3.32 \(\int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left ((c+d x)^m (a+i a \cot (e+f x))^2,x\right ) \]

[Out]

Unintegrable((d*x+c)^m*(a+I*a*cot(f*x+e))^2,x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + d*x)^m*(a + I*a*Cot[e + f*x])^2,x]

[Out]

Defer[Int][(c + d*x)^m*(a + I*a*Cot[e + f*x])^2, x]

Rubi steps

\begin {align*} \int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx &=\int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx\\ \end {align*}

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Mathematica [A]  time = 10.38, size = 0, normalized size = 0.00 \[ \int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x])^2,x]

[Out]

Integrate[(c + d*x)^m*(a + I*a*Cot[e + f*x])^2, x]

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fricas [A]  time = 0.71, size = 0, normalized size = 0.00 \[ \frac {2 i \, {\left (d x + c\right )}^{m} a^{2} + {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )} {\rm integral}\left (\frac {{\left (4 \, a^{2} d f x + 4 \, a^{2} c f + 2 i \, a^{2} d m\right )} {\left (d x + c\right )}^{m}}{d f x + c f - {\left (d f x + c f\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}, x\right )}{f e^{\left (2 i \, f x + 2 i \, e\right )} - f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*(a+I*a*cot(f*x+e))^2,x, algorithm="fricas")

[Out]

(2*I*(d*x + c)^m*a^2 + (f*e^(2*I*f*x + 2*I*e) - f)*integral((4*a^2*d*f*x + 4*a^2*c*f + 2*I*a^2*d*m)*(d*x + c)^
m/(d*f*x + c*f - (d*f*x + c*f)*e^(2*I*f*x + 2*I*e)), x))/(f*e^(2*I*f*x + 2*I*e) - f)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \cot \left (f x + e\right ) + a\right )}^{2} {\left (d x + c\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*(a+I*a*cot(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((I*a*cot(f*x + e) + a)^2*(d*x + c)^m, x)

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maple [A]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +i a \cot \left (f x +e \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m*(a+I*a*cot(f*x+e))^2,x)

[Out]

int((d*x+c)^m*(a+I*a*cot(f*x+e))^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} - \int -\frac {{\left (d x + c\right )}^{m} a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (d x + c\right )}^{m} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d x + c\right )}^{m} a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 4 \, {\left (d x + c\right )}^{m} a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (d x + c\right )}^{m} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, {\left (d x + c\right )}^{m} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (d x + c\right )}^{m} a^{2} - 2 \, {\left (2 \, {\left (d x + c\right )}^{m} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (d x + c\right )}^{m} a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right )}{2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \cos \left (4 \, f x + 4 \, e\right ) - \cos \left (4 \, f x + 4 \, e\right )^{2} - 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} - \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) - 1}\,{d x} - i \, \int -\frac {4 \, {\left ({\left (d x + c\right )}^{m} a^{2} \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (d x + c\right )}^{m} a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )}}{2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \cos \left (4 \, f x + 4 \, e\right ) - \cos \left (4 \, f x + 4 \, e\right )^{2} - 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} - \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*(a+I*a*cot(f*x+e))^2,x, algorithm="maxima")

[Out]

(d*x + c)^(m + 1)*a^2/(d*(m + 1)) - integrate(-((d*x + c)^m*a^2*cos(4*f*x + 4*e)^2 + 4*(d*x + c)^m*a^2*cos(2*f
*x + 2*e)^2 + (d*x + c)^m*a^2*sin(4*f*x + 4*e)^2 - 4*(d*x + c)^m*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(d*
x + c)^m*a^2*sin(2*f*x + 2*e)^2 + 4*(d*x + c)^m*a^2*cos(2*f*x + 2*e) - 3*(d*x + c)^m*a^2 - 2*(2*(d*x + c)^m*a^
2*cos(2*f*x + 2*e) + (d*x + c)^m*a^2)*cos(4*f*x + 4*e))/(2*(2*cos(2*f*x + 2*e) - 1)*cos(4*f*x + 4*e) - cos(4*f
*x + 4*e)^2 - 4*cos(2*f*x + 2*e)^2 - sin(4*f*x + 4*e)^2 + 4*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) - 4*sin(2*f*x +
2*e)^2 + 4*cos(2*f*x + 2*e) - 1), x) - I*integrate(-4*((d*x + c)^m*a^2*sin(4*f*x + 4*e) - 2*(d*x + c)^m*a^2*si
n(2*f*x + 2*e))/(2*(2*cos(2*f*x + 2*e) - 1)*cos(4*f*x + 4*e) - cos(4*f*x + 4*e)^2 - 4*cos(2*f*x + 2*e)^2 - sin
(4*f*x + 4*e)^2 + 4*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) - 4*sin(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e) - 1), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cot(e + f*x)*1i)^2*(c + d*x)^m,x)

[Out]

int((a + a*cot(e + f*x)*1i)^2*(c + d*x)^m, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - a^{2} \left (\int \left (c + d x\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (c + d x\right )^{m} \cot {\left (e + f x \right )}\right )\, dx + \int \left (- \left (c + d x\right )^{m}\right )\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m*(a+I*a*cot(f*x+e))**2,x)

[Out]

-a**2*(Integral((c + d*x)**m*cot(e + f*x)**2, x) + Integral(-2*I*(c + d*x)**m*cot(e + f*x), x) + Integral(-(c
+ d*x)**m, x))

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