Optimal. Leaf size=43 \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x)-\frac {1}{3} \sqrt {5} \sinh ^{-1}\left (\sqrt {5} \cos (3 x)\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {277, 215} \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x)-\frac {1}{3} \sqrt {5} \sinh ^{-1}\left (\sqrt {5} \cos (3 x)\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 277
Rubi steps
\begin {align*} \int \sqrt {1+5 \cos ^2(3 x)} \sec (3 x) \tan (3 x) \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+5 x^2}}{x^2} \, dx,x,\cos (3 x)\right )\right )\\ &=\frac {1}{3} \sqrt {1+5 \cos ^2(3 x)} \sec (3 x)-\frac {5}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+5 x^2}} \, dx,x,\cos (3 x)\right )\\ &=-\frac {1}{3} \sqrt {5} \sinh ^{-1}\left (\sqrt {5} \cos (3 x)\right )+\frac {1}{3} \sqrt {1+5 \cos ^2(3 x)} \sec (3 x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 43, normalized size = 1.00 \[ \frac {1}{3} \sqrt {5 \cos ^2(3 x)+1} \sec (3 x)-\frac {1}{3} \sqrt {5} \sinh ^{-1}\left (\sqrt {5} \cos (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 122, normalized size = 2.84 \[ \frac {\sqrt {5} \cos \left (3 \, x\right ) \log \left (80000 \, \cos \left (3 \, x\right )^{8} + 32000 \, \cos \left (3 \, x\right )^{6} + 4000 \, \cos \left (3 \, x\right )^{4} + 160 \, \cos \left (3 \, x\right )^{2} - 8 \, {\left (2000 \, \sqrt {5} \cos \left (3 \, x\right )^{7} + 600 \, \sqrt {5} \cos \left (3 \, x\right )^{5} + 50 \, \sqrt {5} \cos \left (3 \, x\right )^{3} + \sqrt {5} \cos \left (3 \, x\right )\right )} \sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1} + 1\right ) + 8 \, \sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}}{24 \, \cos \left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1} \sec \left (3 \, x\right ) \tan \left (3 \, x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 65, normalized size = 1.51 \[ \frac {\sqrt {\frac {\sec ^{2}\left (3 x \right )+5}{\sec \left (3 x \right )^{2}}}\, \sec \left (3 x \right ) \left (\sqrt {\sec ^{2}\left (3 x \right )+5}-\sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {\sec ^{2}\left (3 x \right )+5}}\right )\right )}{3 \sqrt {\sec ^{2}\left (3 x \right )+5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 35, normalized size = 0.81 \[ -\frac {1}{3} \, \sqrt {5} \operatorname {arsinh}\left (\sqrt {5} \cos \left (3 \, x\right )\right ) + \frac {\sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.26, size = 36, normalized size = 0.84 \[ \frac {\sqrt {\frac {5\,\cos \left (6\,x\right )}{2}+\frac {7}{2}}}{3\,\cos \left (3\,x\right )}+\frac {\sqrt {5}\,\mathrm {asin}\left (\sqrt {5}\,\cos \left (3\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \]
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 \[ \int \sqrt {5 \cos ^{2}{\left (3 x \right )} + 1} \tan {\left (3 x \right )} \sec {\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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