Optimal. Leaf size=36 \[ \frac {x^{m+1}}{m+1}-\frac {2 x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{m+1} \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 64
Rule 80
Rule 6126
Rule 6167
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*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.75 \[ \frac {x^{m+1} (1-2 \, _2F_1(1,m+1;m+2;-a x))}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x - 1\right )} x^{m}}{a x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x - 1\right )} x^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 93, normalized size = 2.58 \[ a^{-1-m} \left (\frac {x^{m} a^{m} \left (a m x -m -1\right )}{\left (1+m \right ) m}+x^{m} a^{m} \Phi \left (-a x , 1, m\right )\right )-a^{-1-m} \left (\frac {x^{m} a^{m}}{m}+\frac {x^{m} a^{m} \left (-1-m \right ) \Phi \left (-a x , 1, m\right )}{1+m}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x - 1\right )} x^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.69, size = 119, normalized size = 3.31 \[ \frac {a m x^{2} x^{m} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {2 a x^{2} x^{m} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac {m x x^{m} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac {x x^{m} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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