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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{(a+b \cot (e+f x))^2}{c+d x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0533979, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(a+b \cot (e+f x))^2}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 18.7796, size = 0, normalized size = 0. \[ \int \frac{(a+b \cot (e+f x))^2}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 1.752, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\cot \left ( fx+e \right ) \right ) ^{2}}{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \cot \left (f x + e\right )^{2} + 2 \, a b \cot \left (f x + e\right ) + a^{2}}{d x + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \cot{\left (e + f x \right )}\right )^{2}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \cot \left (f x + e\right ) + a\right )}^{2}}{d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)