\(\int e^{n \text {arctanh}(a x)} (c-a^2 c x^2)^p \, dx\) [1400]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (n-2 p),1+\frac {n}{2}+p,2+\frac {n}{2}+p,\frac {1}{2} (1+a x)\right )}{a (2+n+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04 \[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{\frac {n}{2}+p} (1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-p,1-\frac {n}{2}+p,2-\frac {n}{2}+p,\frac {1}{2} (1-a x)\right )}{a \left (1-\frac {n}{2}+p\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, 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.

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

\(\Big \downarrow \) 6693

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

\(\Big \downarrow \) 6690

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

-((2^(1 + n/2 + p)*(1 - a*x)^(1 - n/2 + p)*(c - a^2*c*x^2)^p*Hypergeometri 
c2F1[-1/2*n - p, 1 - n/2 + p, 2 - n/2 + p, (1 - a*x)/2])/(a*(2 - n + 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((-c*(a*x - 1)*(a*x + 1))**p*exp(n*atanh(a*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atanh(a*x)*n)*( - a**2*c*x**2 + c)**p,x)