Integrand size = 25, antiderivative size = 25 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}},x\right ) \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \\ \end{align*}
Not integrable
Time = 1.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]
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Not integrable
Time = 0.46 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 3.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 4.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {(f x)^m \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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