Optimal. Leaf size=7 \[ \frac{1}{2} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.0119241, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4288, 3770} \[ \frac{1}{2} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 4288
Rule 3770
Rubi steps
\begin{align*} \int \csc (2 x) \sin (x) \, dx &=\frac{1}{2} \int \sec (x) \, dx\\ &=\frac{1}{2} \tanh ^{-1}(\sin (x))\\ \end{align*}
Mathematica [B] time = 0.0062136, size = 37, normalized size = 5.29 \[ \frac{1}{2} \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 9, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54641, size = 47, normalized size = 6.71 \begin{align*} \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.3467, size = 59, normalized size = 8.43 \begin{align*} \frac{1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{4} \, \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 [B] time = 6.40099, size = 15, normalized size = 2.14 \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13954, size = 34, normalized size = 4.86 \begin{align*} \frac{1}{8} \, \log \left ({\left | \frac{1}{\sin \left (x\right )} + \sin \left (x\right ) + 2 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | \frac{1}{\sin \left (x\right )} + \sin \left (x\right ) - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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