Integrand size = 12, antiderivative size = 36 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m}}{1+m}-\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{1+m} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6261, 81, 66} \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1}}{m+1}-\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-a x)}{m+1} \]
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Rule 66
Rule 81
Rule 6261
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} x^m \, dx \\ & = -\int \frac {x^m (1-a x)}{1+a x} \, dx \\ & = \frac {x^{1+m}}{1+m}-2 \int \frac {x^m}{1+a x} \, dx \\ & = \frac {x^{1+m}}{1+m}-\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{1+m} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} (1-2 \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x))}{1+m} \]
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Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.49 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.58
method | result | size |
meijerg | \(a^{-m -1} \left (\frac {x^{m} a^{m} \left (a m x -m -1\right )}{\left (1+m \right ) m}+x^{m} a^{m} \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )-a^{-m -1} \left (\frac {x^{m} a^{m}}{m}+\frac {x^{m} a^{m} \left (-m -1\right ) \operatorname {LerchPhi}\left (-a x , 1, m\right )}{1+m}\right )\) | \(93\) |
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\[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int { \frac {{\left (a x - 1\right )} x^{m}}{a x + 1} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.22 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {a m x^{m + 2} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {2 a x^{m + 2} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac {m x^{m + 1} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac {x^{m + 1} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \]
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\[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int { \frac {{\left (a x - 1\right )} x^{m}}{a x + 1} \,d x } \]
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\[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int { \frac {{\left (a x - 1\right )} x^{m}}{a x + 1} \,d x } \]
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Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int \frac {x^m\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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