\(\int e^{n \text {arctanh}(a x)} (c-a^2 c x^2) \, dx\) [1346]

Optimal result

Integrand size = 20, antiderivative size = 63 \[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {2^{2-\frac {n}{2}} c (1+a x)^{\frac {4+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\frac {4+n}{2},\frac {6+n}{2},\frac {1}{2} (1+a x)\right )}{a (4+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int e^{n \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {2^{2+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (-4+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 6690

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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