Integrand size = 22, antiderivative size = 127 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (64-20 n^2+n^4\right )}-\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )} \]
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Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {(n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac {24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^4-20 n^2+64\right )} \]
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Rule 6318
Rule 6320
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac {12 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{c \left (16-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac {24 \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c^2 \left (64-20 n^2+n^4\right )} \\ & = \frac {24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (64-20 n^2+n^4\right )}-\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (n^4-4 a n^3 x+24 \left (-1+a^2 x^2\right )^2-8 a n x \left (-5+3 a^2 x^2\right )+4 n^2 \left (-4+3 a^2 x^2\right )\right )}{a c^3 n \left (-16+n^2\right ) \left (-4+n^2\right ) \left (-1+a^2 x^2\right )^2} \]
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Time = 17.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3} n +12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 a n x -16 n^{2}+24\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{2} c^{3} a \left (n^{2}-16\right ) \left (n^{2}-4\right ) n}\) | \(101\) |
risch | \(\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3} n +12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 a n x -16 n^{2}+24\right ) \left (a x -1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{\left (a^{2} x^{2}-1\right )^{2} c^{3} a \left (n^{2}-16\right ) \left (n^{2}-4\right ) n}\) | \(112\) |
parallelrisch | \(\frac {24 x^{4} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{4}+40 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a n -48 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}-24 a^{3} x^{3} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n +12 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n^{2}-4 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a \,n^{3}+24 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}+{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{4}-16 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{2}}{c^{3} \left (a^{2} x^{2}-1\right )^{2} a \left (n^{2}-16\right ) \left (n^{2}-4\right ) n}\) | \(159\) |
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Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.37 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {{\left (24 \, a^{4} x^{4} - 24 \, a^{3} n x^{3} + n^{4} + 12 \, {\left (a^{2} n^{2} - 4 \, a^{2}\right )} x^{2} - 16 \, n^{2} - 4 \, {\left (a n^{3} - 10 \, a n\right )} x + 24\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{5} - 20 \, a c^{3} n^{3} + 64 \, a c^{3} n + {\left (a^{5} c^{3} n^{5} - 20 \, a^{5} c^{3} n^{3} + 64 \, a^{5} c^{3} n\right )} x^{4} - 2 \, {\left (a^{3} c^{3} n^{5} - 20 \, a^{3} c^{3} n^{3} + 64 \, a^{3} c^{3} n\right )} x^{2}} \]
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Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]
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Time = 4.73 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.51 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {24\,x^4}{a\,c^3\,n\,\left (n^4-20\,n^2+64\right )}-\frac {4\,x\,\left (n^2-10\right )}{a^4\,c^3\,\left (n^4-20\,n^2+64\right )}-\frac {24\,x^3}{a^2\,c^3\,\left (n^4-20\,n^2+64\right )}+\frac {n^4-16\,n^2+24}{a^5\,c^3\,n\,\left (n^4-20\,n^2+64\right )}+\frac {x^2\,\left (12\,n^2-48\right )}{a^3\,c^3\,n\,\left (n^4-20\,n^2+64\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^4}+x^4-\frac {2\,x^2}{a^2}\right )} \]
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