Integrand size = 25, antiderivative size = 25 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Int}\left ((f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ),x\right ) \]
[Out]
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 = {} \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \\ \end{align*}
Not integrable
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
[In]
[Out]
Not integrable
Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \left (f x \right )^{m} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
[In]
[Out]
Not integrable
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
[In]
[Out]
Not integrable
Time = 8.73 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
[In]
[Out]
Not integrable
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
[In]
[Out]
Not integrable
Time = 4.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
[In]
[Out]