\(\int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx\) [55]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a+b \cot (e+f x))},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)/(a+b*cot(f*x+e)),x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx \]

[In]

Int[1/((c + d*x)*(a + b*Cot[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)*(a + b*Cot[e + f*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx \]

[In]

Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right ) \left (a +b \cot \left (f x +e \right )\right )}d x\]

[In]

int(1/(d*x+c)/(a+b*cot(f*x+e)),x)

[Out]

int(1/(d*x+c)/(a+b*cot(f*x+e)),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \cot \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d*x + a*c + (b*d*x + b*c)*cot(f*x + e)), x)

Sympy [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int \frac {1}{\left (a + b \cot {\left (e + f x \right )}\right ) \left (c + d x\right )}\, dx \]

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x)

[Out]

Integral(1/((a + b*cot(e + f*x))*(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 0.94 (sec) , antiderivative size = 279, normalized size of antiderivative = 13.95 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \cot \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \cot \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(b*cot(f*x + e) + a)), x)

Mupad [N/A]

Not integrable

Time = 12.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \cot (e+f x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a + b*cot(e + f*x))*(c + d*x)),x)

[Out]

int(1/((a + b*cot(e + f*x))*(c + d*x)), x)