Optimal. Leaf size=272 \[ \frac{2^{\frac{n+1}{2}} n x \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^{\frac{3-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-a x)\right )}{\left (n^2-4 n+3\right ) \sqrt{1-a^2 x^2}}+\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a x+1}{1-a x}\right )}{(1-n) \sqrt{1-a^2 x^2}}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{3-n}{2}}}{(1-n) \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.259879, antiderivative size = 302, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6160, 6150, 105, 69, 131} \[ \frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1}{2} (-n-1)} \, _2F_1\left (1,\frac{1}{2} (-n-1);\frac{1-n}{2};\frac{1-a x}{a x+1}\right )}{(n+1) \sqrt{1-a^2 x^2}}-\frac{2^{\frac{n+3}{2}} x \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^{\frac{1}{2} (-n-1)} \, _2F_1\left (\frac{1}{2} (-n-1),\frac{1}{2} (-n-1);\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{(n+1) \sqrt{1-a^2 x^2}}+\frac{2^{\frac{n+3}{2}} x \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^{\frac{1-n}{2}} \, _2F_1\left (\frac{1}{2} (-n-1),\frac{1-n}{2};\frac{3-n}{2};\frac{1}{2} (1-a x)\right )}{(1-n) \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 6160
Rule 6150
Rule 105
Rule 69
Rule 131
Rubi steps
\begin{align*} \int e^{n \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{n \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}}}{x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}}}{x} \, dx}{\sqrt{1-a^2 x^2}}-\frac{\left (a \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int (1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{2^{\frac{3+n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)^{\frac{1-n}{2}} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{1-n}{2};\frac{3-n}{2};\frac{1}{2} (1-a x)\right )}{(1-n) \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}}}{x} \, dx}{\sqrt{1-a^2 x^2}}-\frac{\left (a \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1+n}{2}} \, _2F_1\left (1,\frac{1}{2} (-1-n);\frac{1-n}{2};\frac{1-a x}{1+a x}\right )}{(1+n) \sqrt{1-a^2 x^2}}-\frac{2^{\frac{3+n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)^{\frac{1}{2} (-1-n)} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{1}{2} (-1-n);\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{(1+n) \sqrt{1-a^2 x^2}}+\frac{2^{\frac{3+n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} x (1-a x)^{\frac{1-n}{2}} \, _2F_1\left (\frac{1}{2} (-1-n),\frac{1-n}{2};\frac{3-n}{2};\frac{1}{2} (1-a x)\right )}{(1-n) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.172003, size = 208, normalized size = 0.76 \[ \frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^{\frac{1}{2} (-n-1)} \left ((n-1) (a x+1)^{\frac{n+1}{2}} \text{Hypergeometric2F1}\left (1,-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )+2^{\frac{n+1}{2}} \left ((n+1) (a x-1) \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )-(n-1) \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{1}{2},-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )\right )\right )}{\left (n^2-1\right ) \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.116, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}\sqrt{c-{\frac{c}{{a}^{2}{x}^{2}}}}\, 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 \sqrt{c - \frac{c}{a^{2} x^{2}}} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{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 )^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}, x\right ) \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 \sqrt{c - \frac{c}{a^{2} x^{2}}} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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