Optimal. Leaf size=11 \[ x+\frac{\cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.0528004, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4391, 2735, 2648} \[ x+\frac{\cos (x)}{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\tan (x)}{\sec (x)+\tan (x)} \, dx &=\int \frac{\sin (x)}{1+\sin (x)} \, dx\\ &=x-\int \frac{1}{1+\sin (x)} \, dx\\ &=x+\frac{\cos (x)}{1+\sin (x)}\\ \end{align*}
Mathematica [B] time = 0.0336051, size = 25, normalized size = 2.27 \[ x-\frac{2 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 13, normalized size = 1.2 \begin{align*} 2\, \left ( \tan \left ( x/2 \right ) +1 \right ) ^{-1}+x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66223, size = 38, normalized size = 3.45 \begin{align*} \frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.464028, size = 88, normalized size = 8. \begin{align*} \frac{{\left (x + 1\right )} \cos \left (x\right ) +{\left (x - 1\right )} \sin \left (x\right ) + x + 1}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \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{\tan{\left (x \right )}}{\tan{\left (x \right )} + \sec{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12496, size = 16, normalized size = 1.45 \begin{align*} x + \frac{2}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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