Optimal. Leaf size=242 \[ -\frac{a^2 \left (n^2+1\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{1-a x}{a x+1}\right )}{(1-n) \sqrt{c-a^2 c x^2}}-\frac{a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1-n}{2}}}{2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1-n}{2}}}{2 x^2 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.273594, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6153, 6150, 129, 151, 12, 131} \[ -\frac{a^2 \left (n^2+1\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \, _2F_1\left (1,\frac{1-n}{2};\frac{3-n}{2};\frac{1-a x}{a x+1}\right )}{(1-n) \sqrt{c-a^2 c x^2}}-\frac{a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1-n}{2}}}{2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1-n}{2}}}{2 x^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 129
Rule 151
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^3 \sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^3 \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^3} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x^2 \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}} \left (-a n-a^2 x\right )}{x^2} \, dx}{2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{a^2 \left (1+n^2\right ) (1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{1}{2}+\frac{n}{2}}}{x} \, dx}{2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x \sqrt{c-a^2 c x^2}}+\frac{\left (a^2 \left (1+n^2\right ) \sqrt{1-a^2 x^2}\right ) \int \frac{(1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{1}{2}+\frac{n}{2}}}{x} \, dx}{2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1+n}{2}} \sqrt{1-a^2 x^2}}{2 x \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (1+n^2\right ) (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 = 0.0842499, size = 134, normalized size = 0.55 \[ \frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2}-\frac{n}{2}} (a x+1)^{\frac{n-1}{2}} \left (2 a^2 \left (n^2+1\right ) x^2 \text{Hypergeometric2F1}\left (1,\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )-(n-1) (a x+1) (a n x+1)\right )}{2 (n-1) x^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.208, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{3}}{\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^{3}}\,{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^{5} - c x^{3}}, 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^{3} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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