\(\int \frac {(f x)^m (a+b \csc ^{-1}(c x))}{(d+e x^2)^2} \, dx\) [169]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*arccsc(c*x))/(e*x^2+d)^2,x)

Rubi [N/A]

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 = {} \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx \]

[In]

Int[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]

[Out]

Defer[Int][((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.73 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx \]

[In]

Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]

[Out]

Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 3.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{2}}d x\]

[In]

int((f*x)^m*(a+b*arccsc(c*x))/(e*x^2+d)^2,x)

[Out]

int((f*x)^m*(a+b*arccsc(c*x))/(e*x^2+d)^2,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*arccsc(c*x) + a)*(f*x)^m/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((f*x)**m*(a+b*acsc(c*x))/(e*x**2+d)**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

integrate((b*arccsc(c*x) + a)*(f*x)^m/(e*x^2 + d)^2, x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)*(f*x)^m/(e*x^2 + d)^2, x)

Mupad [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

int(((f*x)^m*(a + b*asin(1/(c*x))))/(d + e*x^2)^2,x)

[Out]

int(((f*x)^m*(a + b*asin(1/(c*x))))/(d + e*x^2)^2, x)