\(\int \frac {(c e+d e x)^3}{a+b \text {arccosh}(c+d x)} \, dx\) [49]

Optimal result

Integrand size = 23, antiderivative size = 145 \[ \int \frac {(c e+d e x)^3}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {e^3 \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{4 b d}-\frac {e^3 \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b d} \] Output:

-1/4*e^3*Chi(2*(a+b*arccosh(d*x+c))/b)*sinh(2*a/b)/b/d-1/8*e^3*Chi(4*(a+b* 
arccosh(d*x+c))/b)*sinh(4*a/b)/b/d+1/4*e^3*cosh(2*a/b)*Shi(2*(a+b*arccosh( 
d*x+c))/b)/b/d+1/8*e^3*cosh(4*a/b)*Shi(4*(a+b*arccosh(d*x+c))/b)/b/d
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.75 \[ \int \frac {(c e+d e x)^3}{a+b \text {arccosh}(c+d x)} \, dx=\frac {e^3 \left (-2 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-\text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{8 b d} \] Input:

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

Output:

(e^3*(-2*CoshIntegral[2*(a/b + ArcCosh[c + d*x])]*Sinh[(2*a)/b] - CoshInte 
gral[4*(a/b + ArcCosh[c + d*x])]*Sinh[(4*a)/b] + 2*Cosh[(2*a)/b]*SinhInteg 
ral[2*(a/b + ArcCosh[c + d*x])] + Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcC 
osh[c + d*x])]))/(8*b*d)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6411, 27, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^3*(-1/4*(CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(2*a)/b]) - 
(CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(4*a)/b])/8 + (Cosh[(2* 
a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/4 + (Cosh[(4*a)/b]*Sin 
hIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/8))/(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 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 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 [A] (verified)

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{16 b}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{8 b}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{8 b}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{16 b}}{d}\)	\(134\)
default	\(\frac {\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{16 b}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{8 b}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{8 b}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{16 b}}{d}\)	\(134\)

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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