Optimal. Leaf size=51 \[ \frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{a x}+1\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1} \]
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Rubi [A] time = 0.123224, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6192, 6196, 37} \[ \frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{a x}+1\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6196
Rule 37
Rubi steps
\begin{align*} \int e^{2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{2 p \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^p x^{2 p} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \operatorname{Subst}\left (\int x^{-2-2 p} \left (1+\frac{x}{a}\right )^{2 p} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (1+\frac{1}{a x}\right )^{1+2 p} x \left (c-a^2 c x^2\right )^p}{1+2 p}\\ \end{align*}
Mathematica [A] time = 0.0644862, size = 36, normalized size = 0.71 \[ \frac{(a x+1) \left (c-a^2 c x^2\right )^p e^{2 p \coth ^{-1}(a x)}}{2 a p+a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 38, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax+1 \right ){{\rm e}^{2\,p{\rm arccoth} \left (ax\right )}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{a \left ( 1+2\,p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.055, size = 49, normalized size = 0.96 \begin{align*} \frac{{\left (a \left (-c\right )^{p} x - \left (-c\right )^{p}\right )}{\left (a x - 1\right )}^{2 \, p}}{a{\left (2 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.401, size = 89, normalized size = 1.75 \begin{align*} \frac{{\left (a x - 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{p}}{2 \, a p + a} \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{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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