Optimal. Leaf size=14 \[ \frac {1}{2} \log (\tan (x))-\frac {\tan (x)}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12} \[ \frac {1}{2} \log (\tan (x))-\frac {\tan (x)}{2} \]
Antiderivative was successfully verified.
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Rule 12
Rubi steps
\begin {align*} \int \csc (2 x) (1-\tan (x)) \, dx &=\operatorname {Subst}\left (\int \frac {1}{2} \left (-1+\frac {1}{x}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-1+\frac {1}{x}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \log (\tan (x))-\frac {\tan (x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.50 \[ -\frac {\tan (x)}{2}+\frac {1}{2} \log (\sin (x))-\frac {1}{2} \log (\cos (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc (2 x) (1-\tan (x)) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.20, size = 32, normalized size = 2.29 \[ \frac {1}{4} \, \log \left (\frac {\tan \relax (x)^{2}}{\tan \relax (x)^{2} + 1}\right ) - \frac {1}{4} \, \log \left (\frac {1}{\tan \relax (x)^{2} + 1}\right ) - \frac {1}{2} \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 11, normalized size = 0.79 \[ \frac {1}{2} \, \log \left ({\left | \tan \relax (x) \right |}\right ) - \frac {1}{2} \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {\ln \left (\tan \relax (x )\right )}{2}-\frac {\tan \relax (x )}{2}\) | \(11\) |
norman | \(\frac {\ln \left (\tan \relax (x )\right )}{2}-\frac {\tan \relax (x )}{2}\) | \(11\) |
risch | \(-\frac {i}{1+{\mathrm e}^{2 i x}}-\frac {\ln \left (1+{\mathrm e}^{2 i x}\right )}{2}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{2}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 47, normalized size = 3.36 \[ -\frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1} - \frac {1}{4} \, \log \left (\cos \left (2 \, x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\cos \left (2 \, x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 10, normalized size = 0.71 \[ \frac {\ln \left (\mathrm {tan}\relax (x)\right )}{2}-\frac {\mathrm {tan}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.48, size = 27, normalized size = 1.93 \[ \frac {\log {\left (\cos {\left (2 x \right )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\left (2 x \right )} + 1 \right )}}{4} - \frac {\sin {\relax (x )}}{2 \cos {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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