Optimal. Leaf size=101 \[ -\frac{2 \sqrt{1-a^2 x^2} (1-a x)^{\frac{1-n}{2}} (a x+1)^{\frac{n-1}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{1-a x}{a x+1}\right )}{(1-n) \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.230984, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 131} \[ -\frac{2 \sqrt{1-a^2 x^2} (1-a x)^{\frac{1-n}{2}} (a x+1)^{\frac{n-1}{2}} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{1-a x}{a x+1}\right )}{(1-n) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x \sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{n \tanh ^{-1}(a x)}}{x \sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{1}{2}+\frac{n}{2}}}{x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{2 (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-1+n)} \sqrt{1-a^2 x^2} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{1-a x}{1+a x}\right )}{(1-n) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0482765, size = 99, normalized size = 0.98 \[ \frac{2 \sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2}-\frac{n}{2}} (a x+1)^{\frac{n-1}{2}} \text{Hypergeometric2F1}\left (1,\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )}{(n-1) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.205, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{x}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}}\, 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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{\sqrt{-a^{2} c x^{2} + c} x}\,{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} c x^{2} + c} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{3} - c x}, 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{e^{n \operatorname{atanh}{\left (a x \right )}}}{x \sqrt{- c \left (a x - 1\right ) \left (a x + 1\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 \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{\sqrt{-a^{2} c x^{2} + c} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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