Integrand size = 22, antiderivative size = 96 \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\frac {\arctan \left (\frac {\sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}}\right )}{10 \sqrt {10} e}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3195, 3194, 2728, 210} \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\frac {\arctan \left (\frac {\sin \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )}{\sqrt {2} \sqrt {\cos \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )-1}}\right )}{10 \sqrt {10} e}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}} \]
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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)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}-\frac {1}{20} \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)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}-\frac {1}{20} \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)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}+\frac {\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 )}{10 e} \\ & = \frac {\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 )}{10 \sqrt {10} e}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\frac {\left (\frac {1}{250}-\frac {i}{125}\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left ((-1+i) \sqrt {-20-15 i} \text {arctanh}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (3+\tan \left (\frac {1}{4} (d+e x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^2+(5+10 i) \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )\right )}{e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \]
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Time = 0.77 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {\left (-\sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right ) \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+\sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right )+2 \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}}{100 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\) | \(118\) |
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (81) = 162\).
Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.19 \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=-\frac {{\left (13 \, \sqrt {10} \cos \left (e x + d\right )^{2} - 9 \, {\left (\sqrt {10} \cos \left (e x + d\right ) - 2 \, \sqrt {10}\right )} \sin \left (e x + d\right ) - \sqrt {10} \cos \left (e x + d\right ) - 14 \, \sqrt {10}\right )} \arctan \left (-\frac {{\left (3 \, \sqrt {10} \cos \left (e x + d\right ) + \sqrt {10} \sin \left (e x + d\right ) + 3 \, \sqrt {10}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{10 \, {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}\right ) + 10 \, \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5} {\left (3 \, \cos \left (e x + d\right ) + \sin \left (e x + d\right ) + 3\right )}}{100 \, {\left (13 \, e \cos \left (e x + d\right )^{2} - e \cos \left (e x + d\right ) - 9 \, {\left (e \cos \left (e x + d\right ) - 2 \, e\right )} \sin \left (e x + d\right ) - 14 \, e\right )}} \]
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\[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\int \frac {1}{\left (3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx=\int \frac {1}{{\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5\right )}^{3/2}} \,d x \]
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