Integrand size = 23, antiderivative size = 23 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2},x\right ) \]
[Out]
Not integrable
Time = 0.05 (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 = {} \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx \\ \end{align*}
Not integrable
Time = 2.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx \]
[In]
[Out]
Not integrable
Time = 4.67 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{e \,x^{2}+d}d x\]
[In]
[Out]
Not integrable
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]
[In]
[Out]
Not integrable
Time = 35.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
[In]
[Out]
Not integrable
Time = 0.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.82 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \]
[In]
[Out]