3.9 \(\int \frac{\cot ^2(a+b x)}{x} \, dx\)

Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{\cot ^2(a+b x)}{x},x\right ) \]

[Out]

Unintegrable[Cot[a + b*x]^2/x, x]

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Rubi [A]  time = 0.029607, 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 = {} \[ \int \frac{\cot ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + b*x]^2/x,x]

[Out]

Defer[Int][Cot[a + b*x]^2/x, x]

Rubi steps

\begin{align*} \int \frac{\cot ^2(a+b x)}{x} \, dx &=\int \frac{\cot ^2(a+b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 3.5839, size = 0, normalized size = 0. \[ \int \frac{\cot ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[a + b*x]^2/x,x]

[Out]

Integrate[Cot[a + b*x]^2/x, x]

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Maple [A]  time = 0.235, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b x \cos \left (2 \, b x + 2 \, a\right )^{2} \log \left (x\right ) + b x \log \left (x\right ) \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) \log \left (x\right ) + b x \log \left (x\right ) - \frac{{\left (b^{2} x \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b^{2} x \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x\right )} \int \frac{\sin \left (b x + a\right )}{{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )} x^{2}}\,{d x}}{b^{2}} + \frac{{\left (b^{2} x \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b^{2} x \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x\right )} \int \frac{\sin \left (b x + a\right )}{{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )} x^{2}}\,{d x}}{b^{2}} + 2 \, \sin \left (2 \, b x + 2 \, a\right )}{b x \cos \left (2 \, b x + 2 \, a\right )^{2} + b x \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) + b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cot(b*x + a)^2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(a + b*x)**2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(b*x + a)^2/x, x)