Optimal. Leaf size=20 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}}-\sin (x) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 206
Rule 321
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*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}}-\sin (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 2.06, size = 38, normalized size = 1.90 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} - 2 \, \sqrt {2} \sin \relax (x) - 3}{2 \, \cos \relax (x)^{2} - 1}\right ) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \relax (x) \tan \left (2 \, x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 18, normalized size = 0.90 \[ -\sin \relax (x )+\frac {\arctanh \left (\sin \relax (x ) \sqrt {2}\right ) \sqrt {2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 141, normalized size = 7.05 \[ \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) + \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 17, normalized size = 0.85 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )}{2}-\sin \relax (x) \]
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 \[ \int \sin {\relax (x )} \tan {\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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