Optimal. Leaf size=31 \[ \frac{x^{m+1} F_1\left (m+1;-\frac{1}{4},\frac{1}{4};m+2;a x,-a x\right )}{m+1} \]
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Rubi [A] time = 0.0247624, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6126, 133} \[ \frac{x^{m+1} F_1\left (m+1;-\frac{1}{4},\frac{1}{4};m+2;a x,-a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 133
Rubi steps
\begin{align*} \int e^{-\frac{1}{2} \tanh ^{-1}(a x)} x^m \, dx &=\int \frac{x^m \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac{x^{1+m} F_1\left (1+m;-\frac{1}{4},\frac{1}{4};2+m;a x,-a x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.29763, size = 0, normalized size = 0. \[ \int e^{-\frac{1}{2} \tanh ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}}}}\, dx \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{x^{m}}{\sqrt{\frac{a x + 1}{\sqrt{-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{\sqrt{-a^{2} x^{2} + 1} x^{m} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx \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{x^{m}}{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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