Optimal. Leaf size=48 \[ \frac{295 \cos (x)}{243 \sqrt{9-4 \cos ^2(x)}}-\frac{55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}-\frac{1}{2} \sin ^{-1}\left (\frac{2 \cos (x)}{3}\right ) \]
[Out]
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Rubi [A] time = 0.123762, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{295 \cos (x)}{243 \sqrt{9-4 \cos ^2(x)}}-\frac{55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}-\frac{1}{2} \sin ^{-1}\left (\frac{2 \cos (x)}{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sin[5*x]/(5*Cos[x]^2 + 9*Sin[x]^2)^(5/2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(sin(5*x)/(5*cos(x)**2+9*sin(x)**2)**(5/2),x)
[Out]
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Mathematica [C] time = 0.412996, size = 63, normalized size = 1.31 \[ \frac{2550 \cos (x)-590 \cos (3 x)+243 i (7-2 \cos (2 x))^{3/2} \log \left (\sqrt{7-2 \cos (2 x)}+2 i \cos (x)\right )}{486 (7-2 \cos (2 x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sin[5*x]/(5*Cos[x]^2 + 9*Sin[x]^2)^(5/2),x]
[Out]
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Maple [A] time = 0.106, size = 53, normalized size = 1.1 \[{\frac{26\,\cos \left ( x \right ) }{27} \left ( 9-4\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{214\,\cos \left ( x \right ) }{243}{\frac{1}{\sqrt{9-4\, \left ( \cos \left ( x \right ) \right ) ^{2}}}}}-{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{3}}{3} \left ( 9-4\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}\arcsin \left ({\frac{2\,\cos \left ( x \right ) }{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(sin(5*x)/(5*cos(x)^2+9*sin(x)^2)^(5/2),x)
[Out]
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Maxima [A] time = 1.52975, size = 93, normalized size = 1.94 \[ -2 \,{\left (\frac{2 \, \cos \left (x\right )^{2}}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac{3}{2}}}\right )} \cos \left (x\right ) + \frac{52 \, \cos \left (x\right )}{243 \, \sqrt{-4 \, \cos \left (x\right )^{2} + 9}} + \frac{26 \, \cos \left (x\right )}{27 \,{\left (-4 \, \cos \left (x\right )^{2} + 9\right )}^{\frac{3}{2}}} - \frac{1}{2} \, \arcsin \left (\frac{2}{3} \, \cos \left (x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251594, size = 217, normalized size = 4.52 \[ -\frac{243 \,{\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (-\frac{2 \,{\left (16 \, \cos \left (x\right )^{3} - 27 \, \cos \left (x\right )\right )} \sin \left (x\right ) +{\left (16 \, \cos \left (x\right )^{3} - 17 \, \cos \left (x\right )\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9}}{32 \, \cos \left (x\right )^{4} -{\left (16 \, \cos \left (x\right )^{2} - 9\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9} \sin \left (x\right ) - 70 \, \cos \left (x\right )^{2} + 18}\right ) + 243 \,{\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )} \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) + 80 \,{\left (59 \, \cos \left (x\right )^{3} - 108 \, \cos \left (x\right )\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9}}{972 \,{\left (16 \, \cos \left (x\right )^{4} - 72 \, \cos \left (x\right )^{2} + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(5*x)/(5*cos(x)**2+9*sin(x)**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212165, size = 54, normalized size = 1.12 \[ -\frac{20 \,{\left (59 \, \cos \left (x\right )^{2} - 108\right )} \sqrt{-4 \, \cos \left (x\right )^{2} + 9} \cos \left (x\right )}{243 \,{\left (4 \, \cos \left (x\right )^{2} - 9\right )}^{2}} - \frac{1}{2} \, \arcsin \left (\frac{2}{3} \, \cos \left (x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(5*x)/(5*cos(x)^2 + 9*sin(x)^2)^(5/2),x, algorithm="giac")
[Out]