Optimal. Leaf size=25 \[ -\frac{\log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e} \]
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Rubi [A] time = 0.0210431, 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 = {3121, 31} \[ -\frac{\log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e} \]
Antiderivative was successfully verified.
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Rule 3121
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{\operatorname{Subst}\left (\int \frac{1}{2 a+2 c x} \, dx,x,\cot \left (\frac{1}{2} (d+e x)\right )\right )}{e}\\ &=-\frac{\log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e}\\ \end{align*}
Mathematica [A] time = 0.151511, size = 50, normalized size = 2. \[ \frac{\log \left (\sin \left (\frac{1}{2} (d+e x)\right )\right )-\log \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )}{2 c e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 42, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,ce}\ln \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }+{\frac{1}{2\,ce}\ln \left ( \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 [B] time = 0.986877, size = 73, normalized size = 2.92 \begin{align*} -\frac{\frac{\log \left (c + \frac{a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c} - \frac{\log \left (\frac{\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c}}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13042, size = 159, normalized size = 6.36 \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 = 1.46485, size = 95, normalized size = 3.8 \begin{align*} \begin{cases} \frac{\tilde{\infty } x}{\sin{\left (d \right )}} & \text{for}\: a = 0 \wedge c = 0 \wedge e = 0 \\- \frac{1}{2 a e \tan{\left (\frac{d}{2} + \frac{e x}{2} \right )}} & \text{for}\: c = 0 \\\frac{\log{\left (\tan{\left (\frac{d}{2} + \frac{e x}{2} \right )} \right )}}{2 c e} & \text{for}\: a = 0 \\\frac{x}{- 2 a \cos{\left (d \right )} + 2 a + 2 c \sin{\left (d \right )}} & \text{for}\: e = 0 \\- \frac{\log{\left (\tan{\left (\frac{d}{2} + \frac{e x}{2} \right )} + \frac{c}{a} \right )}}{2 c e} + \frac{\log{\left (\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 = 1.14731, size = 57, normalized size = 2.28 \begin{align*} -\frac{1}{2} \,{\left (\frac{\log \left ({\left | a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + c \right |}\right )}{c} - \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) \right |}\right )}{c}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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