\(\int \frac {(d+e x)^3}{a+b \text {arccosh}(c x)} \, dx\) [27]

Optimal result

Integrand size = 18, antiderivative size = 394 \[ \int \frac {(d+e x)^3}{a+b \text {arccosh}(c x)} \, dx=-\frac {d^3 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {3 d e^2 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b c^3}-\frac {3 d^2 e \text {Chi}\left (\frac {2 a}{b}+2 \text {arccosh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b c^2}-\frac {e^3 \text {Chi}\left (\frac {2 a}{b}+2 \text {arccosh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{4 b c^4}-\frac {3 d e^2 \text {Chi}\left (\frac {3 a}{b}+3 \text {arccosh}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b c^3}-\frac {e^3 \text {Chi}\left (\frac {4 a}{b}+4 \text {arccosh}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b c^4}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )}{b c}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )}{4 b c^3}+\frac {3 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arccosh}(c x)\right )}{2 b c^2}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arccosh}(c x)\right )}{4 b c^4}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arccosh}(c x)\right )}{4 b c^3}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \text {arccosh}(c x)\right )}{8 b c^4} \] Output:

-d^3*Chi(a/b+arccosh(c*x))*sinh(a/b)/b/c-3/4*d*e^2*Chi(a/b+arccosh(c*x))*s 
inh(a/b)/b/c^3-3/2*d^2*e*Chi(2*a/b+2*arccosh(c*x))*sinh(2*a/b)/b/c^2-1/4*e 
^3*Chi(2*a/b+2*arccosh(c*x))*sinh(2*a/b)/b/c^4-3/4*d*e^2*Chi(3*a/b+3*arcco 
sh(c*x))*sinh(3*a/b)/b/c^3-1/8*e^3*Chi(4*a/b+4*arccosh(c*x))*sinh(4*a/b)/b 
/c^4+d^3*cosh(a/b)*Shi(a/b+arccosh(c*x))/b/c+3/4*d*e^2*cosh(a/b)*Shi(a/b+a 
rccosh(c*x))/b/c^3+3/2*d^2*e*cosh(2*a/b)*Shi(2*a/b+2*arccosh(c*x))/b/c^2+1 
/4*e^3*cosh(2*a/b)*Shi(2*a/b+2*arccosh(c*x))/b/c^4+3/4*d*e^2*cosh(3*a/b)*S 
hi(3*a/b+3*arccosh(c*x))/b/c^3+1/8*e^3*cosh(4*a/b)*Shi(4*a/b+4*arccosh(c*x 
))/b/c^4
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^3}{a+b \text {arccosh}(c x)} \, dx=\frac {-2 c d \left (4 c^2 d^2+3 e^2\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-2 e \left (6 c^2 d^2+e^2\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-6 c d e^2 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-e^3 \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+8 c^3 d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+6 c d e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+12 c^2 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+2 e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+6 c d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{8 b c^4} \] Input:

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

Output:

(-2*c*d*(4*c^2*d^2 + 3*e^2)*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 2 
*e*(6*c^2*d^2 + e^2)*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - 
6*c*d*e^2*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - e^3*CoshInt 
egral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + 8*c^3*d^3*Cosh[a/b]*SinhInte 
gral[a/b + ArcCosh[c*x]] + 6*c*d*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[ 
c*x]] + 12*c^2*d^2*e*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 
2*e^3*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 6*c*d*e^2*Cosh[ 
(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + e^3*Cosh[(4*a)/b]*SinhInte 
gral[4*(a/b + ArcCosh[c*x])])/(8*b*c^4)
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.96, 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.

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

\(\Big \downarrow \) 6380

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-((c^3*d^3*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/b) - (3*c*d*e^2*Co 
shIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(4*b) - (3*c^2*d^2*e*CoshIntegra 
l[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(2*b) - (e^3*CoshIntegral[(2*a) 
/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(4*b) - (3*c*d*e^2*CoshIntegral[(3*a)/ 
b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(4*b) - (e^3*CoshIntegral[(4*a)/b + 4*A 
rcCosh[c*x]]*Sinh[(4*a)/b])/(8*b) + (c^3*d^3*Cosh[a/b]*SinhIntegral[a/b + 
ArcCosh[c*x]])/b + (3*c*d*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/ 
(4*b) + (3*c^2*d^2*e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]]) 
/(2*b) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(4*b) 
+ (3*c*d*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(4*b) + 
 (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(8*b))/c^4
 

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 = 1.22 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right )}{16 c^{3} b}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right )}{16 c^{3} b}+\frac {3 e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) d^{2}}{4 c b}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{8 c^{3} b}-\frac {3 e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) d^{2}}{4 c b}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{8 c^{3} b}+\frac {3 d \,e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {3 d \,e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d^{3} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {3 d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {d^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {3 d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}}{c}\)	\(394\)
default	\(\frac {\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right )}{16 c^{3} b}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right )}{16 c^{3} b}+\frac {3 e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) d^{2}}{4 c b}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{8 c^{3} b}-\frac {3 e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) d^{2}}{4 c b}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{8 c^{3} b}+\frac {3 d \,e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {3 d \,e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d^{3} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {3 d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {d^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {3 d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}}{c}\)	\(394\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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