Optimal. Leaf size=12 \[ \frac {2}{\frac {\cot (x)}{x}+1} \]
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Rubi [A] time = 0.10, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 32
Rule 6711
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*}
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Mathematica [A] time = 0.23, size = 14, normalized size = 1.17 \[ \frac {2 x \sin (x)}{x \sin (x)+\cos (x)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 x+\sin (2 x)}{(\cos (x)+x \sin (x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.25, size = 13, normalized size = 1.08 \[ -\frac {2 \, \cos \relax (x)}{x \sin \relax (x) + \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 10, normalized size = 0.83 \[ -\frac {2}{x \tan \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.83, size = 44, normalized size = 3.67
method | result | size |
risch | \(-\frac {2 i}{x +i}-\frac {4 i x}{\left (x +i\right ) \left (x \,{\mathrm e}^{2 i x}-x +i {\mathrm e}^{2 i x}+i\right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 78, normalized size = 6.50 \[ -\frac {2 \, {\left (\cos \left (2 \, x\right )^{2} + 2 \, x \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}}{{\left (x^{2} + 1\right )} \cos \left (2 \, x\right )^{2} + {\left (x^{2} + 1\right )} \sin \left (2 \, x\right )^{2} + x^{2} - 2 \, {\left (x^{2} - 1\right )} \cos \left (2 \, x\right ) + 4 \, x \sin \left (2 \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {2\,x+\sin \left (2\,x\right )}{{\left (\cos \relax (x)+x\,\sin \relax (x)\right )}^2} \,d x \]
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 \[ \int \frac {2 x + \sin {\left (2 x \right )}}{\left (x \sin {\relax (x )} + \cos {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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