\(\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx\) [38]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.24 (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 = {} \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 66.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int \left (c + d x\right )^{m} \cos {\left (a + b x \right )} \csc ^{2}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral((c + d*x)**m*cos(a + b*x)*csc(a + b*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 21.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^m}{{\sin \left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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