\(\int \frac {d+e x}{(a+b \text {arccosh}(c x))^2} \, dx\) [33]

Optimal result

Integrand size = 16, antiderivative size = 190 \[ \int \frac {d+e x}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^2} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.75 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.41 \[ \int \frac {d+e x}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {b c d \sqrt {\frac {-1+c x}{1+c x}}+b c^2 d x \sqrt {\frac {-1+c x}{1+c x}}+b c e x \sqrt {\frac {-1+c x}{1+c x}}+b c^2 e x^2 \sqrt {\frac {-1+c x}{1+c x}}-c d (a+b \text {arccosh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-e (a+b \text {arccosh}(c x)) \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+a c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+b c d \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+a e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+b e \text {arccosh}(c x) \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{b^2 c^2 (a+b \text {arccosh}(c x))} \] Input:

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

Output:

-((b*c*d*Sqrt[(-1 + c*x)/(1 + c*x)] + b*c^2*d*x*Sqrt[(-1 + c*x)/(1 + c*x)] 
 + b*c*e*x*Sqrt[(-1 + c*x)/(1 + c*x)] + b*c^2*e*x^2*Sqrt[(-1 + c*x)/(1 + c 
*x)] - c*d*(a + b*ArcCosh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] 
 - e*(a + b*ArcCosh[c*x])*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x] 
)] + a*c*d*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + b*c*d*ArcCosh[c*x] 
*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + a*e*Sinh[(2*a)/b]*SinhIntegr 
al[2*(a/b + ArcCosh[c*x])] + b*e*ArcCosh[c*x]*Sinh[(2*a)/b]*SinhIntegral[2 
*(a/b + ArcCosh[c*x])])/(b^2*c^2*(a + b*ArcCosh[c*x])))
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6379, 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.

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

\(\Big \downarrow \) 6379

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d + e*x)/(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*Sqrt 
[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d*Cosh[a/b]*CoshIn 
tegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (e*Cosh[(2*a)/b]*CoshIntegral[(2 
*(a + b*ArcCosh[c*x]))/b])/(b^2*c^2) - (d*Sinh[a/b]*SinhIntegral[(a + b*Ar 
cCosh[c*x])/b])/(b^2*c) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[ 
c*x]))/b])/(b^2*c^2)
 

Defintions of rubi rules used

rule 2009

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

rule 6379

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_S 
ymbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; 
 FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(\frac {\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b}-\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {\left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right ) e}{4 c \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right )}{4 c b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\)	\(285\)
default	\(\frac {\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b}-\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {\left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right ) e}{4 c \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right )}{4 c b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\)	\(285\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^3*e*x^4 + c^3*d*x^3 - c*e*x^2 - c*d*x + (c^2*e*x^3 + c^2*d*x^2 - e*x - 
 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((2*c^5*e*x^5 
 + c^5*d*x^4 - 4*c^3*e*x^3 - 2*c^3*d*x^2 + (2*c^3*e*x^3 + c^3*d*x^2 + c*d) 
*(c*x + 1)*(c*x - 1) + 2*c*e*x + (4*c^4*e*x^4 + 2*c^4*d*x^3 - 4*c^2*e*x^2 
- c^2*d*x + e)*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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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