3.2.45 \(\int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx\) [145]

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

[Out]

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

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Rubi [A]
time = 0.30, 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 = {} \begin {gather*} \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \end {gather*}

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 = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int x \cos (k \csc (x)) \cot (x) \csc (x) \, dx \end {gather*}

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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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x*Cos[k/Sin[x]]*Cos[x]/Sin[x]^2,x]')

[Out]

Timed out

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Maple [A]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {x \cos \left (x \right ) \cos \left (\frac {k}{\sin \left (x \right )}\right )}{\sin \left (x \right )^{2}}\, dx\]

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] Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (13) = 26\).
time = 0.26, size = 240, normalized size = 17.14 \begin {gather*} -\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 {gather*}

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 = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \cos {\left (x \right )} \cos {\left (\frac {k}{\sin {\left (x \right )}} \right )}}{\sin ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*cos(x)*cos(k/sin(x))/sin(x)**2, x)

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Giac [A] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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

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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