\(\int \frac {e^{n \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx\) [1337]

Optimal result

Integrand size = 23, antiderivative size = 94 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c n}+\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a^2 c n} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.79 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {(1-a x)^{-n/2} \left (-(1+a x)^{n/2}+2^{1+\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{a^2 c n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6700, 88, 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.

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

(-((1 + a*x)^(n/2)/(a^2*n*(1 - a*x)^(n/2))) + (2^(1 + n/2)*Hypergeometric2 
F1[-1/2*n, -1/2*n, 1 - n/2, (1 - a*x)/2])/(a^2*n*(1 - a*x)^(n/2)))/c
 

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 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 6700

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-Integral(x*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - int((e**(atanh(a*x)*n)*x)/(a**2*x**2 - 1),x))/c