Optimal. Leaf size=23 \[ -6+\frac {1}{5} (-2-x)+x+\frac {3 x^3 \log (4)}{\log (x)} \]
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Rubi [A]
time = 0.08, antiderivative size = 17, normalized size of antiderivative = 0.74, number of steps
used = 8, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {12, 6874, 2343,
2346, 2209} \begin {gather*} \frac {3 x^3 \log (4)}{\log (x)}+\frac {4 x}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-15 x^2 \log (4)+45 x^2 \log (4) \log (x)+4 \log ^2(x)}{\log ^2(x)} \, dx\\ &=\frac {1}{5} \int \left (4-\frac {15 x^2 \log (4)}{\log ^2(x)}+\frac {45 x^2 \log (4)}{\log (x)}\right ) \, dx\\ &=\frac {4 x}{5}-(3 \log (4)) \int \frac {x^2}{\log ^2(x)} \, dx+(9 \log (4)) \int \frac {x^2}{\log (x)} \, dx\\ &=\frac {4 x}{5}+\frac {3 x^3 \log (4)}{\log (x)}-(9 \log (4)) \int \frac {x^2}{\log (x)} \, dx+(9 \log (4)) \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4 x}{5}+9 \text {Ei}(3 \log (x)) \log (4)+\frac {3 x^3 \log (4)}{\log (x)}-(9 \log (4)) \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4 x}{5}+\frac {3 x^3 \log (4)}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 17, normalized size = 0.74 \begin {gather*} \frac {4 x}{5}+\frac {3 x^3 \log (4)}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.06, size = 37, normalized size = 1.61
method | result | size |
risch | \(\frac {4 x}{5}+\frac {6 x^{3} \ln \left (2\right )}{\ln \left (x \right )}\) | \(16\) |
norman | \(\frac {\frac {4 x \ln \left (x \right )}{5}+6 x^{3} \ln \left (2\right )}{\ln \left (x \right )}\) | \(19\) |
default | \(-18 \ln \left (2\right ) \expIntegral \left (1, -3 \ln \left (x \right )\right )-6 \ln \left (2\right ) \left (-\frac {x^{3}}{\ln \left (x \right )}-3 \expIntegral \left (1, -3 \ln \left (x \right )\right )\right )+\frac {4 x}{5}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.32, size = 23, normalized size = 1.00 \begin {gather*} 18 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) \log \left (2\right ) - 18 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) \log \left (2\right ) + \frac {4}{5} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (15 \, x^{3} \log \left (2\right ) + 2 \, x \log \left (x\right )\right )}}{5 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.65 \begin {gather*} \frac {6 x^{3} \log {\left (2 \right )}}{\log {\left (x \right )}} + \frac {4 x}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 0.65 \begin {gather*} \frac {6 \, x^{3} \log \left (2\right )}{\log \left (x\right )} + \frac {4}{5} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 15, normalized size = 0.65 \begin {gather*} \frac {4\,x}{5}+\frac {6\,x^3\,\ln \left (2\right )}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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