Optimal. Leaf size=22 \[ -1+\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2} \]
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Rubi [A] time = 0.25, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 6, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {6688, 12, 2360, 2297, 2299, 2178} \begin {gather*} \frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2297
Rule 2299
Rule 2360
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (3+\sqrt [5]{e}\right )-\sqrt [5]{e} \log (-x)}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {2 \left (3+\sqrt [5]{e}\right )-\sqrt [5]{e} \log (-x)}{\left (6-\sqrt [5]{e} \log (-x)\right )^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {-6+2 \left (3+\sqrt [5]{e}\right )}{\left (6-\sqrt [5]{e} \log (-x)\right )^3}+\frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^2} \, dx+\frac {1}{2} \sqrt [5]{e} \int \frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^3} \, dx\\ &=\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}+\frac {x}{4 \sqrt [5]{e} \left (6-\sqrt [5]{e} \log (-x)\right )}-\frac {1}{4} \int \frac {1}{\left (6-\sqrt [5]{e} \log (-x)\right )^2} \, dx-\frac {\int \frac {1}{6-\sqrt [5]{e} \log (-x)} \, dx}{4 \sqrt [5]{e}}\\ &=\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}+\frac {\int \frac {1}{6-\sqrt [5]{e} \log (-x)} \, dx}{4 \sqrt [5]{e}}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{6-\sqrt [5]{e} x} \, dx,x,\log (-x)\right )}{4 \sqrt [5]{e}}\\ &=-\frac {1}{4} e^{-\frac {2}{5}+\frac {6}{\sqrt [5]{e}}} \text {Ei}\left (-\frac {6-\sqrt [5]{e} \log (-x)}{\sqrt [5]{e}}\right )+\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {e^x}{6-\sqrt [5]{e} x} \, dx,x,\log (-x)\right )}{4 \sqrt [5]{e}}\\ &=\frac {x}{4 \left (6-\sqrt [5]{e} \log (-x)\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 19, normalized size = 0.86 \begin {gather*} \frac {x}{4 \left (-6+\sqrt [5]{e} \log (-x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 24, normalized size = 1.09 \begin {gather*} \frac {x}{4 \, {\left (e^{\frac {2}{5}} \log \left (-x\right )^{2} - 12 \, e^{\frac {1}{5}} \log \left (-x\right ) + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 24, normalized size = 1.09 \begin {gather*} \frac {x}{4 \, {\left (e^{\frac {2}{5}} \log \left (-x\right )^{2} - 12 \, e^{\frac {1}{5}} \log \left (-x\right ) + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 15, normalized size = 0.68
method | result | size |
norman | \(\frac {x}{4 \left ({\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )-6\right )^{2}}\) | \(15\) |
risch | \(\frac {x}{4 \left ({\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )-6\right )^{2}}\) | \(15\) |
derivativedivides | \(\frac {x}{4 \,{\mathrm e}^{\frac {2}{5}} \ln \left (-x \right )^{2}-48 \,{\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )+144}\) | \(27\) |
default | \(\frac {x}{4 \,{\mathrm e}^{\frac {2}{5}} \ln \left (-x \right )^{2}-48 \,{\mathrm e}^{\frac {1}{5}} \ln \left (-x \right )+144}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.03, size = 24, normalized size = 1.09 \begin {gather*} \frac {x}{4\,\left ({\mathrm {e}}^{2/5}\,{\ln \left (-x\right )}^2-12\,{\mathrm {e}}^{1/5}\,\ln \left (-x\right )+36\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 27, normalized size = 1.23 \begin {gather*} \frac {x}{4 e^{\frac {2}{5}} \log {\left (- x \right )}^{2} - 48 e^{\frac {1}{5}} \log {\left (- x \right )} + 144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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