\(\int e^{-3 \text {arctanh}(a x)} (c-a^2 c x^2)^p \, dx\) [1302]

Optimal result

Integrand size = 22, antiderivative size = 84 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{\frac {5}{2}+p} (1+a x)^{-\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-\frac {1}{2}+p,\frac {1}{2}+p,\frac {1}{2} (1+a x)\right )}{a (1-2 p)} \] Output:

-2^(5/2+p)*(a*x+1)^(-1/2+p)*(-a^2*c*x^2+c)^p*hypergeom([-1/2+p, -3/2-p],[1 
/2+p],1/2*a*x+1/2)/a/(1-2*p)/((-a^2*x^2+1)^p)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{-\frac {3}{2}+p} (1-a x)^{\frac {5}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,\frac {5}{2}+p,\frac {7}{2}+p,\frac {1}{2} (1-a x)\right )}{a \left (\frac {5}{2}+p\right )} \] Input:

Integrate[(c - a^2*c*x^2)^p/E^(3*ArcTanh[a*x]),x]
 

Output:

-((2^(-3/2 + p)*(1 - a*x)^(5/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/ 
2 - p, 5/2 + p, 7/2 + p, (1 - a*x)/2])/(a*(5/2 + p)*(1 - a^2*x^2)^p))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6693, 6690, 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.

\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\)

\(\Big \downarrow \) 6693

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{-3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^pdx\)

\(\Big \downarrow \) 6690

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int (1-a x)^{p+\frac {3}{2}} (a x+1)^{p-\frac {3}{2}}dx\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2^{p-\frac {1}{2}} (1-a x)^{p+\frac {5}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,p+\frac {5}{2},p+\frac {7}{2},\frac {1}{2} (1-a x)\right )}{a (2 p+5)}\)

Input:

Int[(c - a^2*c*x^2)^p/E^(3*ArcTanh[a*x]),x]
 

Output:

-((2^(-1/2 + p)*(1 - a*x)^(5/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/ 
2 - p, 5/2 + p, 7/2 + p, (1 - a*x)/2])/(a*(5 + 2*p)*(1 - a^2*x^2)^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 6690

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

rule 6693

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

Maple [F]

\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x +1\right )^{3}}d x\]

Input:

int((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Output:

int((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Fricas [F]

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

integrate((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

\[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{\left (a x + 1\right )^{3}}\, dx \] Input:

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

Output:

Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 
1)**3, x)
 

Maxima [F]

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

integrate((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

integrate((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac 
")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \] Input:

int(((c - a^2*c*x^2)^p*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
 

Output:

int(((c - a^2*c*x^2)^p*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3, x)
 

Reduce [F]

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

int((-a^2*c*x^2+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Output:

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