\(\int e^{n \text {arctanh}(a x)} (c-a^2 c x^2)^2 \, dx\) [1332]

Optimal result

Integrand size = 22, antiderivative size = 70 \[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {2^{3+\frac {n}{2}} c^2 (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (-2-\frac {n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (6-n)} \] Output:

-2^(3+1/2*n)*c^2*(-a*x+1)^(3-1/2*n)*hypergeom([-2-1/2*n, 3-1/2*n],[4-1/2*n 
],-1/2*a*x+1/2)/a/(6-n)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {2^{3+\frac {n}{2}} c^2 (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (-2-\frac {n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (-6+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {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.

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

\(\Big \downarrow \) 6690

\(\displaystyle c^2 \int (1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n+4}{2}}dx\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-4),3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (6-n)}\)

Input:

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

Output:

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

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*(-(a*x + 1)/(a*x - 1))^(1/2*n 
), x)
 

Sympy [F]

\[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=c^{2} \left (\int \left (- 2 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx + \int a^{4} x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int e^{n \operatorname {atanh}{\left (a x \right )}}\, dx\right ) \] Input:

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

Output:

c**2*(Integral(-2*a**2*x**2*exp(n*atanh(a*x)), x) + Integral(a**4*x**4*exp 
(n*atanh(a*x)), x) + Integral(exp(n*atanh(a*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=c^{2} \left (\int e^{\mathit {atanh} \left (a x \right ) n}d x +\left (\int e^{\mathit {atanh} \left (a x \right ) n} x^{4}d x \right ) a^{4}-2 \left (\int e^{\mathit {atanh} \left (a x \right ) n} x^{2}d x \right ) a^{2}\right ) \] Input:

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

Output:

c**2*(int(e**(atanh(a*x)*n),x) + int(e**(atanh(a*x)*n)*x**4,x)*a**4 - 2*in 
t(e**(atanh(a*x)*n)*x**2,x)*a**2)