Integrand size = 24, antiderivative size = 75 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=\frac {a \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{4 c^3 e}-\frac {c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))} \]
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Time = 0.05 (sec) , antiderivative size = 75, 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, 12, 3200, 31} \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=\frac {a \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{4 c^3 e}-\frac {a \sin (d+e x)+c \cos (d+e x)}{4 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))} \]
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Rule 12
Rule 31
Rule 3200
Rule 3208
Rubi steps \begin{align*} \text {integral}& = -\frac {c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}+\frac {\int -\frac {2 a}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{4 c^2} \\ & = -\frac {c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}-\frac {a \int \frac {1}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{2 c^2} \\ & = -\frac {c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))}+\frac {a \text {Subst}\left (\int \frac {1}{2 a+2 c x} \, dx,x,\cot \left (\frac {1}{2} (d+e x)\right )\right )}{2 c^2 e} \\ & = \frac {a \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{4 c^3 e}-\frac {c \cos (d+e x)+a \sin (d+e x)}{4 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(75)=150\).
Time = 5.41 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.05 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=-\frac {\sin \left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left (\cos (d+e x) \left (a^2+2 c^2-2 a^2 \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right )+2 a^2 \log \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )\right )+a \left (a \left (-1+2 \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right )-2 \log \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )\right )+c \left (1+2 \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right )-2 \log \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )\right ) \sin (d+e x)\right )\right )}{4 c^3 e (a-a \cos (d+e x)+c \sin (d+e x))^2} \]
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Time = 0.99 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}+c^{2}}{2 c^{2} a \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}+\frac {a \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{c^{3}}-\frac {1}{2 c^{2} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}-\frac {a \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{c^{3}}}{4 e}\) | \(88\) |
default | \(\frac {-\frac {a^{2}+c^{2}}{2 c^{2} a \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}+\frac {a \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{c^{3}}-\frac {1}{2 c^{2} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}-\frac {a \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{c^{3}}}{4 e}\) | \(88\) |
norman | \(\frac {-\frac {1}{8 c e}-\frac {\left (2 a^{2}+c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{8 a \,c^{2} e}}{\tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{4 c^{3} e}+\frac {a \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{4 c^{3} e}\) | \(109\) |
risch | \(-\frac {i \left (-i a \,{\mathrm e}^{i \left (e x +d \right )}+i a +c \right )}{2 c^{2} e \left (c \,{\mathrm e}^{2 i \left (e x +d \right )}-i a \,{\mathrm e}^{2 i \left (e x +d \right )}-c +2 i a \,{\mathrm e}^{i \left (e x +d \right )}-i a \right )}+\frac {a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i c -a}{i c +a}\right )}{4 c^{3} e}-\frac {a \ln \left ({\mathrm e}^{i \left (e x +d \right )}-1\right )}{4 c^{3} e}\) | \(135\) |
parallelrisch | \(\frac {-2 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right ) a^{2}+2 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right ) a^{2}-2 \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right ) a c +2 \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right ) a c -\cot \left (\frac {e x}{2}+\frac {d}{2}\right ) c^{2}+2 \tan \left (\frac {e x}{2}+\frac {d}{2}\right ) a^{2}+\tan \left (\frac {e x}{2}+\frac {d}{2}\right ) c^{2}}{8 \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right ) c^{3} e}\) | \(150\) |
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (72) = 144\).
Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.16 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=\frac {2 \, c^{2} \cos \left (e x + d\right ) + 2 \, a c \sin \left (e x + d\right ) + {\left (a^{2} \cos \left (e x + d\right ) - a c \sin \left (e x + d\right ) - a^{2}\right )} \log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} - \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) - {\left (a^{2} \cos \left (e x + d\right ) - a c \sin \left (e x + d\right ) - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right )}{8 \, {\left (a c^{3} e \cos \left (e x + d\right ) - c^{4} e \sin \left (e x + d\right ) - a c^{3} e\right )}} \]
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Timed out. \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=-\frac {\frac {a c + \frac {{\left (2 \, a^{2} + c^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{\frac {a c^{3} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {a^{2} c^{2} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}} - \frac {2 \, a \log \left (c + \frac {a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{3}} + \frac {2 \, a \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{3}}}{8 \, e} \]
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Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=\frac {\frac {2 \, a \log \left ({\left | a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + c \right |}\right )}{c^{3}} - \frac {2 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) \right |}\right )}{c^{3}} - \frac {2 \, a^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a c}{{\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )\right )} a c^{2}}}{8 \, e} \]
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Time = 27.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{c}+1\right )}{2\,c^3\,e}-\frac {\frac {1}{c}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a^2+c^2\right )}{a\,c^2}}{e\,\left (8\,a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+8\,c\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )} \]
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