\(\int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx\) [139]

Optimal result

Integrand size = 18, antiderivative size = 257 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.92 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.32 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {4 b c^2 d \sqrt {\frac {-1+c x}{1+c x}}+4 b c^3 d x \sqrt {\frac {-1+c x}{1+c x}}+4 b c^2 e x^2 \sqrt {\frac {-1+c x}{1+c x}}+4 b c^3 e x^3 \sqrt {\frac {-1+c x}{1+c x}}-\left (4 c^2 d+e\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 e (a+b \text {arccosh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 a c^2 d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+a e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+4 b c^2 d \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+b e \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 a e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b e \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{4 b^2 c^3 (a+b \text {arccosh}(c x))} \] Input:

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

Output:

-1/4*(4*b*c^2*d*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^3*d*x*Sqrt[(-1 + c*x)/( 
1 + c*x)] + 4*b*c^2*e*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^3*e*x^3*Sqrt[ 
(-1 + c*x)/(1 + c*x)] - (4*c^2*d + e)*(a + b*ArcCosh[c*x])*Cosh[a/b]*CoshI 
ntegral[a/b + ArcCosh[c*x]] - 3*e*(a + b*ArcCosh[c*x])*Cosh[(3*a)/b]*CoshI 
ntegral[3*(a/b + ArcCosh[c*x])] + 4*a*c^2*d*Sinh[a/b]*SinhIntegral[a/b + A 
rcCosh[c*x]] + a*e*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 4*b*c^2*d* 
ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + b*e*ArcCosh[c*x] 
*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 3*a*e*Sinh[(3*a)/b]*SinhInte 
gral[3*(a/b + ArcCosh[c*x])] + 3*b*e*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegr 
al[3*(a/b + ArcCosh[c*x])])/(b^2*c^3*(a + b*ArcCosh[c*x]))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 6324

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (e*x^2*Sq 
rt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d*Cosh[a/b]*Cosh 
Integral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (e*Cosh[a/b]*CoshIntegral[(a + 
 b*ArcCosh[c*x])/b])/(4*b^2*c^3) + (3*e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + 
 b*ArcCosh[c*x]))/b])/(4*b^2*c^3) - (d*Sinh[a/b]*SinhIntegral[(a + b*ArcCo 
sh[c*x])/b])/(b^2*c) - (e*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/ 
(4*b^2*c^3) - (3*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b]) 
/(4*b^2*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 = 0.42 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.81

method	result	size
derivativedivides	\(\frac {\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b^{2}}-\frac {d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b^{2}}}{c}\)	\(465\)
default	\(\frac {\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) e}{8 c^{2} b^{2}}-\frac {d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) e}{8 c^{2} b^{2}}}{c}\)	\(465\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^3*e*x^5 + (c^3*d - c*e)*x^3 - c*d*x + (c^2*e*x^4 + (c^2*d - e)*x^2 - d 
)*sqrt(c*x + 1)*sqrt(c*x - 1))/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)* 
a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - 
 b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate((3*c^5*e*x^6 + 
 (c^5*d - 6*c^3*e)*x^4 + (3*c^3*e*x^4 + (c^3*d - c*e)*x^2 + c*d)*(c*x + 1) 
*(c*x - 1) - (2*c^3*d - 3*c*e)*x^2 + (6*c^4*e*x^5 + (2*c^4*d - 7*c^2*e)*x^ 
3 - (c^2*d - 2*e)*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + c*d)/(a*b*c^5*x^4 + (c* 
x + 1)*(c*x - 1)*a*b*c^3*x^2 - 2*a*b*c^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a* 
b*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)* 
b^2*c^3*x^2 - 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x 
 + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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