Integrand size = 25, antiderivative size = 172 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 172, 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} \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
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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}} \, 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 {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}{a^3}+\frac {1}{2 a^3 (-1+a x)^2}+\frac {5}{4 a^3 (-1+a x)}-\frac {1}{4 a^3 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.43 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (x+\frac {1}{2 a-2 a^2 x}+\frac {5 \log (1-a x)}{4 a}-\frac {\log (1+a x)}{4 a}\right )}{\left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.55 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.57
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-4 a^{2} x^{2}+a \ln \left (a x +1\right ) x -5 a \ln \left (a x -1\right ) x +4 a x -\ln \left (a x +1\right )+5 \ln \left (a x -1\right )+2\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} a^{4}}\) | \(98\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.40 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {{\left (4 \, a^{2} x^{2} - 4 \, a x - {\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt {-a^{2} c}}{4 \, {\left (a^{6} c^{2} x - a^{5} c^{2}\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\sqrt {\frac {a x - 1}{a x + 1}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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