\(\int \frac {(d+e x)^2}{a+b \text {arccosh}(c x)} \, dx\) [28]

Optimal result

Integrand size = 18, antiderivative size = 245 \[ \int \frac {(d+e x)^2}{a+b \text {arccosh}(c x)} \, dx=-\frac {d^2 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {e^2 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b c^3}-\frac {d e \text {Chi}\left (\frac {2 a}{b}+2 \text {arccosh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b c^2}-\frac {e^2 \text {Chi}\left (\frac {3 a}{b}+3 \text {arccosh}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b c^3}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )}{b c}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )}{4 b c^3}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arccosh}(c x)\right )}{b c^2}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arccosh}(c x)\right )}{4 b c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^2}{a+b \text {arccosh}(c x)} \, dx=\frac {-\left (\left (4 c^2 d^2+e^2\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )\right )-4 c d e \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-e^2 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+4 c^2 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+4 c d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{4 b c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6380, 7293, 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.

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

\(\Big \downarrow \) 6380

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 6380

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {d e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c b}-\frac {d e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c b}}{c}\)	\(254\)
default	\(\frac {\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {d e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c b}-\frac {d e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c b}}{c}\)	\(254\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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