\(\int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx\) [284]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Int}\left (x^m \csc ^3(a+b x) \sec ^2(a+b x),x\right ) \]

[Out]

CannotIntegrate(x^m*csc(b*x+a)^3*sec(b*x+a)^2,x)

Rubi [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 20, 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 = {} \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx \]

[In]

Int[x^m*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

Defer[Int][x^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 46.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx \]

[In]

Integrate[x^m*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

Integrate[x^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int x^{m} \csc \left (x b +a \right )^{3} \sec \left (x b +a \right )^{2}d x\]

[In]

int(x^m*csc(b*x+a)^3*sec(b*x+a)^2,x)

[Out]

int(x^m*csc(b*x+a)^3*sec(b*x+a)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { x^{m} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^m*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(x^m*csc(b*x + a)^3*sec(b*x + a)^2, x)

Sympy [F(-1)]

Timed out. \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Timed out} \]

[In]

integrate(x**m*csc(b*x+a)**3*sec(b*x+a)**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { x^{m} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^m*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate(x^m*csc(b*x + a)^3*sec(b*x + a)^2, x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { x^{m} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^m*csc(b*x+a)^3*sec(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^m*csc(b*x + a)^3*sec(b*x + a)^2, x)

Mupad [N/A]

Not integrable

Time = 26.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int \frac {x^m}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3} \,d x \]

[In]

int(x^m/(cos(a + b*x)^2*sin(a + b*x)^3),x)

[Out]

int(x^m/(cos(a + b*x)^2*sin(a + b*x)^3), x)