Integrand size = 25, antiderivative size = 42 \[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {2+n}{2},1-\frac {n}{2},2+m,a x,-a x\right )}{c (1+m)} \] Output:
x^(1+m)*AppellF1(1+m,1+1/2*n,1-1/2*n,2+m,a*x,-a*x)/c/(1+m)
Leaf count is larger than twice the leaf count of optimal. \(106\) vs. \(2(42)=84\).
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.52 \[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\frac {e^{n \text {arctanh}(a x)} \left (-1+e^{-2 \text {arctanh}(a x)}\right )^m \left (1+e^{-2 \text {arctanh}(a x)}\right )^m \left (-e^{-4 \text {arctanh}(a x)} \left (-1+e^{2 \text {arctanh}(a x)}\right )^2\right )^{-m} x^m \operatorname {AppellF1}\left (-\frac {n}{2},m,-m,1-\frac {n}{2},-e^{-2 \text {arctanh}(a x)},e^{-2 \text {arctanh}(a x)}\right )}{a c n} \] Input:
Integrate[(E^(n*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2),x]
Output:
(E^(n*ArcTanh[a*x])*(-1 + E^(-2*ArcTanh[a*x]))^m*(1 + E^(-2*ArcTanh[a*x])) ^m*x^m*AppellF1[-1/2*n, m, -m, 1 - n/2, -E^(-2*ArcTanh[a*x]), E^(-2*ArcTan h[a*x])])/(a*c*(-((-1 + E^(2*ArcTanh[a*x]))^2/E^(4*ArcTanh[a*x])))^m*n)
Time = 0.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6700, 150}
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.
\(\displaystyle \int \frac {x^m e^{n \text {arctanh}(a x)}}{c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\int x^m (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}dx}{c}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {n+2}{2},\frac {2-n}{2},m+2,a x,-a x\right )}{c (m+1)}\) |
Input:
Int[(E^(n*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2),x]
Output:
(x^(1 + m)*AppellF1[1 + m, (2 + n)/2, (2 - n)/2, 2 + m, a*x, -(a*x)])/(c*( 1 + m))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
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])
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{m}}{-a^{2} c \,x^{2}+c}d x\]
Input:
int(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c),x)
Output:
int(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c),x)
\[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\int { -\frac {x^{m} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c),x, algorithm="fricas")
Output:
integral(-x^m*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)
\[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=- \frac {\int \frac {x^{m} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \] Input:
integrate(exp(n*atanh(a*x))*x**m/(-a**2*c*x**2+c),x)
Output:
-Integral(x**m*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c
\[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\int { -\frac {x^{m} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c),x, algorithm="maxima")
Output:
-integrate(x^m*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)
\[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\int { -\frac {x^{m} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c),x, algorithm="giac")
Output:
integrate(-x^m*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a^2\,c\,x^2} \,d x \] Input:
int((x^m*exp(n*atanh(a*x)))/(c - a^2*c*x^2),x)
Output:
int((x^m*exp(n*atanh(a*x)))/(c - a^2*c*x^2), x)
\[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n}-\left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n}}{x}d x \right ) m}{a c n} \] Input:
int(exp(n*atanh(a*x))*x^m/(-a^2*c*x^2+c),x)
Output:
(x**m*e**(atanh(a*x)*n) - int((x**m*e**(atanh(a*x)*n))/x,x)*m)/(a*c*n)