Integrand size = 10, antiderivative size = 74 \[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{1+m}+\frac {x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \]
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Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6307, 822, 371} \[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-1),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{m+1}+\frac {x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \]
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Rule 371
Rule 822
Rule 6307
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} \left (1+\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )-\frac {\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{1+m}+\frac {x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.73 \[ \int e^{\coth ^{-1}(a x)} x^m \, dx=x^{1+m} \left (-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {-\frac {1}{a^2}+x^2} \operatorname {AppellF1}\left (m,-\frac {1}{2},\frac {1}{2},1+m,-a x,a x\right )}{m \sqrt {-1+a x} \sqrt {\frac {1+a x}{a^2}} \sqrt {1-a^2 x^2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{1+m}\right ) \]
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\[\int \frac {x^{m}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]
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\[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\int { \frac {x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\int \frac {x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\int { \frac {x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\int { \frac {x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} x^m \, dx=\int \frac {x^m}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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