\(\int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^2} \, dx\) [35]

Optimal result

Integrand size = 22, antiderivative size = 161 \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c (a+b \text {arccosh}(c x))}+\frac {3 d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c}-\frac {3 d \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c}-\frac {3 d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c}+\frac {3 d \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {d \left (\frac {4 b \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}{a+b \text {arccosh}(c x)}+3 \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )\right )}{4 b^2 c} \] Input:

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

Output:

(d*((4*b*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3)/(a + b*ArcCosh[c*x]) + 
3*(Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - Cosh[(3*a)/b]*CoshIntegral 
[3*(a/b + ArcCosh[c*x])] - Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + Si 
nh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])])))/(4*b^2*c)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6319, 6368, 5971, 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.

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

\(\Big \downarrow \) 6319

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*(a + b*ArcCosh[c*x])) - (3*d*(-1 
/4*(Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b]) + (Cosh[(3*a)/b]*CoshI 
ntegral[(3*(a + b*ArcCosh[c*x]))/b])/4 + (Sinh[a/b]*SinhIntegral[(a + b*Ar 
cCosh[c*x])/b])/4 - (Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b 
])/4))/(b^2*c)
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6319

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*A 
rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si 
mp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[x*(1 + c*x)^(p - 1/2)*(- 
1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, 
d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(149)=298\).

Time = 0.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.98

method	result	size
derivativedivides	\(\frac {-\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) d}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {3 d \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 b^{2}}+\frac {3 \left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{8 b^{2}}-\frac {3 d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}+\frac {d \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {3 d \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{c}\)	\(318\)
default	\(\frac {-\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) d}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {3 d \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 b^{2}}+\frac {3 \left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{8 b^{2}}-\frac {3 d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}+\frac {d \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {3 d \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{c}\)	\(318\)

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-d*(Integral(c**2*x**2/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2), x) 
+ Integral(-1/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d*( - int(x**2/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**2 + in 
t(1/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x))