3.15 \(\int \frac {\cot ^3(a+b x)}{x^2} \, dx\)

Optimal. Leaf size=15 \[ \text {Int}\left (\frac {\cot ^3(a+b x)}{x^2},x\right ) \]

[Out]

Unintegrable(cot(b*x+a)^3/x^2,x)

________________________________________________________________________________________

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 = {} \[ \int \frac {\cot ^3(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + b*x]^3/x^2,x]

[Out]

Defer[Int][Cot[a + b*x]^3/x^2, x]

Rubi steps

\begin {align*} \int \frac {\cot ^3(a+b x)}{x^2} \, dx &=\int \frac {\cot ^3(a+b x)}{x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 5.27, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^3(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[a + b*x]^3/x^2,x]

[Out]

Integrate[Cot[a + b*x]^3/x^2, x]

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cot \left (b x + a\right )^{3}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3/x^2,x, algorithm="fricas")

[Out]

integral(cot(b*x + a)^3/x^2, x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (b x + a\right )^{3}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)^3/x^2,x, algorithm="giac")

[Out]

integrate(cot(b*x + a)^3/x^2, x)

________________________________________________________________________________________

maple [A]  time = 3.79, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}\left (b x +a \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)^3/x^2,x)

[Out]

int(cot(b*x+a)^3/x^2,x)

________________________________________________________________________________________

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 ) - 2 \, {\left (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 ) - 2 \, {\left (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^{3} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{3} \sin \left (2 \, b x + 2 \, a\right )^{2} - 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3} - 2 \, {\left (2 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) - b^{2} x^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )\right )} {\left (b^{2} \int \frac {x^{2} \sin \left (b x + a\right )}{x^{4} \cos \left (b x + a\right )^{2} + x^{4} \sin \left (b x + a\right )^{2} + 2 \, x^{4} \cos \left (b x + a\right ) + x^{4}}\,{d x} - 3 \, \int \frac {\sin \left (b x + a\right )}{x^{4} \cos \left (b x + a\right )^{2} + x^{4} \sin \left (b x + a\right )^{2} + 2 \, x^{4} \cos \left (b x + a\right ) + x^{4}}\,{d x}\right )}}{b^{2}} + \frac {{\left (b^{2} x^{3} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{3} \sin \left (2 \, b x + 2 \, a\right )^{2} - 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3} - 2 \, {\left (2 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) - b^{2} x^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )\right )} {\left (b^{2} \int \frac {x^{2} \sin \left (b x + a\right )}{x^{4} \cos \left (b x + a\right )^{2} + x^{4} \sin \left (b x + a\right )^{2} - 2 \, x^{4} \cos \left (b x + a\right ) + x^{4}}\,{d x} - 3 \, \int \frac {\sin \left (b x + a\right )}{x^{4} \cos \left (b x + a\right )^{2} + x^{4} \sin \left (b x + a\right )^{2} - 2 \, x^{4} \cos \left (b x + a\right ) + x^{4}}\,{d x}\right )}}{b^{2}} - 2 \, \sin \left (2 \, b x + 2 \, a\right )}{b^{2} x^{3} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right )^{2} - 4 \, b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{3} \sin \left (2 \, b x + 2 \, a\right )^{2} - 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3} - 2 \, {\left (2 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) - b^{2} x^{3}\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^2,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^3*cos(4*b*x + 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b
*x + 4*a)^2 - 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 - 4*b^2*x^3*cos(2*b*x
+ 2*a) + b^2*x^3 - 2*(2*b^2*x^3*cos(2*b*x + 2*a) - b^2*x^3)*cos(4*b*x + 4*a))*integrate((b^2*x^2 - 3)*sin(b*x
+ a)/(b^2*x^4*cos(b*x + a)^2 + b^2*x^4*sin(b*x + a)^2 + 2*b^2*x^4*cos(b*x + a) + b^2*x^4), x) + (b^2*x^3*cos(
4*b*x + 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 - 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*
b*x + 2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 - 4*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 - 2*(2*b^2*x^3*cos(2*b*x + 2*
a) - b^2*x^3)*cos(4*b*x + 4*a))*integrate((b^2*x^2 - 3)*sin(b*x + a)/(b^2*x^4*cos(b*x + a)^2 + b^2*x^4*sin(b*x
+ a)^2 - 2*b^2*x^4*cos(b*x + a) + b^2*x^4), x) - 2*(b*x*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) - 1)*sin(4*b*x +
4*a) - 2*sin(2*b*x + 2*a))/(b^2*x^3*cos(4*b*x + 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b*x + 4*
a)^2 - 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 - 4*b^2*x^3*cos(2*b*x + 2*a)
+ b^2*x^3 - 2*(2*b^2*x^3*cos(2*b*x + 2*a) - b^2*x^3)*cos(4*b*x + 4*a))

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^3/x^2,x)

[Out]

int(cot(a + b*x)^3/x^2, x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)**3/x**2,x)

[Out]

Integral(cot(a + b*x)**3/x**2, x)

________________________________________________________________________________________