Optimal. Leaf size=11 \[ \frac {\cos (x)}{1-\sin (x)} \]
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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2727}
\begin {gather*} \frac {\cos (x)}{1-\sin (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rubi steps
\begin {align*} \int \frac {1}{1-\sin (x)} \, dx &=\frac {\cos (x)}{1-\sin (x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(11)=22\).
time = 0.01, size = 25, normalized size = 2.27 \begin {gather*} \frac {2 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.83, size = 10, normalized size = 0.91 \begin {gather*} \frac {-2}{-1+\text {Tan}\left [\frac {x}{2}\right ]} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 11, normalized size = 1.00
method | result | size |
default | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
norman | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
risch | \(\frac {2}{{\mathrm e}^{i x}-i}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 1.36 \begin {gather*} -\frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 17, normalized size = 1.55 \begin {gather*} \frac {\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 8, normalized size = 0.73 \begin {gather*} - \frac {2}{\tan {\left (\frac {x}{2} \right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\frac {2}{\tan \left (\frac {x}{2}\right )-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 10, normalized size = 0.91 \begin {gather*} -\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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