Integrand size = 25, antiderivative size = 217 \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {11 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1-a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}+\frac {5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1+a x)}{16 \left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.20 (sec) , antiderivative size = 217, 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} \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {11 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \log (1-a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}+\frac {5 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \log (a x+1)}{16 \left (c-a^2 c x^2\right )^{5/2}} \]
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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 )^{5/2} x^5\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x} \, dx}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = \frac {\left (a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {x^4}{(-1+a x)^3 (1+a x)^2} \, dx}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = \frac {\left (a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (\frac {1}{4 a^4 (-1+a x)^3}+\frac {3}{4 a^4 (-1+a x)^2}+\frac {11}{16 a^4 (-1+a x)}-\frac {1}{8 a^4 (1+a x)^2}+\frac {5}{16 a^4 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {11 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1-a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}+\frac {5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1+a x)}{16 \left (c-a^2 c x^2\right )^{5/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.39 \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \left (-\frac {2 \left (-6+3 a x+5 a^2 x^2\right )}{(-1+a x)^2 (1+a x)}+11 \log (1-a x)+5 \log (1+a x)\right )}{16 \left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.55 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (12+11 a^{3} \ln \left (a x -1\right ) x^{3}+5 a^{3} \ln \left (a x +1\right ) x^{3}-11 a^{2} \ln \left (a x -1\right ) x^{2}-5 a^{2} \ln \left (a x +1\right ) x^{2}-10 a^{2} x^{2}-11 a \ln \left (a x -1\right ) x -5 a \ln \left (a x +1\right ) x -6 a x +11 \ln \left (a x -1\right )+5 \ln \left (a x +1\right )\right )}{16 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \left (a^{2} x^{2}-1\right ) c^{3} a^{5} \left (a x +1\right )}\) | \(169\) |
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Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.56 \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {{\left (10 \, a^{2} x^{2} + 6 \, a x - 5 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) - 11 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) - 12\right )} \sqrt {-a^{2} c}}{16 \, {\left (a^{9} c^{3} x^{3} - a^{8} c^{3} x^{2} - a^{7} c^{3} x + a^{6} c^{3}\right )}} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^4}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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