Optimal. Leaf size=13 \[ \log (\sin (x))-\log (1-\sin (x)) \]
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Rubi [A] time = 0.0620936, antiderivative size = 13, normalized size of antiderivative = 1., 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 = {4391, 2707, 36, 29, 31} \[ \log (\sin (x))-\log (1-\sin (x)) \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2707
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\csc (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac{\cot (x)}{1-\sin (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,-\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,-\sin (x)\right )-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,-\sin (x)\right )\\ &=-\log (1-\sin (x))+\log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0220258, size = 22, normalized size = 1.69 \[ \log (\sin (x))-2 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 8, normalized size = 0.6 \begin{align*} -\ln \left ( -1+\csc \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01512, size = 34, normalized size = 2.62 \begin{align*} -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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487989, size = 49, normalized size = 3.77 \begin{align*} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\left (x \right )}}{- \tan{\left (x \right )} + \sec{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12615, size = 19, normalized size = 1.46 \begin{align*} -\log \left (-\sin \left (x\right ) + 1\right ) + \log \left ({\left | \sin \left (x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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