\(\int \frac {(c-a^2 c x^2)^3}{\text {arccosh}(a x)^2} \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}+\frac {35 c^3 \text {Chi}(\text {arccosh}(a x))}{64 a}-\frac {63 c^3 \text {Chi}(3 \text {arccosh}(a x))}{64 a}+\frac {35 c^3 \text {Chi}(5 \text {arccosh}(a x))}{64 a}-\frac {7 c^3 \text {Chi}(7 \text {arccosh}(a x))}{64 a} \] Output:

c^3*(a*x-1)^(7/2)*(a*x+1)^(7/2)/a/arccosh(a*x)+35/64*c^3*Chi(arccosh(a*x)) 
/a-63/64*c^3*Chi(3*arccosh(a*x))/a+35/64*c^3*Chi(5*arccosh(a*x))/a-7/64*c^ 
3*Chi(7*arccosh(a*x))/a
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(257\) vs. \(2(98)=196\).

Time = 0.45 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.62 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\frac {c^3 \left (-64 \sqrt {\frac {-1+a x}{1+a x}}-64 a x \sqrt {\frac {-1+a x}{1+a x}}+192 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}}+192 a^3 x^3 \sqrt {\frac {-1+a x}{1+a x}}-192 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-192 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}+64 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}}+64 a^7 x^7 \sqrt {\frac {-1+a x}{1+a x}}+35 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))-63 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))+35 \text {arccosh}(a x) \text {Chi}(5 \text {arccosh}(a x))-7 \text {arccosh}(a x) \text {Chi}(7 \text {arccosh}(a x))\right )}{64 a \text {arccosh}(a x)} \] Input:

Integrate[(c - a^2*c*x^2)^3/ArcCosh[a*x]^2,x]
 

Output:

(c^3*(-64*Sqrt[(-1 + a*x)/(1 + a*x)] - 64*a*x*Sqrt[(-1 + a*x)/(1 + a*x)] + 
 192*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)] + 192*a^3*x^3*Sqrt[(-1 + a*x)/(1 + 
 a*x)] - 192*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 192*a^5*x^5*Sqrt[(-1 + a 
*x)/(1 + a*x)] + 64*a^6*x^6*Sqrt[(-1 + a*x)/(1 + a*x)] + 64*a^7*x^7*Sqrt[( 
-1 + a*x)/(1 + a*x)] + 35*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]] - 63*Arc 
Cosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] + 35*ArcCosh[a*x]*CoshIntegral[5*Ar 
cCosh[a*x]] - 7*ArcCosh[a*x]*CoshIntegral[7*ArcCosh[a*x]]))/(64*a*ArcCosh[ 
a*x])
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85, 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.

\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx\)

\(\Big \downarrow \) 6319

\(\displaystyle \frac {c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \text {arccosh}(a x)}-7 a c^3 \int \frac {x (a x-1)^{5/2} (a x+1)^{5/2}}{\text {arccosh}(a x)}dx\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \text {arccosh}(a x)}-\frac {7 c^3 \int \frac {a x (a x-1)^3 (a x+1)^3}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \text {arccosh}(a x)}-\frac {7 c^3 \int \left (-\frac {5 a x}{64 \text {arccosh}(a x)}+\frac {9 \cosh (3 \text {arccosh}(a x))}{64 \text {arccosh}(a x)}-\frac {5 \cosh (5 \text {arccosh}(a x))}{64 \text {arccosh}(a x)}+\frac {\cosh (7 \text {arccosh}(a x))}{64 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \text {arccosh}(a x)}-\frac {7 c^3 \left (-\frac {5}{64} \text {Chi}(\text {arccosh}(a x))+\frac {9}{64} \text {Chi}(3 \text {arccosh}(a x))-\frac {5}{64} \text {Chi}(5 \text {arccosh}(a x))+\frac {1}{64} \text {Chi}(7 \text {arccosh}(a x))\right )}{a}\)

Input:

Int[(c - a^2*c*x^2)^3/ArcCosh[a*x]^2,x]
 

Output:

(c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2))/(a*ArcCosh[a*x]) - (7*c^3*((-5*Cosh 
Integral[ArcCosh[a*x]])/64 + (9*CoshIntegral[3*ArcCosh[a*x]])/64 - (5*Cosh 
Integral[5*ArcCosh[a*x]])/64 + CoshIntegral[7*ArcCosh[a*x]]/64))/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 = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-63 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+35 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-7 \,\operatorname {Chi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-35 \sqrt {a x -1}\, \sqrt {a x +1}+21 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-7 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )+\sinh \left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arccosh}\left (a x \right )}\)	\(107\)
default	\(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-63 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+35 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-7 \,\operatorname {Chi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-35 \sqrt {a x -1}\, \sqrt {a x +1}+21 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-7 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )+\sinh \left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arccosh}\left (a x \right )}\)	\(107\)

Input:

int((-a^2*c*x^2+c)^3/arccosh(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/64/a*c^3*(35*Chi(arccosh(a*x))*arccosh(a*x)-63*Chi(3*arccosh(a*x))*arcco 
sh(a*x)+35*Chi(5*arccosh(a*x))*arccosh(a*x)-7*Chi(7*arccosh(a*x))*arccosh( 
a*x)-35*(a*x-1)^(1/2)*(a*x+1)^(1/2)+21*sinh(3*arccosh(a*x))-7*sinh(5*arcco 
sh(a*x))+sinh(7*arccosh(a*x)))/arccosh(a*x)
 

Fricas [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^3/arccosh(a*x)^2,x, algorithm="fricas")
 

Output:

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arccosh(a*x) 
^2, x)
 

Sympy [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx\right ) \] Input:

integrate((-a**2*c*x**2+c)**3/acosh(a*x)**2,x)
 

Output:

-c**3*(Integral(3*a**2*x**2/acosh(a*x)**2, x) + Integral(-3*a**4*x**4/acos 
h(a*x)**2, x) + Integral(a**6*x**6/acosh(a*x)**2, x) + Integral(-1/acosh(a 
*x)**2, x))
 

Maxima [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^3/arccosh(a*x)^2,x, algorithm="maxima")
 

Output:

(a^9*c^3*x^9 - 4*a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 4*a^3*c^3*x^3 + a*c^3*x + ( 
a^8*c^3*x^8 - 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 4*a^2*c^3*x^2 + c^3)*sqrt(a* 
x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*l 
og(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) - integrate((7*a^10*c^3*x^10 - 29*a 
^8*c^3*x^8 + 46*a^6*c^3*x^6 - 34*a^4*c^3*x^4 + 11*a^2*c^3*x^2 + (7*a^8*c^3 
*x^8 - 20*a^6*c^3*x^6 + 18*a^4*c^3*x^4 - 4*a^2*c^3*x^2 - c^3)*(a*x + 1)*(a 
*x - 1) - c^3 + 7*(2*a^9*c^3*x^9 - 7*a^7*c^3*x^7 + 9*a^5*c^3*x^5 - 5*a^3*c 
^3*x^3 + a*c^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1))/((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]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^3/arccosh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(-(a^2*c*x^2 - c)^3/arccosh(a*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \] Input:

int((c - a^2*c*x^2)^3/acosh(a*x)^2,x)
 

Output:

int((c - a^2*c*x^2)^3/acosh(a*x)^2, x)
 

Reduce [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=c^{3} \left (-\left (\int \frac {x^{6}}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{6}+3 \left (\int \frac {x^{4}}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{4}-3 \left (\int \frac {x^{2}}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}+\int \frac {1}{\mathit {acosh} \left (a x \right )^{2}}d x \right ) \] Input:

int((-a^2*c*x^2+c)^3/acosh(a*x)^2,x)
 

Output:

c**3*( - int(x**6/acosh(a*x)**2,x)*a**6 + 3*int(x**4/acosh(a*x)**2,x)*a**4 
 - 3*int(x**2/acosh(a*x)**2,x)*a**2 + int(1/acosh(a*x)**2,x))