Integrand size = 24, antiderivative size = 166 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {120 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6320, 6319} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {120 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {20 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
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Rule 6319
Rule 6320
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {20 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c \left (25-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {120 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c^2 \left (9-n^2\right ) \left (25-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {120 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 1.97 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.80 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {a^2 e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (-\frac {10 \left (225-34 n^2+n^4\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2250 n}{a \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {340 n^3}{a \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {10 n^5}{a \sqrt {1-\frac {1}{a^2 x^2}} x}+15 \left (25-26 n^2+n^4\right ) \cosh \left (3 \coth ^{-1}(a x)\right )-45 \cosh \left (5 \coth ^{-1}(a x)\right )+50 n^2 \cosh \left (5 \coth ^{-1}(a x)\right )-5 n^4 \cosh \left (5 \coth ^{-1}(a x)\right )-125 n \sinh \left (3 \coth ^{-1}(a x)\right )+130 n^3 \sinh \left (3 \coth ^{-1}(a x)\right )-5 n^5 \sinh \left (3 \coth ^{-1}(a x)\right )+9 n \sinh \left (5 \coth ^{-1}(a x)\right )-10 n^3 \sinh \left (5 \coth ^{-1}(a x)\right )+n^5 \sinh \left (5 \coth ^{-1}(a x)\right )\right )}{16 c^2 (-5+n) (-3+n) (-1+n) (1+n) (3+n) (5+n) \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.52 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (120 a^{5} x^{5}-120 n \,a^{4} x^{4}+60 a^{3} n^{2} x^{3}-20 a^{2} n^{3} x^{2}-300 a^{3} x^{3}+5 a \,n^{4} x +260 n \,x^{2} a^{2}-n^{5}-110 n^{2} x a +30 n^{3}+225 a x -149 n \right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a \left (n^{6}-35 n^{4}+259 n^{2}-225\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(140\) |
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Time = 0.26 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.75 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\left (120 \, a^{5} x^{5} - 120 \, a^{4} n x^{4} - n^{5} + 60 \, {\left (a^{3} n^{2} - 5 \, a^{3}\right )} x^{3} + 30 \, n^{3} - 20 \, {\left (a^{2} n^{3} - 13 \, a^{2} n\right )} x^{2} + 5 \, {\left (a n^{4} - 22 \, a n^{2} + 45 \, a\right )} x - 149 \, n\right )} \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{6} - 35 \, a c^{4} n^{4} + 259 \, a c^{4} n^{2} - {\left (a^{7} c^{4} n^{6} - 35 \, a^{7} c^{4} n^{4} + 259 \, a^{7} c^{4} n^{2} - 225 \, a^{7} c^{4}\right )} x^{6} - 225 \, a c^{4} + 3 \, {\left (a^{5} c^{4} n^{6} - 35 \, a^{5} c^{4} n^{4} + 259 \, a^{5} c^{4} n^{2} - 225 \, a^{5} c^{4}\right )} x^{4} - 3 \, {\left (a^{3} c^{4} n^{6} - 35 \, a^{3} c^{4} n^{4} + 259 \, a^{3} c^{4} n^{2} - 225 \, a^{3} c^{4}\right )} x^{2}} \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 4.81 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.74 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {120\,x^5}{c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}-\frac {120\,n\,x^4}{a\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}+\frac {x^3\,\left (60\,n^2-300\right )}{a^2\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}-\frac {n\,\left (n^4-30\,n^2+149\right )}{a^5\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}+\frac {5\,x\,\left (n^4-22\,n^2+45\right )}{a^4\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}-\frac {20\,n\,x^2\,\left (n^2-13\right )}{a^3\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}\right )}{\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^4}+x^4\,\sqrt {c-a^2\,c\,x^2}-\frac {2\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^2}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]
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