3.145 \(\int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx\)

Optimal. Leaf size=14 \[ \text {Int}(x \cot (x) \csc (x) \cos (k \csc (x)),x) \]

[Out]

CannotIntegrate(x*cos(k*csc(x))*cot(x)*csc(x),x)

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Rubi [A]  time = 0.44, 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 = {} \[ \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Cos[k*Csc[x]]*Cot[x]*Csc[x],x]

[Out]

Defer[Int][x*Cos[k*Csc[x]]*Cot[x]*Csc[x], x]

Rubi steps

\begin {align*} \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx &=\int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x*Cos[k*Csc[x]]*Cot[x]*Csc[x],x]

[Out]

Integrate[x*Cos[k*Csc[x]]*Cot[x]*Csc[x], x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x \cos \relax (x) \cos \left (\frac {k}{\sin \relax (x)}\right )}{\cos \relax (x)^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2,x, algorithm="fricas")

[Out]

integral(-x*cos(x)*cos(k/sin(x))/(cos(x)^2 - 1), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \relax (x) \cos \left (\frac {k}{\sin \relax (x)}\right )}{\sin \relax (x)^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2,x, algorithm="giac")

[Out]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2, x)

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maple [A]  time = 1.21, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \relax (x ) \cos \left (\frac {k}{\sin \relax (x )}\right )}{\sin \relax (x )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x)*cos(k/sin(x))/sin(x)^2,x)

[Out]

int(x*cos(x)*cos(k/sin(x))/sin(x)^2,x)

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maxima [A]  time = 0.47, size = 240, normalized size = 17.14 \[ -\frac {{\left (x e^{\left (\frac {4 \, k \cos \left (2 \, x\right ) \cos \relax (x)}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} + \frac {4 \, k \sin \left (2 \, x\right ) \sin \relax (x)}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} + x e^{\left (\frac {4 \, k \cos \relax (x)}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}\right )} e^{\left (-\frac {2 \, k \cos \left (2 \, x\right ) \cos \relax (x)}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac {2 \, k \sin \left (2 \, x\right ) \sin \relax (x)}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac {2 \, k \cos \relax (x)}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )} \sin \left (\frac {2 \, {\left (k \cos \relax (x) \sin \left (2 \, x\right ) - k \cos \left (2 \, x\right ) \sin \relax (x) + k \sin \relax (x)\right )}}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}{2 \, k} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2,x, algorithm="maxima")

[Out]

-1/2*(x*e^(4*k*cos(2*x)*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) + 4*k*sin(2*x)*sin(x)/(cos(2*x)^2 +
sin(2*x)^2 - 2*cos(2*x) + 1)) + x*e^(4*k*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)))*e^(-2*k*cos(2*x)*
cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) - 2*k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x)
+ 1) - 2*k*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1))*sin(2*(k*cos(x)*sin(2*x) - k*cos(2*x)*sin(x) + k
*sin(x))/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1))/k

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mupad [A]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {x\,\cos \left (\frac {k}{\sin \relax (x)}\right )\,\cos \relax (x)}{{\sin \relax (x)}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*cos(k/sin(x))*cos(x))/sin(x)^2,x)

[Out]

int((x*cos(k/sin(x))*cos(x))/sin(x)^2, x)

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sympy [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(x*cos(x)*cos(k/sin(x))/sin(x)**2,x)

[Out]

Timed out

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