Optimal. Leaf size=60 \[ \frac{x (1-a x)^{-p} F_1\left (1-p;-p-\frac{1}{2},\frac{1}{2};2-p;a x,-a x\right ) \left (c-\frac{c}{a x}\right )^p}{1-p} \]
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Rubi [A] time = 0.0952823, antiderivative size = 60, 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 = {6134, 6129, 133} \[ \frac{x (1-a x)^{-p} F_1\left (1-p;-p-\frac{1}{2},\frac{1}{2};2-p;a x,-a x\right ) \left (c-\frac{c}{a x}\right )^p}{1-p} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 133
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx &=\left (\left (c-\frac{c}{a x}\right )^p x^p (1-a x)^{-p}\right ) \int e^{-\tanh ^{-1}(a x)} x^{-p} (1-a x)^p \, dx\\ &=\left (\left (c-\frac{c}{a x}\right )^p x^p (1-a x)^{-p}\right ) \int \frac{x^{-p} (1-a x)^{\frac{1}{2}+p}}{\sqrt{1+a x}} \, dx\\ &=\frac{\left (c-\frac{c}{a x}\right )^p x (1-a x)^{-p} F_1\left (1-p;-\frac{1}{2}-p,\frac{1}{2};2-p;a x,-a x\right )}{1-p}\\ \end{align*}
Mathematica [F] time = 1.04295, size = 0, normalized size = 0. \[ \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ax+1} \left ( c-{\frac{c}{ax}} \right ) ^{p}\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 (c - \frac{c}{a x}\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 (\frac{a c x - c}{a x}\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{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{p} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{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 (c - \frac{c}{a x}\right )}^{p}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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