Optimal. Leaf size=13 \[ -\log (1-\sin (x))+\log (\sin (x)) \]
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Rubi [A]
time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4476, 2786, 36,
29, 31} \begin {gather*} \log (\sin (x))-\log (1-\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2786
Rule 4476
Rubi steps
\begin {align*} \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac {\cot (x)}{1-\sin (x)} \, dx\\ &=\text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,-\sin (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{x} \, dx,x,-\sin (x)\right )-\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,-\sin (x)\right )\\ &=-\log (1-\sin (x))+\log (\sin (x))\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 22, normalized size = 1.69 \begin {gather*} -2 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 8, normalized size = 0.62
method | result | size |
derivativedivides | \(-\ln \left (-1+\csc \left (x \right )\right )\) | \(8\) |
default | \(-\ln \left (-1+\csc \left (x \right )\right )\) | \(8\) |
risch | \(-2 \ln \left ({\mathrm e}^{i x}-i\right )+\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 1.92 \begin {gather*} -2 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.09, size = 15, normalized size = 1.15 \begin {gather*} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}
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 \begin {gather*} \int \frac {\csc {\left (x \right )}}{- \tan {\left (x \right )} + \sec {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 14, normalized size = 1.08 \begin {gather*} -\log \left (-\sin \left (x\right ) + 1\right ) + \log \left ({\left | \sin \left (x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 15, normalized size = 1.15 \begin {gather*} \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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