Optimal. Leaf size=15 \[ \frac {\cos (x)}{1-\sin (x)}-x \]
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Rubi [A] time = 0.06, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4391, 2735, 2648} \[ \frac {\cos (x)}{1-\sin (x)}-x \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 4391
Rubi steps
\begin {align*} \int \frac {\tan (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac {\sin (x)}{1-\sin (x)} \, dx\\ &=-x+\int \frac {1}{1-\sin (x)} \, dx\\ &=-x+\frac {\cos (x)}{1-\sin (x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.93 \[ \frac {2 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 28, normalized size = 1.87 \[ -\frac {{\left (x - 1\right )} \cos \relax (x) - {\left (x + 1\right )} \sin \relax (x) + x - 1}{\cos \relax (x) - \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.95, size = 14, normalized size = 0.93 \[ -x - \frac {2}{\tan \left (\frac {1}{2} \, x\right ) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 15, normalized size = 1.00 \[ -\frac {2}{\tan \left (\frac {x}{2}\right )-1}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 28, normalized size = 1.87 \[ -\frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1} - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 14, normalized size = 0.93 \[ -x-\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\relax (x )}}{- \tan {\relax (x )} + \sec {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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