Optimal. Leaf size=43 \[ -\frac {4}{3} \text {Ei}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2343, 2346,
2209} \begin {gather*} -\frac {4}{3} \text {ExpIntegralEi}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2343
Rule 2346
Rubi steps
\begin {align*} \int \frac {1}{x^3 \log ^4(x)} \, dx &=-\frac {1}{3 x^2 \log ^3(x)}-\frac {2}{3} \int \frac {1}{x^3 \log ^3(x)} \, dx\\ &=-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}+\frac {2}{3} \int \frac {1}{x^3 \log ^2(x)} \, dx\\ &=-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)}-\frac {4}{3} \int \frac {1}{x^3 \log (x)} \, dx\\ &=-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)}-\frac {4}{3} \text {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {4}{3} \text {Ei}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} -\frac {4}{3} \text {Ei}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 37, normalized size = 0.86
method | result | size |
risch | \(-\frac {2 \ln \left (x \right )^{2}-\ln \left (x \right )+1}{3 x^{2} \ln \left (x \right )^{3}}+\frac {4 \expIntegral \left (1, 2 \ln \left (x \right )\right )}{3}\) | \(31\) |
default | \(-\frac {1}{3 x^{2} \ln \left (x \right )^{3}}+\frac {1}{3 x^{2} \ln \left (x \right )^{2}}-\frac {2}{3 x^{2} \ln \left (x \right )}+\frac {4 \expIntegral \left (1, 2 \ln \left (x \right )\right )}{3}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.49, size = 8, normalized size = 0.19 \begin {gather*} -8 \, \Gamma \left (-3, 2 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 34, normalized size = 0.79 \begin {gather*} -\frac {4 \, x^{2} \log \left (x\right )^{3} \operatorname {log\_integral}\left (\frac {1}{x^{2}}\right ) + 2 \, \log \left (x\right )^{2} - \log \left (x\right ) + 1}{3 \, x^{2} \log \left (x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 32, normalized size = 0.74 \begin {gather*} - \frac {4 \operatorname {Ei}{\left (- 2 \log {\left (x \right )} \right )}}{3} + \frac {- 2 \log {\left (x \right )}^{2} + \log {\left (x \right )} - 1}{3 x^{2} \log {\left (x \right )}^{3}} \end {gather*}
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 \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 = 0.28, size = 29, normalized size = 0.67 \begin {gather*} -\frac {4\,\mathrm {ei}\left (-2\,\ln \left (x\right )\right )}{3}-\frac {\frac {2\,{\ln \left (x\right )}^2}{3}-\frac {\ln \left (x\right )}{3}+\frac {1}{3}}{x^2\,{\ln \left (x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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