Optimal. Leaf size=75 \[ -\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{a x}+1\right )^{2 p+1} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (2,2 p+1,2 (p+1),\frac{1}{a x}+1\right )}{a (2 p+1)} \]
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Rubi [A] time = 0.0841536, antiderivative size = 75, 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 = {6197, 6194, 65} \[ -\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{a x}+1\right )^{2 p+1} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);1+\frac{1}{a x}\right )}{a (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6194
Rule 65
Rubi steps
\begin{align*} \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^p \, dx &=\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p\right ) \int e^{2 p \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^p \, dx\\ &=-\left (\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{2 p}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \left (1+\frac{1}{a x}\right )^{1+2 p} \, _2F_1\left (2,1+2 p;2 (1+p);1+\frac{1}{a x}\right )}{a (1+2 p)}\\ \end{align*}
Mathematica [F] time = 0.462574, size = 0, normalized size = 0. \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.182, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,p{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{p}\, 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{\left (c - \frac{c}{a^{2} x^{2}}\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{a x - 1}{a x + 1}\right )^{p} \left (\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}, 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 \left (- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )\right )^{p} e^{2 p \operatorname{acoth}{\left (a x \right )}}\, 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{\left (c - \frac{c}{a^{2} x^{2}}\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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