Optimal. Leaf size=20 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{\sqrt{2}}-\sin (x) \]
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Rubi [A] time = 0.0230994, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 321, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{\sqrt{2}}-\sin (x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \sin (x) \tan (2 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2}{1-2 x^2} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2}{1-2 x^2} \, dx,x,\sin (x)\right )\\ &=-\sin (x)+\operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{\sqrt{2}}-\sin (x)\\ \end{align*}
Mathematica [A] time = 0.0134385, size = 20, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{\sqrt{2}}-\sin (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 18, normalized size = 0.9 \begin{align*} -\sin \left ( x \right ) +{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53618, size = 190, normalized size = 9.5 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{8} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) + \frac{1}{8} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{8} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.36381, size = 109, normalized size = 5.45 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \, \cos \left (x\right )^{2} - 2 \, \sqrt{2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \sin \left (x\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 \sin{\left (x \right )} \tan{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (x\right ) \tan \left (2 \, x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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