Optimal. Leaf size=944 \[ \text{result too large to display} \]
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Rubi [A] time = 0.632143, antiderivative size = 944, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6192, 6195, 128, 45, 37, 69} \[ -\frac{2^{\frac{n+5}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \, _2F_1\left (\frac{1}{2} (-n-3),\frac{1}{2} (-n-3);\frac{1}{2} (-n-1);\frac{a-\frac{1}{x}}{2 a}\right ) \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n+1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n+3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n+1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6195
Rule 128
Rule 45
Rule 37
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^6} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int x^4 \left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}}-4 a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}+6 a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}}-4 a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{1}{2}+\frac{n}{2}}+a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{3}{2}+\frac{n}{2}}\right ) \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \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 )^{5/2}}+\frac{\left (4 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \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 )^{5/2}}+\frac{\left (4 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{1}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (3 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \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^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \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^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1-n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \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^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.5572, size = 220, normalized size = 0.23 \[ \frac{e^{n \coth ^{-1}(a x)} \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (5 a n^3 x-45 a n x-2 n^2+42\right )+6 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (2 \coth ^{-1}(a x)\right )-n \left (n^2-1\right ) \left (a^2 x^2-1\right ) \cosh \left (3 \coth ^{-1}(a x)\right )\right )-8 \left (n^3-n^2-9 n+9\right ) \left (a^2 x^2-1\right ) e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \coth ^{-1}(a x)}\right )}{4 a c^2 (n-1) (n+1) \left (n^2-9\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.342, 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{5}{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{5}{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^{6} c^{3} x^{7} - 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} - c^{3} 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{5}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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