Optimal. Leaf size=12 \[ \frac{2}{\frac{\cot (x)}{x}+1} \]
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Rubi [A] time = 0.0988092, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6711, 32} \[ \frac{2}{\frac{\cot (x)}{x}+1} \]
Antiderivative was successfully verified.
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Rule 6711
Rule 32
Rubi steps
\begin{align*} \int \frac{2 x+\sin (2 x)}{(\cos (x)+x \sin (x))^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,\frac{\cot (x)}{x}\right )\right )\\ &=\frac{2}{1+\frac{\cot (x)}{x}}\\ \end{align*}
Mathematica [A] time = 0.213904, size = 14, normalized size = 1.17 \[ \frac{2 x \sin (x)}{x \sin (x)+\cos (x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.711, size = 44, normalized size = 3.7 \begin{align*}{\frac{-2\,i}{x+i}}-{\frac{4\,ix}{ \left ( x+i \right ) \left ( x{{\rm e}^{2\,ix}}-x+i{{\rm e}^{2\,ix}}+i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.35291, size = 105, normalized size = 8.75 \begin{align*} -\frac{2 \,{\left (\cos \left (2 \, x\right )^{2} + 2 \, x \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}}{{\left (x^{2} + 1\right )} \cos \left (2 \, x\right )^{2} +{\left (x^{2} + 1\right )} \sin \left (2 \, x\right )^{2} + x^{2} - 2 \,{\left (x^{2} - 1\right )} \cos \left (2 \, x\right ) + 4 \, x \sin \left (2 \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95124, size = 42, normalized size = 3.5 \begin{align*} -\frac{2 \, \cos \left (x\right )}{x \sin \left (x\right ) + \cos \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 \frac{2 x + \sin{\left (2 x \right )}}{\left (x \sin{\left (x \right )} + \cos{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06736, size = 14, normalized size = 1.17 \begin{align*} -\frac{2}{x \tan \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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