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

Optimal. Leaf size=15 \[ \text {Int}\left (\frac {\cot ^2(a+b x)}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + b*x]^2/x^2,x]

[Out]

Defer[Int][Cot[a + b*x]^2/x^2, x]

Rubi steps

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

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Mathematica [A]  time = 3.43, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^2(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[a + b*x]^2/x^2,x]

[Out]

Integrate[Cot[a + b*x]^2/x^2, x]

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fricas [A]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cot \left (b x + a\right )^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cot(b*x + a)^2/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(b*x + a)^2/x^2, x)

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maple [A]  time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}\left (b x +a \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {b x \cos \left (2 \, b x + 2 \, a\right )^{2} + b x \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) + b x + \frac {2 \, {\left (b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2}\right )} \int \frac {\sin \left (b x + a\right )}{{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )} x^{3}}\,{d x}}{b^{2}} - \frac {2 \, {\left (b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2}\right )} \int \frac {\sin \left (b x + a\right )}{{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )} x^{3}}\,{d x}}{b^{2}} - 2 \, \sin \left (2 \, b x + 2 \, a\right )}{b x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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 + 2*(b^2*x^2*cos(2*b*x + 2*a)^
2 + b^2*x^2*sin(2*b*x + 2*a)^2 - 2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2)*integrate(sin(b*x + a)/(b^2*x^3*cos(b*x
+ a)^2 + b^2*x^3*sin(b*x + a)^2 + 2*b^2*x^3*cos(b*x + a) + b^2*x^3), x) - 2*(b^2*x^2*cos(2*b*x + 2*a)^2 + b^2
*x^2*sin(2*b*x + 2*a)^2 - 2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2)*integrate(sin(b*x + a)/(b^2*x^3*cos(b*x + a)^2
+ b^2*x^3*sin(b*x + a)^2 - 2*b^2*x^3*cos(b*x + a) + b^2*x^3), x) - 2*sin(2*b*x + 2*a))/(b*x^2*cos(2*b*x + 2*a
)^2 + b*x^2*sin(2*b*x + 2*a)^2 - 2*b*x^2*cos(2*b*x + 2*a) + b*x^2)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {{\mathrm {cot}\left (a+b\,x\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^2/x^2,x)

[Out]

int(cot(a + b*x)^2/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(a + b*x)**2/x**2, x)

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