Optimal. Leaf size=35 \[ \frac{x^{m+1}}{m+1}-\frac{2 x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,a x)}{m+1} \]
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Rubi [A] time = 0.0412795, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6126, 80, 64} \[ \frac{x^{m+1}}{m+1}-\frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 80
Rule 64
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} x^m \, dx &=-\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 = 0.0089821, size = 26, normalized size = 0.74 \[ \frac{x^{m+1} (1-2 \text{Hypergeometric2F1}(1,m+1,m+2,a x))}{m+1} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.38, size = 106, normalized size = 3. \begin{align*}{\frac{ \left ( -a \right ) ^{-m}}{a} \left ( -{\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( -1-m \right ) }{ \left ( 1+m \right ) m}}-{x}^{m} \left ( -a \right ) ^{m}{\it LerchPhi} \left ( ax,1,m \right ) \right ) }-{\frac{ \left ( -a \right ) ^{-m}}{a} \left ( -{\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( amx+m+1 \right ) }{ \left ( 1+m \right ) m}}+{x}^{m} \left ( -a \right ) ^{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 x + 1\right )} x^{m}}{a x - 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 = 2.29895, size = 100, normalized size = 2.86 \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 )} x^{m}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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