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

Optimal. Leaf size=13 \[ \text{CannotIntegrate}(x \cot (x) \csc (x) \cos (k \csc (x)),x) \]

[Out]

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

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Rubi [A]  time = 0.435403, 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 = {} \[ \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \]

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*}

Mathematica [A]  time = 0.8798, size = 0, normalized size = 0. \[ \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \]

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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Maple [A]  time = 0.789, size = 0, normalized size = 0. \begin{align*} \int{\frac{x\cos \left ( x \right ) }{ \left ( \sin \left ( x \right ) \right ) ^{2}}\cos \left ({\frac{k}{\sin \left ( x \right ) }} \right ) }\, dx \end{align*}

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 = 0.983046, size = 324, normalized size = 24.92 \begin{align*} -\frac{{\left (x e^{\left (\frac{4 \, k \cos \left (2 \, x\right ) \cos \left (x\right )}{\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 \left (x\right )}{\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 \left (x\right )}{\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 \left (x\right )}{\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 \left (x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1} - \frac{2 \, k \cos \left (x\right )}{\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 \left (x\right ) \sin \left (2 \, x\right ) - k \cos \left (2 \, x\right ) \sin \left (x\right ) + k \sin \left (x\right )\right )}}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}\right )}{2 \, k} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x \cos \left (x\right ) \cos \left (\frac{k}{\sin \left (x\right )}\right )}{\cos \left (x\right )^{2} - 1}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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)