Integrand size = 14, antiderivative size = 41 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,\frac {1}{8},-\frac {1}{8},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6308, 138} \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (-m-1,\frac {1}{8},-\frac {1}{8},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]
[In]
[Out]
Rule 138
Rule 6308
Rubi steps \begin{align*} \text {integral}& = -\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,\frac {1}{8},-\frac {1}{8},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \\ \end{align*}
\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx \]
[In]
[Out]
\[\int \frac {x^{m}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}}}d x\]
[In]
[Out]
\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\int { \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\int { \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]
[In]
[Out]
\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\int { \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx=\int \frac {x^m}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}} \,d x \]
[In]
[Out]