\(\int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx\) [45]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\text {Int}\left (\frac {(a+b \cot (e+f x))^2}{c+d x},x\right ) \]

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 25.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (b \cot \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

integral((b^2*cot(f*x + e)^2 + 2*a*b*cot(f*x + e) + a^2)/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 1.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\int \frac {\left (a + b \cot {\left (e + f x \right )}\right )^{2}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 697, normalized size of antiderivative = 34.85 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (b \cot \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (b \cot \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 12.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx=\int \frac {{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2}{c+d\,x} \,d x \]

[In]

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

[Out]

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