Integrand size = 16, antiderivative size = 16 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {(d+e x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{e (1+m)}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {Int}\left (\frac {(d+e x)^{1+m}}{x \sqrt {1-c^2 x^2}},x\right )}{e (1+m)} \]
[Out]
Not integrable
Time = 0.04 (sec) , antiderivative size = 16, 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 = {} \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{e (1+m)}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^{1+m}}{x \sqrt {1-c^2 x^2}} \, dx}{e (1+m)} \\ \end{align*}
Not integrable
Time = 21.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]
[In]
[Out]
Not integrable
Time = 0.40 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int \left (e x +d \right )^{m} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )d x\]
[In]
[Out]
Not integrable
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m} \,d x } \]
[In]
[Out]
Not integrable
Time = 13.56 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \]
[In]
[Out]
Not integrable
Time = 0.84 (sec) , antiderivative size = 222, normalized size of antiderivative = 13.88 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m} \,d x } \]
[In]
[Out]
Not integrable
Time = 4.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^m \,d x \]
[In]
[Out]