\(\int \frac {\cot ^2(a+b x)}{x^2} \, dx\) [10]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\text {Int}\left (\frac {\cot ^2(a+b x)}{x^2},x\right ) \]

[Out]

Unintegrable(cot(b*x+a)^2/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int \frac {\cot ^2(a+b x)}{x^2} \, dx \]

[In]

Int[Cot[a + b*x]^2/x^2,x]

[Out]

Defer[Int][Cot[a + b*x]^2/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^2(a+b x)}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.87 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int \frac {\cot ^2(a+b x)}{x^2} \, dx \]

[In]

Integrate[Cot[a + b*x]^2/x^2,x]

[Out]

Integrate[Cot[a + b*x]^2/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

int(cot(b*x+a)^2/x^2,x)

[Out]

int(cot(b*x+a)^2/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int { \frac {\cot \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(cot(b*x+a)^2/x^2,x, algorithm="fricas")

[Out]

integral(cot(b*x + a)^2/x^2, x)

Sympy [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int \frac {\cot ^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

integrate(cot(b*x+a)**2/x**2,x)

[Out]

Integral(cot(a + b*x)**2/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 364, normalized size of antiderivative = 30.33 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int { \frac {\cot \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(cot(b*x+a)^2/x^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int { \frac {\cot \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(cot(b*x+a)^2/x^2,x, algorithm="giac")

[Out]

integrate(cot(b*x + a)^2/x^2, x)

Mupad [N/A]

Not integrable

Time = 12.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {cot}\left (a+b\,x\right )}^2}{x^2} \,d x \]

[In]

int(cot(a + b*x)^2/x^2,x)

[Out]

int(cot(a + b*x)^2/x^2, x)