Integrand size = 25, antiderivative size = 262 \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^6}{\left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 a (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{a (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 a (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {23 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1-a x)}{16 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {7 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1+a x)}{16 a \left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 262, 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} \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (c-a^2 c x^2\right )^{5/2}}+\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{a (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{8 a (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{8 a (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {23 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \log (1-a x)}{16 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {7 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \log (a x+1)}{16 a \left (c-a^2 c x^2\right )^{5/2}} \]
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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}} \, 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^5}{(-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}{a^5}+\frac {1}{4 a^5 (-1+a x)^3}+\frac {1}{a^5 (-1+a x)^2}+\frac {23}{16 a^5 (-1+a x)}+\frac {1}{8 a^5 (1+a x)^2}-\frac {7}{16 a^5 (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^6}{\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 a (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{a (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 a (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {23 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1-a x)}{16 a \left (c-a^2 c x^2\right )^{5/2}}-\frac {7 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \log (1+a x)}{16 a \left (c-a^2 c x^2\right )^{5/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.38 \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \left (x-\frac {1}{8 a (-1+a x)^2}+\frac {1}{a-a^2 x}-\frac {1}{8 a+8 a^2 x}+\frac {23 \log (1-a x)}{16 a}-\frac {7 \log (1+a x)}{16 a}\right )}{\left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.55 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-16 a^{4} x^{4}+7 a^{3} \ln \left (a x +1\right ) x^{3}-23 a^{3} \ln \left (a x -1\right ) x^{3}+16 a^{3} x^{3}-7 a^{2} \ln \left (a x +1\right ) x^{2}+23 a^{2} \ln \left (a x -1\right ) x^{2}+34 a^{2} x^{2}-7 a \ln \left (a x +1\right ) x +23 a \ln \left (a x -1\right ) x -18 a x +7 \ln \left (a x +1\right )-23 \ln \left (a x -1\right )-12\right )}{16 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \left (a^{2} x^{2}-1\right ) c^{3} a^{6} \left (a x +1\right )}\) | \(185\) |
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Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12\right )} \sqrt {-a^{2} c}}{16 \, {\left (a^{10} c^{3} x^{3} - a^{9} c^{3} x^{2} - a^{8} c^{3} x + a^{7} c^{3}\right )}} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{5}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^5}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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