Optimal. Leaf size=37 \[ \frac{2 x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,-a x)}{m+1}-\frac{x^{m+1}}{m+1} \]
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Rubi [A] time = 0.0249729, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6126, 80, 64} \[ \frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{m+1}-\frac{x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 80
Rule 64
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(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} \, _2F_1(1,1+m;2+m;-a x)}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0074615, size = 27, normalized size = 0.73 \[ \frac{x^{m+1} (2 \text{Hypergeometric2F1}(1,m+1,m+2,-a x)-1)}{m+1} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.36, size = 126, normalized size = 3.4 \begin{align*} -{a}^{-1-m} \left ({\frac{{x}^{m}{a}^{m} \left ({a}^{2}m{x}^{2}-amx-2\,ax-{m}^{2}-3\,m-2 \right ) }{m \left ( 1+m \right ) \left ( ax+1 \right ) }}+{x}^{m}{a}^{m} \left ( 2+m \right ){\it LerchPhi} \left ( -ax,1,m \right ) \right ) +{a}^{-1-m} \left ({\frac{{x}^{m}{a}^{m} \left ( -1-m \right ) }{ \left ( 1+m \right ) \left ( ax+1 \right ) }}+{x}^{m}{a}^{m}m{\it LerchPhi} \left ( -ax,1,m \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a^{2} x^{2} - 1\right )} x^{m}}{{\left (a x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a x - 1\right )} x^{m}}{a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.69069, size = 119, normalized size = 3.22 \begin{align*} - \frac{a m x^{2} x^{m} \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^{2} x^{m} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac{m x x^{m} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac{x x^{m} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a^{2} x^{2} - 1\right )} x^{m}}{{\left (a x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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