Integrand size = 24, antiderivative size = 360 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^4}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^3}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Time = 0.15 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6332, 6328, 90} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^7}{(-1+a x)^5 (1+a x)^2} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^7}+\frac {1}{4 a^7 (-1+a x)^5}+\frac {3}{2 a^7 (-1+a x)^4}+\frac {59}{16 a^7 (-1+a x)^3}+\frac {75}{16 a^7 (-1+a x)^2}+\frac {201}{64 a^7 (-1+a x)}+\frac {1}{32 a^7 (1+a x)^2}-\frac {9}{64 a^7 (1+a x)}\right ) \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^4}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^3}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.32 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (\frac {2 \left (104-207 a x-59 a^2 x^2+309 a^3 x^3-87 a^4 x^4-96 a^5 x^5+32 a^6 x^6\right )}{(-1+a x)^4 (1+a x)}+201 \log (1-a x)-9 \log (1+a x)\right )}{64 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
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Time = 0.06 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.69
method | result | size |
default | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (-64 a^{6} x^{6}+9 \ln \left (a x +1\right ) x^{5} a^{5}-201 \ln \left (a x -1\right ) x^{5} a^{5}+192 a^{5} x^{5}-27 \ln \left (a x +1\right ) x^{4} a^{4}+603 \ln \left (a x -1\right ) x^{4} a^{4}+174 a^{4} x^{4}+18 a^{3} \ln \left (a x +1\right ) x^{3}-402 a^{3} \ln \left (a x -1\right ) x^{3}-618 a^{3} x^{3}+18 a^{2} \ln \left (a x +1\right ) x^{2}-402 a^{2} \ln \left (a x -1\right ) x^{2}+118 a^{2} x^{2}-27 a \ln \left (a x +1\right ) x +603 a \ln \left (a x -1\right ) x +414 a x +9 \ln \left (a x +1\right )-201 \ln \left (a x -1\right )-208\right )}{64 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a^{8} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\) | \(247\) |
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none
Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.58 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {{\left (64 \, a^{6} x^{6} - 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} + 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} - 414 \, a x - 9 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x + 1\right ) + 201 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} - 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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