Integrand size = 25, antiderivative size = 184 \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \]
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Time = 0.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6327, 6328, 90, 213} \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {a^2 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \text {arctanh}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}}+\frac {a^2 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac {a^2 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac {a^2 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}} \]
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Rule 90
Rule 213
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^3} \, 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^2}{(-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^2 (-1+a x)^3}+\frac {1}{4 a^2 (-1+a x)^2}-\frac {1}{8 a^2 (1+a x)^2}-\frac {1}{8 a^2 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {a^2 \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 {a^2 \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 {a^2 \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 {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{8 \left (c-a^2 c x^2\right )^{5/2}} \\ & = -\frac {a^2 \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 {a^2 \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 {a^2 \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 {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.46 \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (2-3 a x-a^2 x^2+(-1+a x)^2 (1+a x) \text {arctanh}(a x)\right )}{8 a^2 c^2 (-1+a x)^2 (1+a x) \sqrt {c-a^2 c x^2}} \]
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Time = 0.55 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a^{3} \ln \left (a x +1\right ) x^{3}-a^{3} \ln \left (a x -1\right ) x^{3}-a^{2} \ln \left (a x +1\right ) x^{2}+a^{2} \ln \left (a x -1\right ) x^{2}-2 a^{2} x^{2}-a \ln \left (a x +1\right ) x +a \ln \left (a x -1\right ) x -6 a x +\ln \left (a x +1\right )-\ln \left (a x -1\right )+4\right )}{16 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \left (a^{2} x^{2}-1\right ) c^{3} a^{3} \left (a x +1\right )}\) | \(164\) |
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Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (a^{4} x^{3} - a^{3} x^{2} - a^{2} x + a\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c} \sqrt {-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{2} x^{2} + 3 \, a x - 2\right )} \sqrt {-a^{2} c}}{16 \, {\left (a^{7} c^{3} x^{3} - a^{6} c^{3} x^{2} - a^{5} c^{3} x + a^{4} c^{3}\right )}} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^2}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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