\(\int (c+d x)^m (a+i a \cot (e+f x)) \, dx\) [33]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\text {Int}\left ((c+d x)^m (a+i a \cot (e+f x)),x\right ) \]

[Out]

Unintegrable((d*x+c)^m*(a+I*a*cot(f*x+e)),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 21, 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 = {} \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\int (c+d x)^m (a+i a \cot (e+f x)) \, dx \]

[In]

Int[(c + d*x)^m*(a + I*a*Cot[e + f*x]),x]

[Out]

Defer[Int][(c + d*x)^m*(a + I*a*Cot[e + f*x]), x]

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

Mathematica [N/A]

Not integrable

Time = 6.59 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\int (c+d x)^m (a+i a \cot (e+f x)) \, dx \]

[In]

Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x]),x]

[Out]

Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

\[\int \left (d x +c \right )^{m} \left (a +i a \cot \left (f x +e \right )\right )d x\]

[In]

int((d*x+c)^m*(a+I*a*cot(f*x+e)),x)

[Out]

int((d*x+c)^m*(a+I*a*cot(f*x+e)),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\int { {\left (i \, a \cot \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \]

[In]

integrate((d*x+c)^m*(a+I*a*cot(f*x+e)),x, algorithm="fricas")

[Out]

integral(-2*(d*x + c)^m*a/(e^(2*I*f*x + 2*I*e) - 1), x)

Sympy [N/A]

Not integrable

Time = 1.83 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=i a \left (\int \left (- i \left (c + d x\right )^{m}\right )\, dx + \int \left (c + d x\right )^{m} \cot {\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((d*x+c)**m*(a+I*a*cot(f*x+e)),x)

[Out]

I*a*(Integral(-I*(c + d*x)**m, x) + Integral((c + d*x)**m*cot(e + f*x), x))

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.76 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\int { {\left (i \, a \cot \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \]

[In]

integrate((d*x+c)^m*(a+I*a*cot(f*x+e)),x, algorithm="maxima")

[Out]

2*I*a*integrate((d*x + c)^m*sin(2*f*x + 2*e)/(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 - 2*cos(2*f*x + 2*e) + 1
), x) + (d*x + c)^(m + 1)*a/(d*(m + 1))

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\int { {\left (i \, a \cot \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \]

[In]

integrate((d*x+c)^m*(a+I*a*cot(f*x+e)),x, algorithm="giac")

[Out]

integrate((I*a*cot(f*x + e) + a)*(d*x + c)^m, x)

Mupad [N/A]

Not integrable

Time = 12.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx=\int \left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^m \,d x \]

[In]

int((a + a*cot(e + f*x)*1i)*(c + d*x)^m,x)

[Out]

int((a + a*cot(e + f*x)*1i)*(c + d*x)^m, x)