Optimal. Leaf size=23 \[ \frac{\log \left (\tan \left (\frac{1}{2} (d+e x)\right )+1\right )}{2 a e} \]
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Rubi [A] time = 0.0213659, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3124, 31} \[ \frac{\log \left (\tan \left (\frac{1}{2} (d+e x)\right )+1\right )}{2 a e} \]
Antiderivative was successfully verified.
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Rule 3124
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{2 a+2 a \cos (d+e x)+2 a \sin (d+e x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a+4 a x} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{e}\\ &=\frac{\log \left (1+\tan \left (\frac{1}{2} (d+e x)\right )\right )}{2 a e}\\ \end{align*}
Mathematica [B] time = 0.0296963, size = 50, normalized size = 2.17 \[ \frac{\frac{\log \left (\sin \left (\frac{1}{2} (d+e x)\right )+\cos \left (\frac{1}{2} (d+e x)\right )\right )}{e}-\frac{\log \left (\cos \left (\frac{1}{2} (d+e x)\right )\right )}{e}}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 21, normalized size = 0.9 \begin{align*}{\frac{1}{2\,ae}\ln \left ( 1+\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987027, size = 38, normalized size = 1.65 \begin{align*} \frac{\log \left (\frac{\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{2 \, a e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03023, size = 89, normalized size = 3.87 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (e x + d\right ) + \frac{1}{2}\right ) - \log \left (\sin \left (e x + d\right ) + 1\right )}{4 \, a e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.719413, size = 36, normalized size = 1.57 \begin{align*} \begin{cases} \frac{\log{\left (\tan{\left (\frac{d}{2} + \frac{e x}{2} \right )} + 1 \right )}}{2 a e} & \text{for}\: e \neq 0 \\\frac{x}{2 a \sin{\left (d \right )} + 2 a \cos{\left (d \right )} + 2 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13198, size = 28, normalized size = 1.22 \begin{align*} \frac{e^{\left (-1\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 1 \right |}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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