Optimal. Leaf size=41 \[ \frac{(1-a x)^{1-2 p}}{a (1-2 p)}+\frac{(1-a x)^{-2 p}}{a p} \]
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Rubi [A] time = 0.0478388, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {6140, 43} \[ \frac{(1-a x)^{1-2 p}}{a (1-2 p)}+\frac{(1-a x)^{-2 p}}{a p} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{2 (1+p) \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{-p} \, dx &=\int (1-a x)^{-1-2 p} (1+a x) \, dx\\ &=\int \left (2 (1-a x)^{-1-2 p}-(1-a x)^{-2 p}\right ) \, dx\\ &=\frac{(1-a x)^{1-2 p}}{a (1-2 p)}+\frac{(1-a x)^{-2 p}}{a p}\\ \end{align*}
Mathematica [A] time = 0.0187566, size = 31, normalized size = 0.76 \[ \frac{(1-a x)^{-2 p} (a p x+p-1)}{a p (2 p-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 59, normalized size = 1.4 \begin{align*} -{\frac{ \left ( apx+p-1 \right ) \left ( ax-1 \right ){{\rm e}^{2\, \left ( 1+p \right ){\it Artanh} \left ( ax \right ) }}}{ \left ( ax+1 \right ) ap \left ( 2\,p-1 \right ) \left ( -{a}^{2}{x}^{2}+1 \right ) ^{p}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00252, size = 54, normalized size = 1.32 \begin{align*} -\frac{a p x + p - 1}{{\left (2 \, \left (-1\right )^{p} p^{2} - \left (-1\right )^{p} p\right )}{\left (a x - 1\right )}^{2 \, p} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14788, size = 158, normalized size = 3.85 \begin{align*} -\frac{{\left (a^{2} p x^{2} - a x - p + 1\right )} \left (\frac{a x + 1}{a x - 1}\right )^{p + 1}}{{\left (2 \, a p^{2} - a p +{\left (2 \, a^{2} p^{2} - a^{2} p\right )} x\right )}{\left (-a^{2} x^{2} + 1\right )}^{p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (\frac{a x + 1}{a x - 1}\right )^{p + 1}}{{\left (-a^{2} x^{2} + 1\right )}^{p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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