Integrand size = 18, antiderivative size = 59 \[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a c n} \] Output:
2^(1+1/2*n)*hypergeom([-1/2*n, -1/2*n],[1-1/2*n],-1/2*a*x+1/2)/a/c/n/((-a* x+1)^(1/2*n))
Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a c n} \] Input:
Integrate[E^(n*ArcTanh[a*x])/(c - a*c*x),x]
Output:
(2^(1 + n/2)*Hypergeometric2F1[-1/2*n, -1/2*n, 1 - n/2, (1 - a*x)/2])/(a*c *n*(1 - a*x)^(n/2))
Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\int (1-a x)^{-\frac {n}{2}-1} (a x+1)^{n/2}dx}{c}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a c n}\) |
Input:
Int[E^(n*ArcTanh[a*x])/(c - a*c*x),x]
Output:
(2^(1 + n/2)*Hypergeometric2F1[-1/2*n, -1/2*n, 1 - n/2, (1 - a*x)/2])/(a*c *n*(1 - a*x)^(n/2))
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]))
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])
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{-a c x +c}d x\]
Input:
int(exp(n*arctanh(a*x))/(-a*c*x+c),x)
Output:
int(exp(n*arctanh(a*x))/(-a*c*x+c),x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c),x, algorithm="fricas")
Output:
integral(-(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a x - 1}\, dx}{c} \] Input:
integrate(exp(n*atanh(a*x))/(-a*c*x+c),x)
Output:
-Integral(exp(n*atanh(a*x))/(a*x - 1), x)/c
\[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c),x, algorithm="maxima")
Output:
-integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c),x, algorithm="giac")
Output:
integrate(-(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a\,c\,x} \,d x \] Input:
int(exp(n*atanh(a*x))/(c - a*c*x),x)
Output:
int(exp(n*atanh(a*x))/(c - a*c*x), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{c-a c x} \, dx=-\frac {\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a x -1}d x}{c} \] Input:
int(exp(n*atanh(a*x))/(-a*c*x+c),x)
Output:
( - int(e**(atanh(a*x)*n)/(a*x - 1),x))/c