Integrand size = 23, antiderivative size = 23 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\text {Int}\left (\frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}},x\right ) \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx \\ \end{align*}
Not integrable
Time = 8.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {a +b \,\operatorname {arcsech}\left (c x \right )}{x^{3} \sqrt {e \,x^{2}+d}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{3}} \,d x } \]
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Not integrable
Time = 7.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{3} \sqrt {d + e x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{3}} \,d x } \]
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Not integrable
Time = 4.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^3\,\sqrt {e\,x^2+d}} \,d x \]
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