Optimal. Leaf size=36 \[ \frac{2 x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,a x)}{m+1}-\frac{x^{m+1}}{m+1} \]
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Rubi [A] time = 0.0245886, antiderivative size = 36, 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 = 0.0068547, size = 26, normalized size = 0.72 \[ \frac{x^{m+1} (2 \text{Hypergeometric2F1}(1,m+1,m+2,a x)-1)}{m+1} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.359, size = 184, normalized size = 5.1 \begin{align*} -{\frac{1}{2} \left ( -{a}^{2} \right ) ^{-{\frac{1}{2}}-{\frac{m}{2}}} \left ( 2\,{\frac{{x}^{1+m} \left ( -{a}^{2} \right ) ^{3/2+m/2} \left ( -3-m \right ) }{ \left ( 1+m \right ) \left ( 3+m \right ){a}^{2}}}+{\frac{{x}^{1+m}}{{a}^{2}} \left ( -{a}^{2} \right ) ^{{\frac{3}{2}}+{\frac{m}{2}}}{\it LerchPhi} \left ({a}^{2}{x}^{2},1,{\frac{1}{2}}+{\frac{m}{2}} \right ) } \right ) }-{\frac{1}{a} \left ( -{a}^{2} \right ) ^{-{\frac{m}{2}}} \left ( -2\,{\frac{{x}^{m} \left ( -{a}^{2} \right ) ^{m/2} \left ( -m-2 \right ) }{m \left ( 2+m \right ) }}-{x}^{m} \left ( -{a}^{2} \right ) ^{{\frac{m}{2}}}{\it LerchPhi} \left ({a}^{2}{x}^{2},1,{\frac{m}{2}} \right ) \right ) }+{\frac{{x}^{1+m}}{1+m} \left ({\frac{1}{2}}+{\frac{m}{2}} \right ){\it LerchPhi} \left ({a}^{2}{x}^{2},1,{\frac{1}{2}}+{\frac{m}{2}} \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 x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{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 [B] time = 3.26814, size = 99, normalized size = 2.75 \begin{align*} \frac{a m x^{2} x^{m} \Phi \left (a x, 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, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac{m x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac{x x^{m} \Phi \left (a x, 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 x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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