\(\int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx\) [309]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Int}\left ((c+d x)^m \csc (a+b x) \sec ^3(a+b x),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 22, 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 = {} \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 21.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \csc \left (b x + a\right ) \sec \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \csc \left (b x + a\right ) \sec \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \csc \left (b x + a\right ) \sec \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\cos \left (a+b\,x\right )}^3\,\sin \left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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