - 3*c^2*d*e^3)*x)*(d*x 
+ c + 1)*(d*x + c - 1) + 4*(21*c^5*d^2*e^3 - 20*c^3*d^2*e^3 + 3*c*d^2*e^3) 
*x^2 + (8*d^6*e^3*x^6 + 48*c*d^5*e^3*x^5 + 8*c^6*e^3 - 10*c^4*e^3 + 3*c^2* 
e^3 + 10*(12*c^2*d^4*e^3 - d^4*e^3)*x^4 + 40*(4*c^3*d^3*e^3 - c*d^3*e^3)*x 
^3 + 3*(40*c^4*d^2*e^3 - 20*c^2*d^2*e^3 + d^2*e^3)*x^2 + 2*(24*c^5*d*e^3 - 
 20*c^3*d*e^3 + 3*c*d*e^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 4*(7*c 
^6*d*e^3 - 10*c^4*d*e^3 + 3*c^2*d*e^3)*x)/(a*b*d^4*x^4 + 4*a*b*c*d^3*x^...
 

Giac [F]

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

integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((c*e + d*e*x)^3/(a + b*acosh(c + d*x))^2,x)
 

Output:

int((c*e + d*e*x)^3/(a + b*acosh(c + d*x))^2, x)
 

Reduce [F]

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

int((d*e*x+c*e)^3/(a+b*acosh(d*x+c))^2,x)
 

Output:

e**3*(int(x**3/(acosh(c + d*x)**2*b**2 + 2*acosh(c + d*x)*a*b + a**2),x)*d 
**3 + 3*int(x**2/(acosh(c + d*x)**2*b**2 + 2*acosh(c + d*x)*a*b + a**2),x) 
*c*d**2 + 3*int(x/(acosh(c + d*x)**2*b**2 + 2*acosh(c + d*x)*a*b + a**2),x 
)*c**2*d + int(1/(acosh(c + d*x)**2*b**2 + 2*acosh(c + d*x)*a*b + a**2),x) 
*c**3)