\(\int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx\) [278]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Int}\left ((c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, 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 = {} \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx \]

[In]

Int[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 27.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx \]

[In]

Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.97 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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