\(\int \frac {(c-a^2 c x^2)^2}{\text {arccosh}(a x)^2} \, dx\) [24]

Optimal result

Integrand size = 20, antiderivative size = 82 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\frac {5 c^2 \text {Chi}(\text {arccosh}(a x))}{8 a}-\frac {15 c^2 \text {Chi}(3 \text {arccosh}(a x))}{16 a}+\frac {5 c^2 \text {Chi}(5 \text {arccosh}(a x))}{16 a} \] Output:

-c^2*(a*x-1)^(5/2)*(a*x+1)^(5/2)/a/arccosh(a*x)+5/8*c^2*Chi(arccosh(a*x))/ 
a-15/16*c^2*Chi(3*arccosh(a*x))/a+5/16*c^2*Chi(5*arccosh(a*x))/a
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(82)=164\).

Time = 0.32 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.37 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=-\frac {c^2 \left (16 \sqrt {\frac {-1+a x}{1+a x}}+16 a x \sqrt {\frac {-1+a x}{1+a x}}-32 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}}-32 a^3 x^3 \sqrt {\frac {-1+a x}{1+a x}}+16 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}+16 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}-10 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))+15 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))-5 \text {arccosh}(a x) \text {Chi}(5 \text {arccosh}(a x))\right )}{16 a \text {arccosh}(a x)} \] Input:

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

Output:

-1/16*(c^2*(16*Sqrt[(-1 + a*x)/(1 + a*x)] + 16*a*x*Sqrt[(-1 + a*x)/(1 + a* 
x)] - 32*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)] - 32*a^3*x^3*Sqrt[(-1 + a*x)/( 
1 + a*x)] + 16*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] + 16*a^5*x^5*Sqrt[(-1 + 
a*x)/(1 + a*x)] - 10*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]] + 15*ArcCosh[ 
a*x]*CoshIntegral[3*ArcCosh[a*x]] - 5*ArcCosh[a*x]*CoshIntegral[5*ArcCosh[ 
a*x]]))/(a*ArcCosh[a*x])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.

\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx\)

\(\Big \downarrow \) 6319

\(\displaystyle 5 a c^2 \int \frac {x (a x-1)^{3/2} (a x+1)^{3/2}}{\text {arccosh}(a x)}dx-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \text {arccosh}(a x)}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {5 c^2 \int \frac {a x (a x-1)^2 (a x+1)^2}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \text {arccosh}(a x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {5 c^2 \int \left (\frac {a x}{8 \text {arccosh}(a x)}-\frac {3 \cosh (3 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}+\frac {\cosh (5 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a}-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \text {arccosh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 c^2 \left (\frac {1}{8} \text {Chi}(\text {arccosh}(a x))-\frac {3}{16} \text {Chi}(3 \text {arccosh}(a x))+\frac {1}{16} \text {Chi}(5 \text {arccosh}(a x))\right )}{a}-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \text {arccosh}(a x)}\)

Input:

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

Output:

-((c^2*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2))/(a*ArcCosh[a*x])) + (5*c^2*(CoshI 
ntegral[ArcCosh[a*x]]/8 - (3*CoshIntegral[3*ArcCosh[a*x]])/16 + CoshIntegr 
al[5*ArcCosh[a*x]]/16))/a
 

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 [A] (verified)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-15 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-10 \sqrt {a x -1}\, \sqrt {a x +1}+5 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{16 a \,\operatorname {arccosh}\left (a x \right )}\)	\(87\)
default	\(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-15 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-10 \sqrt {a x -1}\, \sqrt {a x +1}+5 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{16 a \,\operatorname {arccosh}\left (a x \right )}\)	\(87\)

Input:

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

Output:

1/16/a*c^2*(10*Chi(arccosh(a*x))*arccosh(a*x)-15*Chi(3*arccosh(a*x))*arcco 
sh(a*x)+5*Chi(5*arccosh(a*x))*arccosh(a*x)-10*(a*x-1)^(1/2)*(a*x+1)^(1/2)+ 
5*sinh(3*arccosh(a*x))-sinh(5*arccosh(a*x)))/arccosh(a*x)
 

Fricas [F]

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

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

Output:

integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)/arccosh(a*x)^2, x)
 

Sympy [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx\right ) \] Input:

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

Output:

c**2*(Integral(-2*a**2*x**2/acosh(a*x)**2, x) + Integral(a**4*x**4/acosh(a 
*x)**2, x) + Integral(acosh(a*x)**(-2), x))
 

Maxima [F]

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

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

Output:

-(a^7*c^2*x^7 - 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 - a*c^2*x + (a^6*c^2*x^6 - 3 
*a^4*c^2*x^4 + 3*a^2*c^2*x^2 - c^2)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 
 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x 
 - 1))) + integrate((5*a^8*c^2*x^8 - 16*a^6*c^2*x^6 + 18*a^4*c^2*x^4 - 8*a 
^2*c^2*x^2 + (5*a^6*c^2*x^6 - 9*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*(a*x + 
1)*(a*x - 1) + 5*(2*a^7*c^2*x^7 - 5*a^5*c^2*x^5 + 4*a^3*c^2*x^3 - a*c^2*x) 
*sqrt(a*x + 1)*sqrt(a*x - 1) + c^2)/((a^4*x^4 + (a*x + 1)*(a*x - 1)*a^2*x^ 
2 - 2*a^2*x^2 + 2*(a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)*log(a*x 
 + sqrt(a*x + 1)*sqrt(a*x - 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=c^{2} \left (\left (\int \frac {x^{4}}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{4}-2 \left (\int \frac {x^{2}}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}+\int \frac {1}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) \] Input:

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

Output:

c**2*(int(x**4/acosh(a*x)**2,x)*a**4 - 2*int(x**2/acosh(a*x)**2,x)*a**2 + 
int(1/acosh(a*x)**2,x))