Integrand size = 12, antiderivative size = 65 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2738, 213} \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
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Rule 213
Rule 2738
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {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*}
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d}-\frac {\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
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Time = 0.55 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.51
method | result | size |
parallelrisch | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{4 d}\) | \(33\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{4}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{4}}{d}\) | \(34\) |
default | \(\frac {-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{4}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{4}}{d}\) | \(34\) |
norman | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{4 d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{4 d}\) | \(36\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{4 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{4 d}\) | \(40\) |
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=-\frac {\log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right )}{8 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.65 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=\begin {cases} \frac {\log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{4 d} - \frac {\log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{4 d} & \text {for}\: d \neq 0 \\\frac {x}{- 5 \cos {\left (c \right )} - 3} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=-\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{4 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.52 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=-\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{4 \, d} \]
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Time = 13.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.26 \[ \int \frac {1}{-3-5 \cos (c+d x)} \, dx=-\frac {\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{2\,d} \]
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