\(\int \frac {\cot ^2(a+b x)}{c+d x} \, dx\) [109]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\text {Int}\left (\frac {\cot ^2(a+b x)}{c+d x},x\right ) \]

[Out]

Unintegrable(cot(b*x+a)^2/(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 = {} \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cot ^2(a+b x)}{c+d x} \, dx \]

[In]

Int[Cot[a + b*x]^2/(c + d*x),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.46 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cot ^2(a+b x)}{c+d x} \, dx \]

[In]

Integrate[Cot[a + b*x]^2/(c + d*x),x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.66 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

integral(cot(b*x + a)^2/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cot ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 581, normalized size of antiderivative = 36.31 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 22.93 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^2(a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {cot}\left (a+b\,x\right )}^2}{c+d\,x} \,d x \]

[In]

int(cot(a + b*x)^2/(c + d*x),x)

[Out]

int(cot(a + b*x)^2/(c + d*x), x)