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