3.14 \(\int \frac {\cot ^3(a+b x)}{x} \, dx\)

Optimal. Leaf size=15 \[ \text {Int}\left (\frac {\cot ^3(a+b x)}{x},x\right ) \]

[Out]

Unintegrable(cot(b*x+a)^3/x,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 = {} \[ \int \frac {\cot ^3(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + b*x]^3/x,x]

[Out]

Defer[Int][Cot[a + b*x]^3/x, x]

Rubi steps

\begin {align*} \int \frac {\cot ^3(a+b x)}{x} \, dx &=\int \frac {\cot ^3(a+b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 6.51, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^3(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[a + b*x]^3/x,x]

[Out]

Integrate[Cot[a + b*x]^3/x, x]

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cot \left (b x + a\right )^{3}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3/x,x, algorithm="fricas")

[Out]

integral(cot(b*x + a)^3/x, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (b x + a\right )^{3}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3/x,x, algorithm="giac")

[Out]

integrate(cot(b*x + a)^3/x, x)

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maple [A]  time = 2.92, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}\left (b x +a \right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^3/x,x)

[Out]

int(cot(b*x+a)^3/x,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {4 \, b x \cos \left (2 \, b x + 2 \, a\right )^{2} + 4 \, b x \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) - {\left (2 \, b x \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) - {\left (2 \, b x \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) - \frac {{\left (b^{2} x^{2} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2} - 2 \, {\left (2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) - b^{2} x^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )\right )} {\left (b^{2} \int \frac {x^{2} \sin \left (b x + a\right )}{x^{3} \cos \left (b x + a\right )^{2} + x^{3} \sin \left (b x + a\right )^{2} + 2 \, x^{3} \cos \left (b x + a\right ) + x^{3}}\,{d x} - \int \frac {\sin \left (b x + a\right )}{x^{3} \cos \left (b x + a\right )^{2} + x^{3} \sin \left (b x + a\right )^{2} + 2 \, x^{3} \cos \left (b x + a\right ) + x^{3}}\,{d x}\right )}}{b^{2}} + \frac {{\left (b^{2} x^{2} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2} - 2 \, {\left (2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) - b^{2} x^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )\right )} {\left (b^{2} \int \frac {x^{2} \sin \left (b x + a\right )}{x^{3} \cos \left (b x + a\right )^{2} + x^{3} \sin \left (b x + a\right )^{2} - 2 \, x^{3} \cos \left (b x + a\right ) + x^{3}}\,{d x} - \int \frac {\sin \left (b x + a\right )}{x^{3} \cos \left (b x + a\right )^{2} + x^{3} \sin \left (b x + a\right )^{2} - 2 \, x^{3} \cos \left (b x + a\right ) + x^{3}}\,{d x}\right )}}{b^{2}} - \sin \left (2 \, b x + 2 \, a\right )}{b^{2} x^{2} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2} - 2 \, {\left (2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) - b^{2} x^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3/x,x, algorithm="maxima")

[Out]

-(4*b*x*cos(2*b*x + 2*a)^2 + 4*b*x*sin(2*b*x + 2*a)^2 - 2*b*x*cos(2*b*x + 2*a) - (2*b*x*cos(2*b*x + 2*a) - sin
(2*b*x + 2*a))*cos(4*b*x + 4*a) - (b^2*x^2*cos(4*b*x + 4*a)^2 + 4*b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(4*b
*x + 4*a)^2 - 4*b^2*x^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^2*sin(2*b*x + 2*a)^2 - 4*b^2*x^2*cos(2*b*x
+ 2*a) + b^2*x^2 - 2*(2*b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(4*b*x + 4*a))*integrate((b^2*x^2 - 1)*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) + (b^2*x^2*cos(
4*b*x + 4*a)^2 + 4*b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(4*b*x + 4*a)^2 - 4*b^2*x^2*sin(4*b*x + 4*a)*sin(2*
b*x + 2*a) + 4*b^2*x^2*sin(2*b*x + 2*a)^2 - 4*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2 - 2*(2*b^2*x^2*cos(2*b*x + 2*
a) - b^2*x^2)*cos(4*b*x + 4*a))*integrate((b^2*x^2 - 1)*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*x*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) - 1)*sin(4*b*x +
4*a) - sin(2*b*x + 2*a))/(b^2*x^2*cos(4*b*x + 4*a)^2 + 4*b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(4*b*x + 4*a)
^2 - 4*b^2*x^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^2*sin(2*b*x + 2*a)^2 - 4*b^2*x^2*cos(2*b*x + 2*a) +
b^2*x^2 - 2*(2*b^2*x^2*cos(2*b*x + 2*a) - b^2*x^2)*cos(4*b*x + 4*a))

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mupad [A]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^3/x,x)

[Out]

int(cot(a + b*x)^3/x, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}{\left (a + b x \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)**3/x,x)

[Out]

Integral(cot(a + b*x)**3/x, x)

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