Optimal. Leaf size=11 \[ \cot (x)-\frac {1}{2} \cos (2 x) \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3767, 8, 2638} \[ \cot (x)-\frac {1}{2} \cos (2 x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2638
Rule 3767
Rubi steps
\begin {align*} \int \left (-\csc ^2(x)+\sin (2 x)\right ) \, dx &=-\int \csc ^2(x) \, dx+\int \sin (2 x) \, dx\\ &=-\frac {1}{2} \cos (2 x)+\operatorname {Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\frac {1}{2} \cos (2 x)+\cot (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \cot (x)-\frac {1}{2} \cos (2 x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 22, normalized size = 2.00 \[ -\frac {{\left (2 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x) - 2 \, \cos \relax (x)}{2 \, \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 11, normalized size = 1.00 \[ \frac {1}{\tan \relax (x)} - \frac {1}{2} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 10, normalized size = 0.91 \[ -\frac {\cos \left (2 x \right )}{2}+\cot \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 11, normalized size = 1.00 \[ \frac {1}{\tan \relax (x)} - \frac {1}{2} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 14, normalized size = 1.27 \[ \frac {\cos \relax (x)}{\sin \relax (x)}-{\cos \relax (x)}^2 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 12, normalized size = 1.09 \[ - \frac {\cos {\left (2 x \right )}}{2} + \frac {\cos {\relax (x )}}{\sin {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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