Optimal. Leaf size=57 \[ \frac {8 \sin \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}-\frac {4 \log (3) \cos \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2281, 4433} \[ \frac {8 \sin \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}-\frac {4 \log (3) \cos \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]
Antiderivative was successfully verified.
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Rule 2281
Rule 4433
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx &=\frac {3^{3 x/4} \int 3^{-3 x/4} \cos \left (\frac {3 x}{2}\right ) \, dx}{\sqrt [4]{3^{3 x}}}\\ &=-\frac {4 \cos \left (\frac {3 x}{2}\right ) \log (3)}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}+\frac {8 \sin \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 37, normalized size = 0.65 \[ -\frac {4 \left (\log (3) \cos \left (\frac {3 x}{2}\right )-2 \sin \left (\frac {3 x}{2}\right )\right )}{3 \sqrt [4]{27^x} \left (4+\log ^2(3)\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {3 x}{2}\right )}{\sqrt [4]{3^{3 x}}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 39, normalized size = 0.68 \[ -\frac {4 \, {\left (\frac {\cos \left (\frac {3}{2} \, x\right ) \log \relax (3)}{\log \relax (3)^{2} + 4} - \frac {2 \, \sin \left (\frac {3}{2} \, x\right )}{\log \relax (3)^{2} + 4}\right )}}{3 \cdot 3^{\frac {3}{4} \, x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 37, normalized size = 0.65
method | result | size |
risch | \(-\frac {2 \left (2 \cos \left (\frac {3 x}{2}\right ) \ln \relax (3)-4 \sin \left (\frac {3 x}{2}\right )\right )}{3 \left (2 i+\ln \relax (3)\right ) \left (-2 i+\ln \relax (3)\right ) \left (27^{x}\right )^{\frac {1}{4}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 31, normalized size = 0.54 \[ -\frac {4 \, {\left (\cos \left (\frac {3}{2} \, x\right ) \log \relax (3) - 2 \, \sin \left (\frac {3}{2} \, x\right )\right )}}{3 \, {\left (\log \relax (3)^{2} + 4\right )} 3^{\frac {3}{4} \, x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 33, normalized size = 0.58 \[ \frac {\frac {3\,\sin \left (\frac {3\,x}{2}\right )}{2}-\frac {3\,\cos \left (\frac {3\,x}{2}\right )\,\ln \relax (3)}{4}}{3^{\frac {3\,x}{4}}\,\left (\frac {9\,{\ln \relax (3)}^2}{16}+\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.45, size = 70, normalized size = 1.23 \[ \frac {8 \sin {\left (\frac {3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log {\relax (3 )}^{2} + 12 \sqrt [4]{3^{3 x}}} - \frac {4 \log {\relax (3 )} \cos {\left (\frac {3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log {\relax (3 )}^{2} + 12 \sqrt [4]{3^{3 x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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