Integrand size = 24, antiderivative size = 207 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))} \]
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Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3208, 3235, 3232, 3203, 31} \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]
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Rule 31
Rule 3203
Rule 3208
Rule 3232
Rule 3235
Rubi steps \begin{align*} \text {integral}& = -\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {\int \frac {-6 a+4 a \cos (d+e x)+4 c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx}{12 c^2} \\ & = -\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {\int \frac {8 \left (5 a^2+2 c^2\right )-20 a^2 \cos (d+e x)-20 a c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx}{96 c^4} \\ & = -\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}-\frac {\left (a \left (5 a^2+3 c^2\right )\right ) \int \frac {1}{2 a+2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{16 c^6} \\ & = -\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}-\frac {\left (a \left (5 a^2+3 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 a+4 c x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{8 c^6 e} \\ & = -\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))} \\ \end{align*}
Time = 4.44 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {\cos ^8\left (\frac {1}{2} (d+e x)\right ) \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right ) \left (3 c^4 \left (a^2+2 c^2\right ) \sec ^4\left (\frac {1}{2} (d+e x)\right )+c^6 \sec ^6\left (\frac {1}{2} (d+e x)\right )-2 \left (37 a^6+36 a^4 c^2+3 a^2 c^4+4 c^6+30 a^6 \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )+18 a^4 c^2 \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )+48 a c^5 \csc ^5(d+e x) \sin ^6\left (\frac {1}{2} (d+e x)\right )+3 a c \left (27 a^4+30 a^2 c^2+c^4+6 a^2 \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )\right ) \tan \left (\frac {1}{2} (d+e x)\right )+6 c^2 \left (6 a^4+11 a^2 c^2+2 c^4+3 a^2 \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (d+e x)\right )+6 a c^3 \left (-3 a^2+\left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )\right ) \tan ^3\left (\frac {1}{2} (d+e x)\right )-6 a^2 c^4 \tan ^4\left (\frac {1}{2} (d+e x)\right )\right )\right )}{24 c^7 e (a+a \cos (d+e x)+c \sin (d+e x))^4} \]
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Time = 1.41 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {-60 a \left (a^{2}+\frac {3 c^{2}}{5}\right ) \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6} c^{6}-3 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5} c^{5}+\left (15 a^{2} c^{4}+9 c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+\left (-180 a^{4} c^{2}-108 a^{2} c^{4}-9 c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\left (-270 a^{5} c -162 a^{3} c^{3}-9 a \,c^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-110 a^{6}-66 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{384 c^{7} e \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}\) | \(215\) |
derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} c^{2}}{3}-2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} c +10 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) a^{2}+3 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) c^{2}}{8 c^{6}}+\frac {3 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}{8 c^{7} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}-\frac {15 a^{4}+18 a^{2} c^{2}+3 c^{4}}{8 c^{7} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{24 c^{7} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}}{16 e}\) | \(221\) |
default | \(\frac {\frac {\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} c^{2}}{3}-2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} c +10 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) a^{2}+3 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) c^{2}}{8 c^{6}}+\frac {3 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}{8 c^{7} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}-\frac {15 a^{4}+18 a^{2} c^{2}+3 c^{4}}{8 c^{7} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{24 c^{7} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}}{16 e}\) | \(221\) |
norman | \(\frac {-\frac {110 a^{6}+66 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{384 c^{7} e}+\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{384 c e}-\frac {a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{128 c^{2} e}+\frac {\left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{128 c^{3} e}-\frac {3 \left (20 a^{4}+12 a^{2} c^{2}+c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{128 c^{5} e}-\frac {3 a \left (30 a^{4}+18 a^{2} c^{2}+c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{128 c^{6} e}}{\left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 c^{7} e}\) | \(226\) |
risch | \(\frac {i \left (-4 c^{5}+45 a^{4} c -15 i a^{5}-3 a^{2} c^{3}-130 i a^{3} c^{2} {\mathrm e}^{3 i \left (e x +d \right )}-24 i a \,c^{4} {\mathrm e}^{3 i \left (e x +d \right )}+12 c^{5} {\mathrm e}^{2 i \left (e x +d \right )}-75 i a^{5} {\mathrm e}^{4 i \left (e x +d \right )}-150 i a^{5} {\mathrm e}^{3 i \left (e x +d \right )}+150 a^{4} c \,{\mathrm e}^{2 i \left (e x +d \right )}+41 i a^{3} c^{2}+12 i a \,c^{4}-75 i a^{5} {\mathrm e}^{i \left (e x +d \right )}-15 i a^{5} {\mathrm e}^{5 i \left (e x +d \right )}-18 a^{2} c^{3} {\mathrm e}^{5 i \left (e x +d \right )}+60 a^{2} c^{3} {\mathrm e}^{2 i \left (e x +d \right )}-45 a^{2} c^{3} {\mathrm e}^{4 i \left (e x +d \right )}-30 a^{4} c \,{\mathrm e}^{5 i \left (e x +d \right )}+30 a^{2} c^{3} {\mathrm e}^{i \left (e x +d \right )}+150 a^{4} c \,{\mathrm e}^{i \left (e x +d \right )}-150 i a^{5} {\mathrm e}^{2 i \left (e x +d \right )}-75 a^{4} c \,{\mathrm e}^{4 i \left (e x +d \right )}+60 i a^{3} c^{2} {\mathrm e}^{i \left (e x +d \right )}+15 i a \,c^{4} {\mathrm e}^{i \left (e x +d \right )}-60 i a^{3} c^{2} {\mathrm e}^{2 i \left (e x +d \right )}-45 i a^{3} c^{2} {\mathrm e}^{4 i \left (e x +d \right )}+9 i a \,c^{4} {\mathrm e}^{5 i \left (e x +d \right )}-12 i a \,c^{4} {\mathrm e}^{2 i \left (e x +d \right )}+6 i a^{3} c^{2} {\mathrm e}^{5 i \left (e x +d \right )}\right )}{48 \left (c \,{\mathrm e}^{2 i \left (e x +d \right )}+i a \,{\mathrm e}^{2 i \left (e x +d \right )}-c +2 i a \,{\mathrm e}^{i \left (e x +d \right )}+i a \right )^{3} c^{6} e}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+1\right )}{32 c^{7} e}+\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+1\right )}{32 c^{5} e}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}-\frac {i c +a}{i c -a}\right )}{32 c^{7} e}-\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}-\frac {i c +a}{i c -a}\right )}{32 c^{5} e}\) | \(593\) |
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Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (198) = 396\).
Time = 0.30 (sec) , antiderivative size = 791, normalized size of antiderivative = 3.82 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {60 \, a^{4} c^{2} + 6 \, a^{2} c^{4} - 2 \, {\left (45 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - 4 \, c^{6}\right )} \cos \left (e x + d\right )^{3} - 12 \, {\left (10 \, a^{4} c^{2} + a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{4} c^{2} - 2 \, a^{2} c^{4} - 2 \, c^{6}\right )} \cos \left (e x + d\right ) - 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} + {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \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 ) + 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} + {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) + 2 \, {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 6 \, a c^{5} + {\left (15 \, a^{5} c - 41 \, a^{3} c^{3} - 12 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (10 \, a^{5} c - 9 \, a^{3} c^{3} - a c^{5}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{192 \, {\left ({\left (a^{3} c^{7} - 3 \, a c^{9}\right )} e \cos \left (e x + d\right )^{3} + 3 \, {\left (a^{3} c^{7} - a c^{9}\right )} e \cos \left (e x + d\right )^{2} + 3 \, {\left (a^{3} c^{7} + a c^{9}\right )} e \cos \left (e x + d\right ) + {\left (a^{3} c^{7} + 3 \, a c^{9}\right )} e + {\left (6 \, a^{2} c^{8} e \cos \left (e x + d\right ) + {\left (3 \, a^{2} c^{8} - c^{10}\right )} e \cos \left (e x + d\right )^{2} + {\left (3 \, a^{2} c^{8} + c^{10}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {\frac {37 \, a^{6} + 39 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6} + \frac {9 \, {\left (9 \, a^{5} c + 10 \, a^{3} c^{3} + a c^{5}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {9 \, {\left (5 \, a^{4} c^{2} + 6 \, a^{2} c^{4} + c^{6}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}}{a^{3} c^{7} + \frac {3 \, a^{2} c^{8} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, a c^{9} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {c^{10} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}} + \frac {\frac {6 \, a c \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac {c^{2} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} - \frac {3 \, {\left (10 \, a^{2} + 3 \, c^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{c^{6}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (a + \frac {c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \]
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Time = 0.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {\frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a \right |}\right )}{c^{7}} - \frac {110 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 66 \, a c^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 285 \, a^{4} c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 144 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - 9 \, c^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 249 \, a^{5} c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 108 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 9 \, a c^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 73 \, a^{6} + 27 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - c^{6}}{{\left (c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a\right )}^{3} c^{7}} - \frac {c^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 6 \, a c^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 30 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 9 \, c^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{c^{12}}}{384 \, e} \]
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Time = 26.99 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3}{384\,c^4\,e}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {3}{128\,c^4}+\frac {5\,a^2}{64\,c^6}\right )}{e}-\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (27\,a^5+30\,a^3\,c^2+3\,a\,c^4\right )+\frac {37\,a^6+39\,a^4\,c^2+3\,a^2\,c^4+c^6}{3\,c}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (15\,a^4\,c+18\,a^2\,c^3+3\,c^5\right )}{e\,\left (128\,a^3\,c^6+384\,a^2\,c^7\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+384\,a\,c^8\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+128\,c^9\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\right )}-\frac {a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{64\,c^5\,e}-\frac {\ln \left (a+c\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (5\,a^3+3\,a\,c^2\right )}{32\,c^7\,e} \]
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