Integrand size = 22, antiderivative size = 236 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6332, 6328, 90} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {c^2 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c^2 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x)^2 (1+a x)^3}{x^5} \, dx}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c^2 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^5+\frac {1}{x^5}+\frac {a}{x^4}-\frac {2 a^2}{x^3}-\frac {2 a^3}{x^2}+\frac {a^4}{x}\right ) \, dx}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = -\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.31 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4 a^5 x^4}-\frac {1}{3 a^4 x^3}+\frac {1}{a^3 x^2}+\frac {2}{a^2 x}+x+\frac {\log (x)}{a}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.41
method | result | size |
default | \(\frac {\left (12 a^{5} x^{5}+12 \ln \left (x \right ) x^{4} a^{4}+24 a^{3} x^{3}+12 a^{2} x^{2}-4 a x -3\right ) {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} x}{12 \left (a x +1\right ) \left (a^{2} x^{2}-1\right )^{2} \sqrt {\frac {a x -1}{a x +1}}}\) | \(96\) |
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Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.31 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {{\left (12 \, a^{5} c^{2} x^{5} + 12 \, a^{4} c^{2} x^{4} \log \left (x\right ) + 24 \, a^{3} c^{2} x^{3} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right )} \sqrt {a^{2} c}}{12 \, a^{6} x^{4}} \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\text {Timed out} \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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