Optimal. Leaf size=25 \[ \frac{\log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e} \]
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Rubi [A] time = 0.0223817, antiderivative size = 25, 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} \[ \frac{\log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e} \]
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 c \sin (d+e x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a+4 c x} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{e}\\ &=\frac{\log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e}\\ \end{align*}
Mathematica [B] time = 0.0533843, size = 57, normalized size = 2.28 \[ \frac{1}{2} \left (\frac{\log \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )}{c e}-\frac{\log \left (\cos \left (\frac{1}{2} (d+e x)\right )\right )}{c e}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 23, normalized size = 0.9 \begin{align*}{\frac{1}{2\,ce}\ln \left ( a+c\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 = 0.997076, size = 39, normalized size = 1.56 \begin{align*} \frac{\log \left (a + \frac{c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{2 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17535, size = 157, normalized size = 6.28 \begin{align*} \frac{\log \left (a c \sin \left (e x + d\right ) + \frac{1}{2} \, a^{2} + \frac{1}{2} \, c^{2} + \frac{1}{2} \,{\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) - \log \left (\frac{1}{2} \, \cos \left (e x + d\right ) + \frac{1}{2}\right )}{4 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20441, size = 63, normalized size = 2.52 \begin{align*} \begin{cases} \frac{x}{2 a \cos{\left (d \right )} + 2 a} & \text{for}\: c = 0 \wedge e = 0 \\\frac{\tan{\left (\frac{d}{2} + \frac{e x}{2} \right )}}{2 a e} & \text{for}\: c = 0 \\\frac{x}{2 a \cos{\left (d \right )} + 2 a + 2 c \sin{\left (d \right )}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{a}{c} + \tan{\left (\frac{d}{2} + \frac{e x}{2} \right )} \right )}}{2 c e} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1631, size = 31, normalized size = 1.24 \begin{align*} \frac{e^{\left (-1\right )} \log \left ({\left | c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a \right |}\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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