\(\int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx\) [170]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b}+\text {Int}\left ((c+d x)^m \cot (a+b x) \csc (a+b x),x\right ) \]

[Out]

CannotIntegrate((d*x+c)^m*cot(b*x+a)*csc(b*x+a),x)+1/2*I*exp(I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-I*b*(d*x+c)/d)/
b/((-I*b*(d*x+c)/d)^m)-1/2*I*(d*x+c)^m*GAMMA(1+m,I*b*(d*x+c)/d)/b/exp(I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)

Rubi [N/A]

Not integrable

Time = 0.34 (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 = {} \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx \]

[In]

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

[Out]

((I/2)*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) - ((I/
2)*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + 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 \cos (a+b x) \, dx+\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx \\ & = -\left (\frac {1}{2} \int e^{-i (a+b x)} (c+d x)^m \, dx\right )-\frac {1}{2} \int e^{i (a+b x)} (c+d x)^m \, dx+\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx \\ & = \frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b}+\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 13.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 22.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\int \left (c + d x\right )^{m} \cos {\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \cot \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \cot \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]

[In]

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

[Out]

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