\(\int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx\) [134]

Optimal result

Integrand size = 18, antiderivative size = 139 \[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=-\frac {\left (4 c^2 d+e\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b c^3}-\frac {e \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b c^3}+\frac {\left (4 c^2 d+e\right ) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^3}+\frac {e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x^2}{a+b \text {arccosh}(c x)} \, dx=\frac {-\left (\left (4 c^2 d+e\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )\right )-e \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+4 c^2 d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{4 b c^3} \] Input:

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

Output:

(-((4*c^2*d + e)*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b]) - e*CoshInteg 
ral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] + 4*c^2*d*Cosh[a/b]*SinhIntegral 
[a/b + ArcCosh[c*x]] + e*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + e*Co 
sh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])])/(4*b*c^3)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6324, 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.

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

\(\Big \downarrow \) 6324

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 6324

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && 
(p > 0 || IGtQ[n, 0])
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b}}{c}\)	\(178\)
default	\(\frac {-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b}}{c}\)	\(178\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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