3.8.36 \(\int e^{n \coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [736]

3.8.36.1 Optimal result

Integrand size = 22, antiderivative size = 81 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {256 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \operatorname {Hypergeometric2F1}\left (8,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]

output

-256*c^3*(1-1/a/x)^(4-1/2*n)*(1+1/a/x)^(-4+1/2*n)*hypergeom([8, 4-1/2*n],[ 
5-1/2*n],(a-1/x)/(a+1/x))/a/(8-n)
 

3.8.36.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(81)=162\).

Time = 2.39 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.30 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 e^{n \coth ^{-1}(a x)} \left (-912 n+58 n^3-n^5-5040 a x+912 a n^2 x-58 a n^4 x+a n^6 x+1368 a^2 n x^2-64 a^2 n^3 x^2+a^2 n^5 x^2+5040 a^3 x^3-152 a^3 n^2 x^3+2 a^3 n^4 x^3-576 a^4 n x^4+6 a^4 n^3 x^4-3024 a^5 x^5+24 a^5 n^2 x^5+120 a^6 n x^6+720 a^7 x^7+e^{2 \coth ^{-1}(a x)} n \left (-1152+576 n+104 n^2-52 n^3-2 n^4+n^5\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+\left (-2304+784 n^2-56 n^4+n^6\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{5040 a} \]

input

Integrate[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]
 

output

-1/5040*(c^3*E^(n*ArcCoth[a*x])*(-912*n + 58*n^3 - n^5 - 5040*a*x + 912*a* 
n^2*x - 58*a*n^4*x + a*n^6*x + 1368*a^2*n*x^2 - 64*a^2*n^3*x^2 + a^2*n^5*x 
^2 + 5040*a^3*x^3 - 152*a^3*n^2*x^3 + 2*a^3*n^4*x^3 - 576*a^4*n*x^4 + 6*a^ 
4*n^3*x^4 - 3024*a^5*x^5 + 24*a^5*n^2*x^5 + 120*a^6*n*x^6 + 720*a^7*x^7 + 
E^(2*ArcCoth[a*x])*n*(-1152 + 576*n + 104*n^2 - 52*n^3 - 2*n^4 + n^5)*Hype 
rgeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (-2304 + 784*n^2 
- 56*n^4 + n^6)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])]))/a
 

3.8.36.3 Rubi [F]

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.

\(\displaystyle \int \left (c-a^2 c x^2\right )^3 e^{n \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

\(\Big \downarrow \) 6745

\(\displaystyle -a^6 c^3 \int \frac {e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\)

\(\Big \downarrow \) 2005

\(\displaystyle -c^3 \int e^{n \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\)

input

Int[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]
 

output

$Aborted
 

3.8.36.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 6745

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symb 
ol] :> Simp[d^p   Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], 
 x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] && In 
tegerQ[p]
 

3.8.36.4 Maple [F]

\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{3}d x\]

input

int(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^3,x)
 

output

int(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^3,x)
 

3.8.36.5 Fricas [F]

\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int { -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]

input

integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^3,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)*((a*x + 1)/( 
a*x - 1))^(1/2*n), x)
 

3.8.36.6 Sympy [F]

\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- c^{3} \left (\int 3 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- 3 a^{4} x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx + \int a^{6} x^{6} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \]

input

integrate(exp(n*acoth(a*x))*(-a**2*c*x**2+c)**3,x)
 

output

-c**3*(Integral(3*a**2*x**2*exp(n*acoth(a*x)), x) + Integral(-3*a**4*x**4* 
exp(n*acoth(a*x)), x) + Integral(a**6*x**6*exp(n*acoth(a*x)), x) + Integra 
l(-exp(n*acoth(a*x)), x))
 

3.8.36.7 Maxima [F]

\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int { -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]

input

integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^3,x, algorithm="maxima")
 

output

-integrate((a^2*c*x^2 - c)^3*((a*x + 1)/(a*x - 1))^(1/2*n), x)
 

3.8.36.8 Giac [F]

\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int { -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]

input

integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^3,x, algorithm="giac")
 

output

integrate(-(a^2*c*x^2 - c)^3*((a*x + 1)/(a*x - 1))^(1/2*n), x)
 

3.8.36.9 Mupad [F(-1)]

Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \]

input

int(exp(n*acoth(a*x))*(c - a^2*c*x^2)^3,x)
 

output

int(exp(n*acoth(a*x))*(c - a^2*c*x^2)^3, x)