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

Optimal. Leaf size=24 \[ \text {Int}\left ((c+d x)^m (a+i a \cot (e+f x)),x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 5.74, size = 0, normalized size = 0.00 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx \]

Verification is Not applicable to the result.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2 \, {\left (d x + c\right )}^{m} a}{e^{\left (2 i \, f x + 2 i \, e\right )} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \cot \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 1.01, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +i a \cot \left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 i \, a \int \frac {{\left (d x + c\right )}^{m} \sin \left (2 \, f x + 2 \, e\right )}{\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} - 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1}\,{d x} + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ i a \left (\int \left (- i \left (c + d x\right )^{m}\right )\, dx + \int \left (c + d x\right )^{m} \cot {\left (e + f x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________