Integrand size = 24, antiderivative size = 111 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {2^{\frac {5}{2}-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \left (c-\frac {c}{a x}\right )^{3/2} \operatorname {AppellF1}\left (\frac {2+n}{2},\frac {1}{2} (-3+n),2,\frac {4+n}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \left (1-\frac {1}{a x}\right )^{3/2}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 111, 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 = {6317, 6314, 141} \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {2^{\frac {5}{2}-\frac {n}{2}} \left (c-\frac {c}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {AppellF1}\left (\frac {n+2}{2},\frac {n-3}{2},2,\frac {n+4}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2) \left (1-\frac {1}{a x}\right )^{3/2}} \]
[In]
[Out]
Rule 141
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-\frac {c}{a x}\right )^{3/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {\left (c-\frac {c}{a x}\right )^{3/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {2^{\frac {5}{2}-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \left (c-\frac {c}{a x}\right )^{3/2} \operatorname {AppellF1}\left (\frac {2+n}{2},\frac {1}{2} (-3+n),2,\frac {4+n}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \left (1-\frac {1}{a x}\right )^{3/2}} \\ \end{align*}
Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {\$Aborted} \]
[In]
[Out]
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (c -\frac {c}{a x}\right )^{\frac {3}{2}}d x\]
[In]
[Out]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
[In]
[Out]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a\,x}\right )}^{3/2} \,d x \]
[In]
[Out]