Optimal. Leaf size=39 \[ \frac {2 \sin ^{-1}\left (\sqrt {\frac {5}{2}} \sin (x)\right )}{5 \sqrt {5}}+\frac {\sin (x)}{10 \sqrt {2-5 \sin ^2(x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4356, 385, 216} \[ \frac {2 \sin ^{-1}\left (\sqrt {\frac {5}{2}} \sin (x)\right )}{5 \sqrt {5}}+\frac {\sin (x)}{10 \sqrt {2-5 \sin ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 385
Rule 4356
Rubi steps
\begin {align*} \int \frac {\cos (x) \cos (2 x)}{\left (2-5 \sin ^2(x)\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1-2 x^2}{\left (2-5 x^2\right )^{3/2}} \, dx,x,\sin (x)\right )\\ &=\frac {\sin (x)}{10 \sqrt {2-5 \sin ^2(x)}}+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-5 x^2}} \, dx,x,\sin (x)\right )\\ &=\frac {2 \sin ^{-1}\left (\sqrt {\frac {5}{2}} \sin (x)\right )}{5 \sqrt {5}}+\frac {\sin (x)}{10 \sqrt {2-5 \sin ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 39, normalized size = 1.00 \[ \frac {1}{50} \left (4 \sqrt {5} \sin ^{-1}\left (\sqrt {\frac {5}{2}} \sin (x)\right )+\frac {5 \sin (x)}{\sqrt {2-5 \sin ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (x) \cos (2 x)}{\left (2-5 \sin ^2(x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.81, size = 100, normalized size = 2.56 \[ -\frac {{\left (5 \, \sqrt {5} \cos \relax (x)^{2} - 3 \, \sqrt {5}\right )} \arctan \left (\frac {{\left (50 \, \sqrt {5} \cos \relax (x)^{4} - 80 \, \sqrt {5} \cos \relax (x)^{2} + 31 \, \sqrt {5}\right )} \sqrt {5 \, \cos \relax (x)^{2} - 3}}{10 \, {\left (25 \, \cos \relax (x)^{4} - 35 \, \cos \relax (x)^{2} + 12\right )} \sin \relax (x)}\right ) - 5 \, \sqrt {5 \, \cos \relax (x)^{2} - 3} \sin \relax (x)}{50 \, {\left (5 \, \cos \relax (x)^{2} - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 38, normalized size = 0.97 \[ \frac {2}{25} \, \sqrt {5} \arcsin \left (\frac {1}{2} \, \sqrt {10} \sin \relax (x)\right ) - \frac {\sqrt {-5 \, \sin \relax (x)^{2} + 2} \sin \relax (x)}{10 \, {\left (5 \, \sin \relax (x)^{2} - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 58, normalized size = 1.49
method | result | size |
default | \(\frac {20 \arcsin \left (\frac {\sin \relax (x ) \sqrt {10}}{2}\right ) \sqrt {5}\, \left (\cos ^{2}\relax (x )\right )+5 \sin \relax (x ) \sqrt {5 \left (\cos ^{2}\relax (x )\right )-3}-12 \arcsin \left (\frac {\sin \relax (x ) \sqrt {10}}{2}\right ) \sqrt {5}}{250 \left (\cos ^{2}\relax (x )\right )-150}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 716, normalized size = 18.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\cos \left (2\,x\right )\,\cos \relax (x)}{{\left (2-5\,{\sin \relax (x)}^2\right )}^{3/2}} \,d x \]
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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