Integrand size = 24, antiderivative size = 322 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 \sqrt {1-\frac {1}{a^2 x^2}} x^8}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 \sqrt {1-\frac {1}{a^2 x^2}} x^7}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.13 (sec) , antiderivative size = 322, 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} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\frac {c^4 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 x^8 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 x^7 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x)^3 (1+a x)^6}{x^9} \, dx}{a^9 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^9-\frac {1}{x^9}-\frac {3 a}{x^8}+\frac {8 a^3}{x^6}+\frac {6 a^4}{x^5}-\frac {6 a^5}{x^4}-\frac {8 a^6}{x^3}+\frac {3 a^8}{x}\right ) \, dx}{a^9 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 \sqrt {1-\frac {1}{a^2 x^2}} x^8}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 \sqrt {1-\frac {1}{a^2 x^2}} x^7}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (\frac {1}{8 a^9 x^8}+\frac {3}{7 a^8 x^7}-\frac {8}{5 a^6 x^5}-\frac {3}{2 a^5 x^4}+\frac {2}{a^4 x^3}+\frac {4}{a^3 x^2}+x+\frac {3 \log (x)}{a}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
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Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\left (280 a^{9} x^{9}+840 a^{8} \ln \left (x \right ) x^{8}+1120 a^{6} x^{6}+560 a^{5} x^{5}-420 a^{4} x^{4}-448 a^{3} x^{3}+120 a x +35\right ) {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {9}{2}} x}{280 \left (a x +1\right )^{3} \left (a^{2} x^{2}-1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(112\) |
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Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.30 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\frac {{\left (280 \, a^{9} c^{4} x^{9} + 840 \, a^{8} c^{4} x^{8} \log \left (x\right ) + 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} - 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x + 35 \, c^{4}\right )} \sqrt {a^{2} c}}{280 \, a^{10} x^{8}} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\text {Timed out} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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