Optimal. Leaf size=57 \[ -\frac {4 \cos \left (\frac {3 x}{2}\right ) \log (3)}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )}+\frac {8 \sin \left (\frac {3 x}{2}\right )}{3 \sqrt [4]{3^{3 x}} \left (4+\log ^2(3)\right )} \]
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Rubi [A]
time = 0.02, 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 = {2319, 4518}
\begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2319
Rule 4518
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 = 0.05, size = 37, normalized size = 0.65 \begin {gather*} -\frac {4 \left (\cos \left (\frac {3 x}{2}\right ) \log (3)-2 \sin \left (\frac {3 x}{2}\right )\right )}{3 \sqrt [4]{27^x} \left (4+\log ^2(3)\right )} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.65, size = 30, normalized size = 0.53 \begin {gather*} \frac {4 \left (-\text {Cos}\left [\frac {3 x}{2}\right ] \text {Log}\left [3\right ]+2 \text {Sin}\left [\frac {3 x}{2}\right ]\right )}{3 \left (4+\text {Log}\left [3\right ]^2\right ) {\left (27^x\right )}^{\frac {1}{4}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.05, size = 37, normalized size = 0.65
method | result | size |
risch | \(-\frac {2 \left (2 \cos \left (\frac {3 x}{2}\right ) \ln \left (3\right )-4 \sin \left (\frac {3 x}{2}\right )\right )}{3 \left (2 i+\ln \left (3\right )\right ) \left (-2 i+\ln \left (3\right )\right ) \left (27^{x}\right )^{\frac {1}{4}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 31, normalized size = 0.54 \begin {gather*} -\frac {4 \, {\left (\cos \left (\frac {3}{2} \, x\right ) \log \left (3\right ) - 2 \, \sin \left (\frac {3}{2} \, x\right )\right )}}{3 \, {\left (\log \left (3\right )^{2} + 4\right )} 3^{\frac {3}{4} \, x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 70, normalized size = 1.23 \begin {gather*} \frac {8 \sin {\left (\frac {3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log {\left (3 \right )}^{2} + 12 \sqrt [4]{3^{3 x}}} - \frac {4 \log {\left (3 \right )} \cos {\left (\frac {3 x}{2} \right )}}{3 \sqrt [4]{3^{3 x}} \log {\left (3 \right )}^{2} + 12 \sqrt [4]{3^{3 x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
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 \begin {gather*} \frac {\frac {3\,\sin \left (\frac {3\,x}{2}\right )}{2}-\frac {3\,\cos \left (\frac {3\,x}{2}\right )\,\ln \left (3\right )}{4}}{3^{\frac {3\,x}{4}}\,\left (\frac {9\,{\ln \left (3\right )}^2}{16}+\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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