Integrand size = 22, antiderivative size = 142 \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=-\frac {3 \arctan \left (\frac {\sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}}\right )}{400 \sqrt {10} e}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}-\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3195, 3194, 2728, 210} \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=-\frac {3 \arctan \left (\frac {\sin \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )}{\sqrt {2} \sqrt {\cos \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )-1}}\right )}{400 \sqrt {10} e}-\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}} \]
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Rule 210
Rule 2728
Rule 3194
Rule 3195
Rubi steps \begin{align*} \text {integral}& = \frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}-\frac {3}{40} \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx \\ & = \frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}-\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}+\frac {3}{800} \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx \\ & = \frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}-\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}+\frac {3}{800} \int \frac {1}{\sqrt {-5+5 \cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}} \, dx \\ & = \frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}-\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-10-x^2} \, dx,x,-\frac {5 \sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {-5+5 \cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}}\right )}{400 e} \\ & = -\frac {3 \arctan \left (\frac {\sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}}\right )}{400 \sqrt {10} e}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}-\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=\frac {\left (\frac {1}{10000}+\frac {i}{20000}\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left ((6+6 i) \sqrt {-20-15 i} \text {arctanh}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (3+\tan \left (\frac {1}{4} (d+e x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^4+(10-5 i) \left (165 \cos \left (\frac {1}{2} (d+e x)\right )-27 \cos \left (\frac {3}{2} (d+e x)\right )+55 \sin \left (\frac {1}{2} (d+e x)\right )-39 \sin \left (\frac {3}{2} (d+e x)\right )\right )\right )}{e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \]
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Time = 0.88 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {\left (3 \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right ) \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )^{2}-6 \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right ) \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+3 \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right )-6 \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+14 \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}}{4000 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\) | \(190\) |
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (123) = 246\).
Time = 0.25 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=\frac {3 \, {\left (79 \, \sqrt {10} \cos \left (e x + d\right )^{3} - 123 \, \sqrt {10} \cos \left (e x + d\right )^{2} + 3 \, {\left (\sqrt {10} \cos \left (e x + d\right )^{2} + 38 \, \sqrt {10} \cos \left (e x + d\right ) - 44 \, \sqrt {10}\right )} \sin \left (e x + d\right ) - 78 \, \sqrt {10} \cos \left (e x + d\right ) + 124 \, \sqrt {10}\right )} \arctan \left (-\frac {{\left (3 \, \sqrt {10} \cos \left (e x + d\right ) + \sqrt {10} \sin \left (e x + d\right ) + 3 \, \sqrt {10}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{10 \, {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}\right ) + 10 \, {\left (27 \, \cos \left (e x + d\right )^{2} + {\left (39 \, \cos \left (e x + d\right ) - 8\right )} \sin \left (e x + d\right ) - 69 \, \cos \left (e x + d\right ) - 96\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{4000 \, {\left (79 \, e \cos \left (e x + d\right )^{3} - 123 \, e \cos \left (e x + d\right )^{2} - 78 \, e \cos \left (e x + d\right ) + 3 \, {\left (e \cos \left (e x + d\right )^{2} + 38 \, e \cos \left (e x + d\right ) - 44 \, e\right )} \sin \left (e x + d\right ) + 124 \, e\right )}} \]
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\[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=\int \frac {1}{\left (3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=\int { \frac {1}{{\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx=\int \frac {1}{{\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5\right )}^{5/2}} \,d x \]
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