Optimal. Leaf size=86 \[ -\frac{2^{p+\frac{1}{2}} (1-a x)^{p+\frac{3}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,p+\frac{3}{2},p+\frac{5}{2},\frac{1}{2} (1-a x)\right )}{a (2 p+3)} \]
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Rubi [A] time = 0.0664606, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6143, 6140, 69} \[ -\frac{2^{p+\frac{1}{2}} (1-a x)^{p+\frac{3}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{1}{2}-p,p+\frac{3}{2};p+\frac{5}{2};\frac{1}{2} (1-a x)\right )}{a (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 69
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{\frac{1}{2}+p} (1+a x)^{-\frac{1}{2}+p} \, dx\\ &=-\frac{2^{\frac{1}{2}+p} (1-a x)^{\frac{3}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{1}{2}-p,\frac{3}{2}+p;\frac{5}{2}+p;\frac{1}{2} (1-a x)\right )}{a (3+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0307036, size = 83, normalized size = 0.97 \[ \frac{(a x-1) (2-2 a x)^{p+\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,p+\frac{3}{2},p+\frac{5}{2},\frac{1}{2} (1-a x)\right )}{a (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.422, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{ax+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{\sqrt{-a^{2} x^{2} + 1}{\left (-a^{2} c x^{2} + c\right )}^{p}}{a x + 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}{\left (-a^{2} c x^{2} + c\right )}^{p}}{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{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{a x + 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{\sqrt{-a^{2} x^{2} + 1}{\left (-a^{2} c x^{2} + c\right )}^{p}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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