Integrand size = 27, antiderivative size = 330 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.25 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6327, 6330, 47, 37} \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {3 a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}} \]
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Rule 37
Rule 47
Rule 6327
Rule 6330
Rubi steps \begin{align*} \text {integral}& = \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^2} \, 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 ) \text {Subst}\left (\int \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 {a \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 {\left (3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {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 {a \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 \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 {\left (6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {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 {a \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 \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 \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 {\left (6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \text {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 \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 \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 \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 \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}} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.33 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (3 \left (10-2 n^2-9 a n x+a n^3 x\right )-6 \left (-1+n^2\right ) \cosh \left (2 \coth ^{-1}(a x)\right )+a n \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )\right )}{4 a^4 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
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Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.28
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{3} x^{3}-7 a^{3} x^{3} n -3 a^{2} n^{2} x^{2}+9 a^{2} x^{2}+6 a n x -6\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a^{4} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x - 3 \, {\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} - 6\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} + {\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \, {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {6}{a^6\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {6\,n\,x}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {x^2\,\left (3\,n^2-9\right )}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x^3\,\left (n^2-7\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]
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