Integrand size = 24, antiderivative size = 56 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {2 x \sqrt {1-a x} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1+n}{2},-\frac {n}{2},\frac {5}{2},a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}} \] Output:
2/3*x*(-a*x+1)^(1/2)*AppellF1(3/2,1/2+1/2*n,-1/2*n,5/2,a*x,-a*x)/(c-c/a/x) ^(1/2)
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\text {\$Aborted} \] Input:
Integrate[E^(n*ArcTanh[a*x])/Sqrt[c - c/(a*x)],x]
Output:
$Aborted
Time = 0.41 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6684, 6679, 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.
\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle \frac {\sqrt {1-a x} \int \frac {e^{n \text {arctanh}(a x)} \sqrt {x}}{\sqrt {1-a x}}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\sqrt {1-a x} \int \sqrt {x} (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{n/2}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {2 x \sqrt {1-a x} \operatorname {AppellF1}\left (\frac {3}{2},\frac {n+1}{2},-\frac {n}{2},\frac {5}{2},a x,-a x\right )}{3 \sqrt {c-\frac {c}{a x}}}\) |
Input:
Int[E^(n*ArcTanh[a*x])/Sqrt[c - c/(a*x)],x]
Output:
(2*x*Sqrt[1 - a*x]*AppellF1[3/2, (1 + n)/2, -1/2*n, 5/2, a*x, -(a*x)])/(3* Sqrt[c - c/(a*x)])
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_.))*(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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\sqrt {c -\frac {c}{a x}}}d x\]
Input:
int(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x)
Output:
int(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a x}}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm="fricas")
Output:
integral(a*x*(-(a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a*c*x - c)/(a*x))/(a*c*x - c), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right )}}\, dx \] Input:
integrate(exp(n*atanh(a*x))/(c-c/a/x)**(1/2),x)
Output:
Integral(exp(n*atanh(a*x))/sqrt(-c*(-1 + 1/(a*x))), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a x}}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm="maxima")
Output:
integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/sqrt(c - c/(a*x)), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {c - \frac {c}{a x}}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm="giac")
Output:
integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/sqrt(c - c/(a*x)), x)
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-\frac {c}{a\,x}}} \,d x \] Input:
int(exp(n*atanh(a*x))/(c - c/(a*x))^(1/2),x)
Output:
int(exp(n*atanh(a*x))/(c - c/(a*x))^(1/2), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {x}\, e^{\mathit {atanh} \left (a x \right ) n}}{\sqrt {a x -1}}d x \right )}{\sqrt {c}} \] Input:
int(exp(n*atanh(a*x))/(c-c/a/x)^(1/2),x)
Output:
(sqrt(a)*int((sqrt(x)*e**(atanh(a*x)*n))/sqrt(a*x - 1),x))/sqrt(c)