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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a+b \cot (e+f x))^2}{c+d x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx \]

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*}

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Mathematica [A]  time = 19.85, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \cot (e+f x))^2}{c+d x} \, dx \]

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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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \cot \left (f x + e\right )^{2} + 2 \, a b \cot \left (f x + e\right ) + a^{2}}{d x + c}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cot \left (f x + e\right ) + a\right )}^{2}}{d x + c}\,{d x} \]

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)

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maple [A]  time = 5.79, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \cot \left (f x +e \right )\right )^{2}}{d x +c}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \cot {\left (e + f x \right )}\right )^{2}}{c + d x}\, dx \]

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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