Optimal. Leaf size=142 \[ -\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}-\frac {3 \tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )-1}}\right )}{400 \sqrt {10} e} \]
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Rubi [A] time = 0.08, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3116, 3115, 2649, 204} \[ -\frac {3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{20 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}-\frac {3 \tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )-1}}\right )}{400 \sqrt {10} e} \]
Antiderivative was successfully verified.
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Rule 204
Rule 2649
Rule 3115
Rule 3116
Rubi steps
\begin {align*} \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, 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}{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-\tan ^{-1}\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 \operatorname {Subst}\left (\int \frac {1}{-10-x^2} \, dx,x,-\frac {5 \sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {-5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{400 e}\\ &=-\frac {3 \tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\tan ^{-1}\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*}
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Mathematica [C] time = 0.38, size = 178, normalized size = 1.25 \[ \frac {\left (\frac {1}{10000}+\frac {i}{20000}\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left ((10-5 i) \left (55 \sin \left (\frac {1}{2} (d+e x)\right )-39 \sin \left (\frac {3}{2} (d+e x)\right )+165 \cos \left (\frac {1}{2} (d+e x)\right )-27 \cos \left (\frac {3}{2} (d+e x)\right )\right )+(6+6 i) \sqrt {-20-15 i} \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^4 \tanh ^{-1}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (\tan \left (\frac {1}{4} (d+e x)\right )+3\right )\right )\right )}{e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 280, normalized size = 1.97 \[ \frac {3 \, {\left (79 \, \sqrt {10} \cos \left (e x + d\right )^{3} - 123 \, \sqrt {10} \cos \left (e x + d\right )^{2} + 3 \, {\left (\sqrt {10} \cos \left (e x + d\right )^{2} + 38 \, \sqrt {10} \cos \left (e x + d\right ) - 44 \, \sqrt {10}\right )} \sin \left (e x + d\right ) - 78 \, \sqrt {10} \cos \left (e x + d\right ) + 124 \, \sqrt {10}\right )} \arctan \left (-\frac {{\left (3 \, \sqrt {10} \cos \left (e x + d\right ) + \sqrt {10} \sin \left (e x + d\right ) + 3 \, \sqrt {10}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{10 \, {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}\right ) + 10 \, {\left (27 \, \cos \left (e x + d\right )^{2} + {\left (39 \, \cos \left (e x + d\right ) - 8\right )} \sin \left (e x + d\right ) - 69 \, \cos \left (e x + d\right ) - 96\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{4000 \, {\left (79 \, e \cos \left (e x + d\right )^{3} - 123 \, e \cos \left (e x + d\right )^{2} - 78 \, e \cos \left (e x + d\right ) + 3 \, {\left (e \cos \left (e x + d\right )^{2} + 38 \, e \cos \left (e x + d\right ) - 44 \, e\right )} \sin \left (e x + d\right ) + 124 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.80, size = 381, normalized size = 2.68 \[ -\frac {1}{162000} \, {\left (\frac {243 \, \sqrt {10} \arctan \left (\frac {1}{10} \, \sqrt {10} {\left (3 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 3 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + i\right )}\right )}{\mathrm {sgn}\left (-3 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )} + \frac {10 \, {\left (15039 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{7} + 6291 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{6} - 579 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{5} + 1645 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{4} + 25365 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{3} - 11367 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} + 4887 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 4887 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3807 i\right )}}{{\left (3 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} + 2 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 2 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 3 i\right )}^{4} \mathrm {sgn}\left (-3 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 190, normalized size = 1.34 \[ \frac {\left (3 \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right ) \left (\sin ^{2}\left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right )-6 \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right ) \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+3 \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right )-6 \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+14 \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}}{4000 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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