\(\int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx\) [45]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Int}\left ((c+d x)^m \cot (a+b x) \csc ^2(a+b x),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.25 (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 = {} \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 23.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^m}{{\sin \left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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