Integrand size = 24, antiderivative size = 275 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{16 (1+a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{12 (1+a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{8 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7 \text {arctanh}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}} \]
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Time = 0.18 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6327, 6328, 46, 213} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {5 a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \text {arctanh}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}}+\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{32 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{32 (a x+1)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{12 (a x+1)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{16 (a x+1)^4 \left (c-a^2 c x^2\right )^{7/2}} \]
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Rule 46
Rule 213
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \, dx}{\left (c-a^2 c x^2\right )^{7/2}} \\ & = \frac {\left (a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{(-1+a x)^2 (1+a x)^5} \, dx}{\left (c-a^2 c x^2\right )^{7/2}} \\ & = \frac {\left (a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (\frac {1}{32 (-1+a x)^2}+\frac {1}{4 (1+a x)^5}+\frac {1}{4 (1+a x)^4}+\frac {3}{16 (1+a x)^3}+\frac {1}{8 (1+a x)^2}-\frac {5}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{7/2}} \\ & = \frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{16 (1+a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{12 (1+a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{8 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac {\left (5 a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{32 \left (c-a^2 c x^2\right )^{7/2}} \\ & = \frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{16 (1+a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{12 (1+a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{8 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7 \text {arctanh}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.36 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (32+15 a x-35 a^2 x^2-45 a^3 x^3-15 a^4 x^4+15 (-1+a x) (1+a x)^4 \text {arctanh}(a x)\right )}{96 c^3 (-1+a x) (1+a x)^4 \sqrt {c-a^2 c x^2}} \]
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Time = 0.52 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (15 \ln \left (a x +1\right ) x^{5} a^{5}-15 \ln \left (a x -1\right ) x^{5} a^{5}+45 \ln \left (a x +1\right ) x^{4} a^{4}-45 \ln \left (a x -1\right ) x^{4} a^{4}-30 a^{4} x^{4}+30 a^{3} \ln \left (a x +1\right ) x^{3}-30 a^{3} \ln \left (a x -1\right ) x^{3}-90 a^{3} x^{3}-30 a^{2} \ln \left (a x +1\right ) x^{2}+30 a^{2} \ln \left (a x -1\right ) x^{2}-70 a^{2} x^{2}-45 a \ln \left (a x +1\right ) x +45 a \ln \left (a x -1\right ) x +30 a x -15 \ln \left (a x +1\right )+15 \ln \left (a x -1\right )+64\right )}{192 \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right ) c^{4} a}\) | \(241\) |
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Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {15 \, {\left (a^{6} x^{5} + 3 \, a^{5} x^{4} + 2 \, a^{4} x^{3} - 2 \, a^{3} x^{2} - 3 \, a^{2} x - a\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c} \sqrt {-c} x + c}{a^{2} x^{2} - 1}\right ) + 2 \, {\left (15 \, a^{4} x^{4} + 45 \, a^{3} x^{3} + 35 \, a^{2} x^{2} - 15 \, a x - 32\right )} \sqrt {-a^{2} c}}{192 \, {\left (a^{7} c^{4} x^{5} + 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x - a^{2} c^{4}\right )}} \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]
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