Optimal. Leaf size=277 \[ -\frac{a^3 2^{\frac{n+1}{2}} x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.373535, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6192, 6195, 89, 79, 69} \[ -\frac{a^3 2^{\frac{n+1}{2}} x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \, _2F_1\left (\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6195
Rule 89
Rule 79
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int x^2 \left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \left (\frac{n}{a}+\frac{(1+n) x}{a^2}\right ) \, dx,x,\frac{1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{\frac{1}{2} (-1+n)} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{2^{\frac{1+n}{2}} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} x^3 \, _2F_1\left (\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.824422, size = 127, normalized size = 0.46 \[ \frac{e^{n \coth ^{-1}(a x)} \left (a x \sqrt{1-\frac{1}{a^2 x^2}} (a n x-1)-2 (n-1) \left (a^2 x^2-1\right ) e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )}{a c (n-1) (n+1) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.309, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{x} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} 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^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, 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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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