Optimal. Leaf size=48 \[ \frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}-\frac {1}{2} \sin ^{-1}\left (\frac {2 \cos (x)}{3}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1157, 385, 216} \[ \frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}-\frac {1}{2} \sin ^{-1}\left (\frac {2 \cos (x)}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 385
Rule 1157
Rubi steps
\begin {align*} \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1-12 x^2+16 x^4}{\left (9-4 x^2\right )^{5/2}} \, dx,x,\cos (x)\right )\\ &=-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {52+108 x^2}{\left (9-4 x^2\right )^{3/2}} \, dx,x,\cos (x)\right )\\ &=-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x^2}} \, dx,x,\cos (x)\right )\\ &=-\frac {1}{2} \sin ^{-1}\left (\frac {2 \cos (x)}{3}\right )-\frac {55 \cos (x)}{27 \left (9-4 \cos ^2(x)\right )^{3/2}}+\frac {295 \cos (x)}{243 \sqrt {9-4 \cos ^2(x)}}\\ \end {align*}
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Mathematica [C] time = 0.29, 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.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (5 x)}{\left (5 \cos ^2(x)+9 \sin ^2(x)\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.52, size = 131, normalized size = 2.73 \[ \frac {243 \, {\left (16 \, \cos \relax (x)^{4} - 72 \, \cos \relax (x)^{2} + 81\right )} \arctan \left (-\frac {81 \, \cos \relax (x) \sin \relax (x) - 4 \, {\left (8 \, \cos \relax (x)^{3} - 9 \, \cos \relax (x)\right )} \sqrt {-4 \, \cos \relax (x)^{2} + 9}}{64 \, \cos \relax (x)^{4} - 225 \, \cos \relax (x)^{2} + 81}\right ) - 243 \, {\left (16 \, \cos \relax (x)^{4} - 72 \, \cos \relax (x)^{2} + 81\right )} \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) - 80 \, {\left (59 \, \cos \relax (x)^{3} - 108 \, \cos \relax (x)\right )} \sqrt {-4 \, \cos \relax (x)^{2} + 9}}{972 \, {\left (16 \, \cos \relax (x)^{4} - 72 \, \cos \relax (x)^{2} + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 40, normalized size = 0.83 \[ -\frac {20 \, {\left (59 \, \cos \relax (x)^{2} - 108\right )} \sqrt {-4 \, \cos \relax (x)^{2} + 9} \cos \relax (x)}{243 \, {\left (4 \, \cos \relax (x)^{2} - 9\right )}^{2}} - \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, \cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 53, normalized size = 1.10
method | result | size |
derivativedivides | \(-\frac {4 \left (\cos ^{3}\relax (x )\right )}{3 \left (9-4 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}+\frac {214 \cos \relax (x )}{243 \sqrt {9-4 \left (\cos ^{2}\relax (x )\right )}}-\frac {\arcsin \left (\frac {2 \cos \relax (x )}{3}\right )}{2}+\frac {26 \cos \relax (x )}{27 \left (9-4 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}\) | \(53\) |
default | \(-\frac {4 \left (\cos ^{3}\relax (x )\right )}{3 \left (9-4 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}+\frac {214 \cos \relax (x )}{243 \sqrt {9-4 \left (\cos ^{2}\relax (x )\right )}}-\frac {\arcsin \left (\frac {2 \cos \relax (x )}{3}\right )}{2}+\frac {26 \cos \relax (x )}{27 \left (9-4 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 69, normalized size = 1.44 \[ -2 \, {\left (\frac {2 \, \cos \relax (x)^{2}}{{\left (-4 \, \cos \relax (x)^{2} + 9\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-4 \, \cos \relax (x)^{2} + 9\right )}^{\frac {3}{2}}}\right )} \cos \relax (x) + \frac {52 \, \cos \relax (x)}{243 \, \sqrt {-4 \, \cos \relax (x)^{2} + 9}} + \frac {26 \, \cos \relax (x)}{27 \, {\left (-4 \, \cos \relax (x)^{2} + 9\right )}^{\frac {3}{2}}} - \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, \cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (5\,x\right )}{{\left (5\,{\cos \relax (x)}^2+9\,{\sin \relax (x)}^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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