Optimal. Leaf size=6 \[ \frac {\tan (x)}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {8} \[ \frac {\tan (x)}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin {align*} \int \csc (2 x) \tan (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{2} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 6, normalized size = 1.00 \[ \frac {\tan (x)}{2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc (2 x) \tan (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.16, size = 4, normalized size = 0.67 \[ \frac {1}{2} \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 4, normalized size = 0.67 \[ \frac {1}{2} \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 5, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\tan \relax (x )}{2}\) | \(5\) |
default | \(\frac {\tan \relax (x )}{2}\) | \(5\) |
norman | \(\frac {\tan \relax (x )}{2}\) | \(5\) |
risch | \(\frac {i}{1+{\mathrm e}^{2 i x}}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 27, normalized size = 4.50 \[ \frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 4, normalized size = 0.67 \[ \frac {\mathrm {tan}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.79, size = 7, normalized size = 1.17 \[ \frac {\sin {\relax (x )}}{2 \cos {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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