\(\int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx\) [50]

Optimal result

Integrand size = 23, antiderivative size = 141 \[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {e^2 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b d}-\frac {e^2 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b d} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.72 \[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=\frac {e^2 \left (-\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{4 b d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 115, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^2*(-1/4*(CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b]) - (CoshInt 
egral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/4 + (Cosh[a/b]*SinhIn 
tegral[(a + b*ArcCosh[c + d*x])/b])/4 + (Cosh[(3*a)/b]*SinhIntegral[(3*(a 
+ b*ArcCosh[c + d*x]))/b])/4))/(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 = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b}}{d}\)	\(130\)
default	\(\frac {\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b}}{d}\)	\(130\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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