\(\int e^{n \text {arctanh}(a x)} (c-a c x)^2 \, dx\) [316]

Optimal result

Integrand size = 18, antiderivative size = 65 \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^2 \, dx=\frac {2^{3-\frac {n}{2}} c^2 (1+a x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-4+n),\frac {2+n}{2},\frac {4+n}{2},\frac {1}{2} (1+a x)\right )}{a (2+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^2 \, dx=\frac {2^{1+\frac {n}{2}} c^2 (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (3-\frac {n}{2},-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (-6+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6679, 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.

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

-((2^(1 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[3 - n/2, -1/2*n, 
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 6679

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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