Optimal. Leaf size=47 \[ -\sin (x)-\frac{1}{6} \log (1-2 \sin (x))-\frac{1}{6} \log (1-\sin (x))+\frac{1}{6} \log (\sin (x)+1)+\frac{1}{6} \log (2 \sin (x)+1) \]
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Rubi [A] time = 0.0519936, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {1279, 1161, 616, 31} \[ -\sin (x)-\frac{1}{6} \log (1-2 \sin (x))-\frac{1}{6} \log (1-\sin (x))+\frac{1}{6} \log (\sin (x)+1)+\frac{1}{6} \log (2 \sin (x)+1) \]
Antiderivative was successfully verified.
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Rule 1279
Rule 1161
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \sin (x) \tan (3 x) \, dx &=\operatorname{Subst}\left (\int \frac{x^2 \left (3-4 x^2\right )}{1-5 x^2+4 x^4} \, dx,x,\sin (x)\right )\\ &=-\sin (x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-4+8 x^2}{1-5 x^2+4 x^4} \, dx,x,\sin (x)\right )\\ &=-\sin (x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}-\frac{x}{2}+x^2} \, dx,x,\sin (x)\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}+\frac{x}{2}+x^2} \, dx,x,\sin (x)\right )\\ &=-\sin (x)-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\sin (x)\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}+x} \, dx,x,\sin (x)\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2}+x} \, dx,x,\sin (x)\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sin (x)\right )\\ &=-\frac{1}{6} \log (1-2 \sin (x))-\frac{1}{6} \log (1-\sin (x))+\frac{1}{6} \log (1+\sin (x))+\frac{1}{6} \log (1+2 \sin (x))-\sin (x)\\ \end{align*}
Mathematica [A] time = 0.0284489, size = 21, normalized size = 0.45 \[ -\sin (x)+\frac{1}{3} \tanh ^{-1}(\sin (x))+\frac{1}{3} \tanh ^{-1}(2 \sin (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.094, size = 38, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{6}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{6}}+{\frac{\ln \left ( 1+2\,\sin \left ( x \right ) \right ) }{6}}-{\frac{\ln \left ( -1+2\,\sin \left ( x \right ) \right ) }{6}}-\sin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) -{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - \cos \left (2 \, x\right ) \cos \left (x\right ) +{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right ) \sin \left (x\right ) + \cos \left (x\right )}{3 \,{\left (2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )}}\,{d x} + \frac{1}{6} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac{1}{6} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) - \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54101, size = 138, normalized size = 2.94 \begin{align*} \frac{1}{6} \, \log \left (2 \, \sin \left (x\right ) + 1\right ) + \frac{1}{6} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{6} \, \log \left (-\sin \left (x\right ) + 1\right ) - \frac{1}{6} \, \log \left (-2 \, \sin \left (x\right ) + 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 (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25093, size = 491, normalized size = 10.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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