\(\int e^{-4 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [743]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{-2+p} (1+a x)^{-3-p} \left (c-a^2 c x^2\right )^{3+p} \operatorname {Hypergeometric2F1}\left (2-p,3+p,4+p,\frac {1}{2} (1-a x)\right )}{a c^3 (3+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12 \[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{-2+p} (1-a x)^{3+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (2-p,3+p,4+p,\frac {1}{2} (1-a x)\right )}{a (3+p)} \] Input:

Integrate[(c - a^2*c*x^2)^p/E^(4*ArcCoth[a*x]),x]
 

Output:

-((2^(-2 + p)*(1 - a*x)^(3 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[2 - p, 
 3 + p, 4 + p, (1 - a*x)/2])/(a*(3 + p)*(1 - a^2*x^2)^p))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6717, 6692, 473, 79}

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.

\(\displaystyle \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle \int e^{-4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^pdx\)

\(\Big \downarrow \) 6692

\(\displaystyle c^2 \int (1-a x)^4 \left (c-a^2 c x^2\right )^{p-2}dx\)

\(\Big \downarrow \) 473

\(\displaystyle c^2 (1-a x)^{1-p} (a c x+c)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \int (1-a x)^{p+2} (a x c+c)^{p-2}dx\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {c 2^{p+2} (1-a x)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \operatorname {Hypergeometric2F1}\left (-p-2,p-1,p,\frac {1}{2} (a x+1)\right )}{a (1-p)}\)

Input:

Int[(c - a^2*c*x^2)^p/E^(4*ArcCoth[a*x]),x]
 

Output:

-((2^(2 + p)*c*(1 - a*x)^(1 - p)*(c - a^2*c*x^2)^(-1 + p)*Hypergeometric2F 
1[-2 - p, -1 + p, p, (1 + a*x)/2])/(a*(1 - p)))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 6692

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[1/c^(n/2)   Int[(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, 
 c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[ 
n/2, 0]
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [F]

\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \left (a x -1\right )^{2}}{\left (a x +1\right )^{2}}d x\]

Input:

int((-a^2*c*x^2+c)^p*(a*x-1)^2/(a*x+1)^2,x)
 

Output:

int((-a^2*c*x^2+c)^p*(a*x-1)^2/(a*x+1)^2,x)
 

Fricas [F]

\[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x - 1\right )}^{2} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^p*(a*x-1)^2/(a*x+1)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x - 1\right )^{2}}{\left (a x + 1\right )^{2}}\, dx \] Input:

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

Output:

Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x - 1)**2/(a*x + 1)**2, x)
 

Maxima [F]

\[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x - 1\right )}^{2} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^p*(a*x-1)^2/(a*x+1)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x - 1\right )}^{2} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((-a^2*c*x^2+c)^p*(a*x-1)^2/(a*x+1)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2} \,d x \] Input:

int(((c - a^2*c*x^2)^p*(a*x - 1)^2)/(a*x + 1)^2,x)
 

Output:

int(((c - a^2*c*x^2)^p*(a*x - 1)^2)/(a*x + 1)^2, x)
 

Reduce [F]

\[ \int e^{-4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (-a^{2} c \,x^{2}+c \right )^{p} a^{2} p \,x^{2}-3 \left (-a^{2} c \,x^{2}+c \right )^{p} a p x -2 \left (-a^{2} c \,x^{2}+c \right )^{p} a x -2 \left (-a^{2} c \,x^{2}+c \right )^{p} p^{2}-4 \left (-a^{2} c \,x^{2}+c \right )^{p} p -2 \left (-a^{2} c \,x^{2}+c \right )^{p}+8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{3} p^{4} x +28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{3} p^{3} x +28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{3} p^{2} x +8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{3} p x +8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{2} p^{4}+28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{2} p^{3}+28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{2} p^{2}+8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}+2 a^{2} p \,x^{2}+a^{2} x^{2}-2 a p x -a x -2 p -1}d x \right ) a^{2} p}{a p \left (2 a p x +a x +2 p +1\right )} \] Input:

int((-a^2*c*x^2+c)^p*(a*x-1)^2/(a*x+1)^2,x)
 

Output:

(( - a**2*c*x**2 + c)**p*a**2*p*x**2 - 3*( - a**2*c*x**2 + c)**p*a*p*x - 2 
*( - a**2*c*x**2 + c)**p*a*x - 2*( - a**2*c*x**2 + c)**p*p**2 - 4*( - a**2 
*c*x**2 + c)**p*p - 2*( - a**2*c*x**2 + c)**p + 8*int((( - a**2*c*x**2 + c 
)**p*x)/(2*a**3*p*x**3 + a**3*x**3 + 2*a**2*p*x**2 + a**2*x**2 - 2*a*p*x - 
 a*x - 2*p - 1),x)*a**3*p**4*x + 28*int((( - a**2*c*x**2 + c)**p*x)/(2*a** 
3*p*x**3 + a**3*x**3 + 2*a**2*p*x**2 + a**2*x**2 - 2*a*p*x - a*x - 2*p - 1 
),x)*a**3*p**3*x + 28*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a** 
3*x**3 + 2*a**2*p*x**2 + a**2*x**2 - 2*a*p*x - a*x - 2*p - 1),x)*a**3*p**2 
*x + 8*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a**3*x**3 + 2*a**2 
*p*x**2 + a**2*x**2 - 2*a*p*x - a*x - 2*p - 1),x)*a**3*p*x + 8*int((( - a* 
*2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a**3*x**3 + 2*a**2*p*x**2 + a**2*x** 
2 - 2*a*p*x - a*x - 2*p - 1),x)*a**2*p**4 + 28*int((( - a**2*c*x**2 + c)** 
p*x)/(2*a**3*p*x**3 + a**3*x**3 + 2*a**2*p*x**2 + a**2*x**2 - 2*a*p*x - a* 
x - 2*p - 1),x)*a**2*p**3 + 28*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x 
**3 + a**3*x**3 + 2*a**2*p*x**2 + a**2*x**2 - 2*a*p*x - a*x - 2*p - 1),x)* 
a**2*p**2 + 8*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a**3*x**3 + 
 2*a**2*p*x**2 + a**2*x**2 - 2*a*p*x - a*x - 2*p - 1),x)*a**2*p)/(a*p*(2*a 
*p*x + a*x + 2*p + 1))