Optimal. Leaf size=22 \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x) \]
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Rubi [A] time = 0.09, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {264} \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x) \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {\sec (3 x) \tan (3 x)}{\sqrt {1+5 \cos ^2(3 x)}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+5 x^2}} \, dx,x,\cos (3 x)\right )\right )\\ &=\frac {1}{3} \sqrt {1+5 \cos ^2(3 x)} \sec (3 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 20, normalized size = 0.91 \[ \frac {\sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 1.55 \[ -\frac {2 \, \sqrt {5}}{3 \, {\left ({\left (\sqrt {5} \cos \left (3 \, x\right ) - \sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}\right )}^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 34, normalized size = 1.55 \[ \frac {\sec ^{2}\left (3 x \right )+5}{3 \sqrt {\frac {\sec ^{2}\left (3 x \right )+5}{\sec \left (3 x \right )^{2}}}\, \sec \left (3 x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 20, normalized size = 0.91 \[ \frac {\sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 18, normalized size = 0.82 \[ \frac {\sqrt {\frac {5\,\cos \left (6\,x\right )}{2}+\frac {7}{2}}}{3\,\cos \left (3\,x\right )} \]
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 \[ \int \frac {\tan {\left (3 x \right )} \sec {\left (3 x \right )}}{\sqrt {5 \cos ^{2}{\left (3 x \right )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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