Optimal. Leaf size=944 \[ -\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}} \]
[Out]
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Rubi [A] time = 0.63, antiderivative size = 944, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 37
Rule 45
Rule 69
Rule 128
Rule 6192
Rule 6195
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*}
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Mathematica [A] time = 1.69, size = 220, normalized size = 0.23 \[ \frac {e^{n \coth ^{-1}(a x)} \left (6 a \left (n^2-1\right ) x \sqrt {1-\frac {1}{a^2 x^2}} \cosh \left (2 \coth ^{-1}(a x)\right )-n \left (n^2-1\right ) \left (a^2 x^2-1\right ) \cosh \left (3 \coth ^{-1}(a x)\right )+a x \sqrt {1-\frac {1}{a^2 x^2}} \left (5 a n^3 x-45 a n x-2 n^2+42\right )\right )-8 \left (n^3-n^2-9 n+9\right ) \left (a^2 x^2-1\right ) e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-e^{2 \coth ^{-1}(a x)}\right )}{4 a c^2 (n-1) (n+1) \left (n^2-9\right ) x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{6} c^{3} x^{7} - 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} - c^{3} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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