Integrand size = 25, antiderivative size = 252 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {7 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6327, 6328, 90} \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {a^5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 a^5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {7 a^5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^4 x^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {a^3 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rule 90
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^6} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{x^3 (-1+a x)^2 (1+a x)} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {2 a^2}{x}+\frac {a^3}{2 (-1+a x)^2}-\frac {7 a^3}{4 (-1+a x)}-\frac {a^3}{4 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {7 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.37 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (-\frac {2}{x^2}-\frac {4 a}{x}+\frac {2 a^2}{1-a x}+8 a^2 \log (x)-7 a^2 \log (1-a x)-a^2 \log (1+a x)\right )}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.55 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a^{3} \ln \left (a x +1\right ) x^{3}-8 a^{3} \ln \left (x \right ) x^{3}+7 a^{3} \ln \left (a x -1\right ) x^{3}-a^{2} \ln \left (a x +1\right ) x^{2}+8 a^{2} \ln \left (x \right ) x^{2}-7 a^{2} \ln \left (a x -1\right ) x^{2}+6 a^{2} x^{2}-2 a x -2\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} x^{2}}\) | \(138\) |
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Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {{\left (6 \, a^{2} x^{2} - 2 \, a x + {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 8 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 2\right )} \sqrt {-a^{2} c}}{4 \, {\left (a^{2} c^{2} x^{3} - a c^{2} x^{2}\right )}} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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