Optimal. Leaf size=22 \[ \frac {x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1} \]
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Rubi [A] time = 0.08, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 64} \[ \frac {x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {x^m}{1-a x} \, dx\\ &=\frac {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 = 22, normalized size = 1.00 \[ \frac {x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{m}}{a x - 1}, x\right ) \]
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 \[ \int -\frac {{\left (a x + 1\right )} x^{m}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 100, normalized size = 4.55 \[ -\frac {\left (-a^{2}\right )^{-\frac {m}{2}} \left (-\frac {2 x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (-m -2\right )}{\left (2+m \right ) m}-x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \Phi \left (a^{2} x^{2}, 1, \frac {m}{2}\right )\right )}{2 a}+\frac {x^{1+m} \left (\frac {1}{2}+\frac {m}{2}\right ) \Phi \left (a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m} \]
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 \[ -\int \frac {{\left (a x + 1\right )} x^{m}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int -\frac {x^m\,\left (a\,x+1\right )}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.69, size = 44, normalized size = 2.00 \[ \frac {m x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac {x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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