Optimal. Leaf size=65 \[ \frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
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Rubi [A] time = 0.0177694, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2659, 207} \[ \frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 2659
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{-3-5 \cos (c+d x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{-8+2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0235016, size = 65, normalized size = 1. \[ \frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 36, normalized size = 0.6 \begin{align*} -{\frac{1}{4\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }+{\frac{1}{4\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37264, size = 65, normalized size = 1. \begin{align*} -\frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61464, size = 138, normalized size = 2.12 \begin{align*} -\frac{\log \left (\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - \log \left (\frac{3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.72806, size = 42, normalized size = 0.65 \begin{align*} \begin{cases} \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 2 \right )}}{4 d} - \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x}{- 5 \cos{\left (c \right )} - 3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20199, size = 46, normalized size = 0.71 \begin{align*} -\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \right |}\right ) - \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \right |}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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