3.8.38 \(\int e^{n \coth ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [738]

3.8.38.1 Optimal result

Integrand size = 20, antiderivative size = 79 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (4,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]

output

-16*c*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^(-2+1/2*n)*hypergeom([4, 2-1/2*n],[3-1 
/2*n],(a-1/x)/(a+1/x))/a/(4-n)
 

3.8.38.2 Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.41 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \left (-n-6 a x+a n^2 x+a^2 n x^2+2 a^3 x^3+e^{2 \coth ^{-1}(a x)} (-2+n) n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+\left (-4+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{6 a} \]

input

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

output

-1/6*(c*E^(n*ArcCoth[a*x])*(-n - 6*a*x + a*n^2*x + a^2*n*x^2 + 2*a^3*x^3 + 
 E^(2*ArcCoth[a*x])*(-2 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2 
*ArcCoth[a*x])] + (-4 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCo 
th[a*x])]))/a
 

3.8.38.3 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.

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

input

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

output

$Aborted
 

3.8.38.3.1 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]
 

3.8.38.4 Maple [F]

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

input

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

output

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

3.8.38.5 Fricas [F]

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

input

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

output

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

3.8.38.6 Sympy [F]

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

input

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

output

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

3.8.38.7 Maxima [F]

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

input

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

output

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

3.8.38.8 Giac [F]

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

input

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

output

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

3.8.38.9 Mupad [F(-1)]

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

input

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

output

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