Integrand size = 25, antiderivative size = 211 \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4}{a \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5}{2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 a^2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {7 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 a^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 a^2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.16 (sec) , antiderivative size = 211, 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} \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{a \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 a^2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {7 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 a^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 a^2 \left (c-a^2 c x^2\right )^{3/2}} \]
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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)} 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^4}{(-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^4}+\frac {x}{a^3}+\frac {1}{2 a^4 (-1+a x)^2}+\frac {7}{4 a^4 (-1+a x)}+\frac {1}{4 a^4 (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}{a \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5}{2 \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^2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {7 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 a^2 \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^2 \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.39 \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (2 \left (\frac {2 x}{a}+x^2+\frac {1}{a^2-a^3 x}\right )+\frac {7 \log (1-a x)}{a^2}+\frac {\log (1+a x)}{a^2}\right )}{4 \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.55 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.50
method | result | size |
default | \(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}+2 a^{2} x^{2}+a \ln \left (a x +1\right ) x +7 a \ln \left (a x -1\right ) x -4 a x -\ln \left (a x +1\right )-7 \ln \left (a x -1\right )-2\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} a^{5}}\) | \(106\) |
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Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36 \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {{\left (2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 4 \, a x + {\left (a x - 1\right )} \log \left (a x + 1\right ) + 7 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt {-a^{2} c}}{4 \, {\left (a^{7} c^{2} x - a^{6} c^{2}\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\sqrt {\frac {a x - 1}{a x + 1}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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