Integrand size = 24, antiderivative size = 33 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=-\frac {\log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{2 b e} \]
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Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3202, 31} \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=-\frac {\log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{2 b e} \]
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Rule 31
Rule 3202
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{2 a+2 b x} \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} (d+e x)\right )\right )}{e} \\ & = -\frac {\log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{2 b e} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(33)=66\).
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.82 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=\frac {1}{2} \left (\frac {\log \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )}{b e}-\frac {\log \left (a \cos \left (\frac {1}{2} (d+e x)\right )+b \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )-b \sin \left (\frac {1}{2} (d+e x)\right )\right )}{b e}\right ) \]
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Time = 0.90 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30
method | result | size |
parallelrisch | \(\frac {\ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )-\ln \left (a +b +\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b e}\) | \(43\) |
norman | \(\frac {\ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b e}-\frac {\ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b e}\) | \(57\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{2 b e}-\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right )}{2 b e}\) | \(57\) |
derivativedivides | \(\frac {\frac {\ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{b}+\frac {\left (-a +b \right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{b \left (a -b \right )}}{2 e}\) | \(66\) |
default | \(\frac {\frac {\ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{b}+\frac {\left (-a +b \right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{b \left (a -b \right )}}{2 e}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=-\frac {\log \left (2 \, a b \cos \left (e x + d\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \sin \left (e x + d\right )\right ) - \log \left (\sin \left (e x + d\right ) + 1\right )}{4 \, b e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (24) = 48\).
Time = 1.00 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.24 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=\begin {cases} \frac {\tilde {\infty } x}{\cos {\left (d \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge e = 0 \\- \frac {1}{a e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + a e} & \text {for}\: b = 0 \\\frac {x}{2 a \sin {\left (d \right )} + 2 a + 2 b \cos {\left (d \right )}} & \text {for}\: e = 0 \\\frac {\log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{2 b e} & \text {for}\: a = b \\\frac {\log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{2 b e} - \frac {\log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{2 b e} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=-\frac {\frac {\log \left (-a - b - \frac {{\left (a - b\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{b} - \frac {\log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{b}}{2 \, e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.39 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=\frac {\log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 2 \, a - 2 \, {\left | b \right |} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 2 \, a + 2 \, {\left | b \right |} \right |}}\right )}{2 \, e {\left | b \right |}} \]
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Time = 27.37 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx=-\frac {\mathrm {atanh}\left (\frac {a+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2}}{b}\right )}{b\,e} \]
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