Integrand size = 24, antiderivative size = 134 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\frac {\left (3 a^2+c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^5 e}-\frac {c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{16 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))} \]
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Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3208, 3232, 3203, 31} \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\frac {3 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{16 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac {\left (3 a^2+c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^5 e}-\frac {c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^2} \]
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Rule 31
Rule 3203
Rule 3208
Rule 3232
Rubi steps \begin{align*} \text {integral}& = -\frac {c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {\int \frac {-4 a+2 a \cos (d+e x)+2 c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx}{8 c^2} \\ & = -\frac {c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{16 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (3 a^2+c^2\right ) \int \frac {1}{2 a+2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{8 c^4} \\ & = -\frac {c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{16 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (3 a^2+c^2\right ) \text {Subst}\left (\int \frac {1}{4 a+4 c x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{4 c^4 e} \\ & = \frac {\left (3 a^2+c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^5 e}-\frac {c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{16 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))} \\ \end{align*}
Time = 2.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=-\frac {4 \left (3 a^2+c^2\right ) \log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right )-4 \left (3 a^2+c^2\right ) \log \left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right )-c^2 \sec ^2\left (\frac {1}{2} (d+e x)\right )+\frac {c^2 \left (a^2+c^2\right )}{\left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right )^2}+\frac {6 c \left (a^2+c^2\right ) \sin \left (\frac {1}{2} (d+e x)\right )}{a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )}+6 a c \tan \left (\frac {1}{2} (d+e x)\right )}{64 c^5 e} \]
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Time = 1.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{2}+3 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{4 c^{4}}-\frac {a^{4}+2 a^{2} c^{2}+c^{4}}{8 c^{5} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}+\frac {a \left (a^{2}+c^{2}\right )}{c^{5} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}+\frac {\left (6 a^{2}+2 c^{2}\right ) \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{4 c^{5}}}{8 e}\) | \(131\) |
default | \(\frac {-\frac {-\frac {c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{2}+3 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{4 c^{4}}-\frac {a^{4}+2 a^{2} c^{2}+c^{4}}{8 c^{5} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}+\frac {a \left (a^{2}+c^{2}\right )}{c^{5} \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}+\frac {\left (6 a^{2}+2 c^{2}\right ) \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{4 c^{5}}}{8 e}\) | \(131\) |
parallelrisch | \(\frac {12 \left (a^{2}+\frac {c^{2}}{3}\right ) \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2} \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4} c^{4}-4 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} c^{3}+8 \left (3 a^{3} c +a \,c^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+18 a^{4}+6 a^{2} c^{2}-c^{4}}{64 c^{5} e \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}\) | \(138\) |
norman | \(\frac {\frac {18 a^{4}+6 a^{2} c^{2}-c^{4}}{64 c^{5} e}+\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{64 c e}-\frac {a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{16 c^{2} e}+\frac {\left (3 a^{2}+c^{2}\right ) a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{8 c^{4} e}}{\left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}+c^{2}\right ) \ln \left (a +c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{16 c^{5} e}\) | \(143\) |
risch | \(\frac {3 i a^{3} {\mathrm e}^{3 i \left (e x +d \right )}+i a \,c^{2} {\mathrm e}^{3 i \left (e x +d \right )}+9 i a^{3} {\mathrm e}^{2 i \left (e x +d \right )}+3 a^{2} c \,{\mathrm e}^{3 i \left (e x +d \right )}+3 i a \,c^{2} {\mathrm e}^{2 i \left (e x +d \right )}+c^{3} {\mathrm e}^{3 i \left (e x +d \right )}+9 i a^{3} {\mathrm e}^{i \left (e x +d \right )}-i a \,c^{2} {\mathrm e}^{i \left (e x +d \right )}+3 i a^{3}-9 a^{2} c \,{\mathrm e}^{i \left (e x +d \right )}-3 i a \,c^{2}+c^{3} {\mathrm e}^{i \left (e x +d \right )}-6 a^{2} c}{8 \left (c \,{\mathrm e}^{2 i \left (e x +d \right )}+i a \,{\mathrm e}^{2 i \left (e x +d \right )}-c +2 i a \,{\mathrm e}^{i \left (e x +d \right )}+i a \right )^{2} c^{4} e}+\frac {3 \ln \left ({\mathrm e}^{i \left (e x +d \right )}-\frac {i c +a}{i c -a}\right ) a^{2}}{16 c^{5} e}+\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}-\frac {i c +a}{i c -a}\right )}{16 c^{3} e}-\frac {3 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+1\right ) a^{2}}{16 c^{5} e}-\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+1\right )}{16 c^{3} e}\) | \(346\) |
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Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (127) = 254\).
Time = 0.26 (sec) , antiderivative size = 433, normalized size of antiderivative = 3.23 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\frac {12 \, a^{2} c^{2} \cos \left (e x + d\right )^{2} - 6 \, a^{2} c^{2} + 2 \, {\left (3 \, a^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right ) + {\left (3 \, a^{4} + 4 \, a^{2} c^{2} + c^{4} + {\left (3 \, a^{4} - 2 \, a^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{4} + a^{2} c^{2}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{3} c + a c^{3} + {\left (3 \, a^{3} c + a c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} + \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) - {\left (3 \, a^{4} + 4 \, a^{2} c^{2} + c^{4} + {\left (3 \, a^{4} - 2 \, a^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{4} + a^{2} c^{2}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{3} c + a c^{3} + {\left (3 \, a^{3} c + a c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) - 2 \, {\left (3 \, a^{3} c - a c^{3} + 3 \, {\left (a^{3} c - a c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{32 \, {\left (2 \, a^{2} c^{5} e \cos \left (e x + d\right ) + {\left (a^{2} c^{5} - c^{7}\right )} e \cos \left (e x + d\right )^{2} + {\left (a^{2} c^{5} + c^{7}\right )} e + 2 \, {\left (a c^{6} e \cos \left (e x + d\right ) + a c^{6} e\right )} \sin \left (e x + d\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\frac {\frac {7 \, a^{4} + 6 \, a^{2} c^{2} - c^{4} + \frac {8 \, {\left (a^{3} c + a c^{3}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{a^{2} c^{5} + \frac {2 \, a c^{6} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {c^{7} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}} - \frac {\frac {6 \, a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - \frac {c \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}}{c^{4}} + \frac {4 \, {\left (3 \, a^{2} + c^{2}\right )} \log \left (a + \frac {c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{5}}}{64 \, e} \]
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Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\frac {\frac {4 \, {\left (3 \, a^{2} + c^{2}\right )} \log \left ({\left | c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a \right |}\right )}{c^{5}} + \frac {c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - 6 \, a c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{c^{6}} - \frac {18 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 6 \, c^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 28 \, a^{3} c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 4 \, a c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 11 \, a^{4} + c^{4}}{{\left (c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a\right )}^{2} c^{5}}}{64 \, e} \]
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Time = 26.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{64\,c^3\,e}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (4\,a^3+4\,a\,c^2\right )+\frac {7\,a^4+6\,a^2\,c^2-c^4}{2\,c}}{e\,\left (32\,a^2\,c^4+64\,a\,c^5\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+32\,c^6\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\right )}-\frac {3\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{32\,c^4\,e}+\frac {\ln \left (a+c\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (3\,a^2+c^2\right )}{16\,c^5\,e} \]
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