\(\int \frac {\cot ^3(a+b x)}{c+d x} \, dx\) [182]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\text {Int}\left (\frac {\cot ^3(a+b x)}{c+d x},x\right ) \]

[Out]

Unintegrable(cot(b*x+a)^3/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int \frac {\cot ^3(a+b x)}{c+d x} \, dx \]

[In]

Int[Cot[a + b*x]^3/(c + d*x),x]

[Out]

Defer[Int][Cot[a + b*x]^3/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^3(a+b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int \frac {\cot ^3(a+b x)}{c+d x} \, dx \]

[In]

Integrate[Cot[a + b*x]^3/(c + d*x),x]

[Out]

Integrate[Cot[a + b*x]^3/(c + d*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\cot \left (x b +a \right )^{3}}{d x +c}d x\]

[In]

int(cot(b*x+a)^3/(d*x+c),x)

[Out]

int(cot(b*x+a)^3/(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(cot(b*x+a)^3/(d*x+c),x, algorithm="fricas")

[Out]

integral(cot(b*x + a)^3/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int \frac {\cot ^{3}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

integrate(cot(b*x+a)**3/(d*x+c),x)

[Out]

Integral(cot(a + b*x)**3/(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 2.64 (sec) , antiderivative size = 1636, normalized size of antiderivative = 102.25 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(cot(b*x+a)^3/(d*x+c),x, algorithm="maxima")

[Out]

-(4*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 4*(b*d*x + b*c)*sin(2*b*x + 2*a)^2 - (2*(b*d*x + b*c)*cos(2*b*x + 2*a)
- d*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c
^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*
b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c
^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^
2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4
*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*integrate((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - d^2)
*sin(b*x + a)/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^
2*c^2*d*x + b^2*c^3)*cos(b*x + a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(b*x + a)^2
+ 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)), x) + (b^2*d^2*x^2 + 2*b^2*c*d*x
+ b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)
*cos(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x
+ b^2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2 + 2*
(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4
*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*integrate((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2
- d^2)*sin(b*x + a)/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2
+ 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(b*x
+ a)^2 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(b*x + a)), x) - (d*cos(2*b*x + 2*a)
+ 2*(b*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(4*b*x + 4*a) - d*sin(2*b*x + 2*a))/(b^2*d^2*x^2 + 2*b^2*c*d*x + b^
2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos
(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^
2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2 + 2*(b^2
*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a)
- 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(cot(b*x+a)^3/(d*x+c),x, algorithm="giac")

[Out]

integrate(cot(b*x + a)^3/(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 24.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^3(a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{c+d\,x} \,d x \]

[In]

int(cot(a + b*x)^3/(c + d*x),x)

[Out]

int(cot(a + b*x)^3/(c + d*x), x)