\(\int e^{-\text {arctanh}(a x)} (c-a^2 c x^2)^p \, dx\) [1262]

Optimal result

Integrand size = 22, antiderivative size = 83 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{\frac {3}{2}+p} (1+a x)^{\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,\frac {1}{2}+p,\frac {3}{2}+p,\frac {1}{2} (1+a x)\right )}{a (1+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {(2-2 a x)^{\frac {1}{2}+p} (-1+a x) \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,\frac {3}{2}+p,\frac {5}{2}+p,\frac {1}{2} (1-a x)\right )}{a (3+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 6693

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

\(\Big \downarrow \) 6690

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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