Integrand size = 24, antiderivative size = 168 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=-\frac {\log \left (1+\tan \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e}-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 (a \cos (d+e x)-a \sin (d+e x))}{96 e \left (a^5+a^5 \cos (d+e x)+a^5 \sin (d+e x)\right )} \]
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Time = 0.20 (sec) , antiderivative size = 168, 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} \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=-\frac {19 (a \cos (d+e x)-a \sin (d+e x))}{96 e \left (a^5 \sin (d+e x)+a^5 \cos (d+e x)+a^5\right )}-\frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{4 a^4 e}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right )^2}-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a \sin (d+e x)+a \cos (d+e x)+a)^3} \]
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Rule 31
Rule 3203
Rule 3208
Rule 3232
Rule 3235
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {\int \frac {-6 a+4 a \cos (d+e x)+4 a \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^3} \, dx}{12 a^2} \\ & = -\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}+\frac {\int \frac {56 a^2-20 a^2 \cos (d+e x)-20 a^2 \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^2} \, dx}{96 a^4} \\ & = -\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 \left (a^3 \cos (d+e x)-a^3 \sin (d+e x)\right )}{96 e \left (a^7+a^7 \cos (d+e x)+a^7 \sin (d+e x)\right )}-\frac {\int \frac {1}{2 a+2 a \cos (d+e x)+2 a \sin (d+e x)} \, dx}{2 a^3} \\ & = -\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 \left (a^3 \cos (d+e x)-a^3 \sin (d+e x)\right )}{96 e \left (a^7+a^7 \cos (d+e x)+a^7 \sin (d+e x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{4 a+4 a x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{a^3 e} \\ & = -\frac {\log \left (1+\tan \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e}-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 \left (a^3 \cos (d+e x)-a^3 \sin (d+e x)\right )}{96 e \left (a^7+a^7 \cos (d+e x)+a^7 \sin (d+e x)\right )} \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {\log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e}-\frac {\log \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e}-\frac {\sec ^2\left (\frac {1}{2} (d+e x)\right )}{64 a^4 e}+\frac {\sin \left (\frac {1}{2} (d+e x)\right )}{96 a^4 e \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^3}+\frac {5}{192 a^4 e \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^2}+\frac {19 \sin \left (\frac {1}{2} (d+e x)\right )}{96 a^4 e \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )}+\frac {19 \tan \left (\frac {1}{2} (d+e x)\right )}{192 a^4 e}+\frac {\sec ^2\left (\frac {1}{2} (d+e x)\right ) \tan \left (\frac {1}{2} (d+e x)\right )}{384 a^4 e} \]
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Time = 0.95 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3}-2 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+13 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-32 \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\frac {12}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {36}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )}-\frac {8}{3 \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}}{128 e \,a^{4}}\) | \(106\) |
default | \(\frac {\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3}-2 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+13 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-32 \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\frac {12}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {36}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )}-\frac {8}{3 \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}}{128 e \,a^{4}}\) | \(106\) |
parallelrisch | \(\frac {-96 \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}-3 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}+24 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}-297 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-441 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-180}{384 e \,a^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}\) | \(112\) |
norman | \(\frac {\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{16 a e}-\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{128 a e}+\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{384 a e}-\frac {15}{32 a e}-\frac {147 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{128 a e}-\frac {99 \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{128 a e}}{a^{3} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {\ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{4 a^{4} e}\) | \(141\) |
risch | \(\frac {\left (-\frac {1}{96}+\frac {i}{96}\right ) \left (60 i {\mathrm e}^{4 i \left (e x +d \right )}+24 \,{\mathrm e}^{5 i \left (e x +d \right )}+152 i {\mathrm e}^{3 i \left (e x +d \right )}+60 \,{\mathrm e}^{4 i \left (e x +d \right )}+111 i {\mathrm e}^{2 i \left (e x +d \right )}-111 \,{\mathrm e}^{2 i \left (e x +d \right )}-19-19 i-90 \,{\mathrm e}^{i \left (e x +d \right )}\right )}{a^{4} e \left (i {\mathrm e}^{i \left (e x +d \right )}+{\mathrm e}^{2 i \left (e x +d \right )}+i+{\mathrm e}^{i \left (e x +d \right )}\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+1\right )}{4 a^{4} e}-\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{4 a^{4} e}\) | \(172\) |
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Time = 0.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {38 \, \cos \left (e x + d\right )^{3} + 66 \, \cos \left (e x + d\right )^{2} + 24 \, {\left (\cos \left (e x + d\right )^{3} - {\left (\cos \left (e x + d\right )^{2} + 3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 3 \, \cos \left (e x + d\right ) - 2\right )} \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) - 24 \, {\left (\cos \left (e x + d\right )^{3} - {\left (\cos \left (e x + d\right )^{2} + 3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 3 \, \cos \left (e x + d\right ) - 2\right )} \log \left (\sin \left (e x + d\right ) + 1\right ) + {\left (38 \, \cos \left (e x + d\right )^{2} - 35\right )} \sin \left (e x + d\right ) - 3 \, \cos \left (e x + d\right ) - 33}{192 \, {\left (a^{4} e \cos \left (e x + d\right )^{3} - 3 \, a^{4} e \cos \left (e x + d\right ) - 2 \, a^{4} e - {\left (a^{4} e \cos \left (e x + d\right )^{2} + 3 \, a^{4} e \cos \left (e x + d\right ) + 2 \, a^{4} e\right )} \sin \left (e x + d\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (156) = 312\).
Time = 9.73 (sec) , antiderivative size = 792, normalized size of antiderivative = 4.71 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Too large to display} \]
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Time = 0.25 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=-\frac {\frac {4 \, {\left (\frac {45 \, \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {27 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + 20\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, a^{4} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}} - \frac {\frac {39 \, \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - \frac {6 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {\sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}}{a^{4}} + \frac {96 \, \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{a^{4}}}{384 \, e} \]
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Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=-\frac {\frac {96 \, \log \left ({\left | \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 1 \right |}\right )}{a^{4}} - \frac {4 \, {\left (44 \, \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 105 \, \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 87 \, \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 24\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 1\right )}^{3}} - \frac {a^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 6 \, a^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 39 \, a^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{a^{12}}}{384 \, e} \]
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Time = 27.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3}{384\,a^4\,e}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{64\,a^4\,e}-\frac {\ln \left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+1\right )}{4\,a^4\,e}+\frac {13\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{128\,a^4\,e}-\frac {9\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+15\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+\frac {20}{3}}{e\,\left (32\,a^4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3+96\,a^4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+96\,a^4\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+32\,a^4\right )} \]
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