Optimal. Leaf size=23 \[ -\left (\left (3 e^{-x^4}-x\right ) (4+x (25+\log (x)))\right ) \]
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Rubi [C] time = 0.63, antiderivative size = 188, normalized size of antiderivative = 8.17, number of steps used = 13, number of rules used = 7, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6742, 2208, 2209, 2218, 2554, 15, 6561} \begin {gather*} 3 x \, _2F_2\left (\frac {1}{4},\frac {1}{4};\frac {5}{4},\frac {5}{4};-x^4\right )-\frac {12}{25} x^5 \, _2F_2\left (\frac {5}{4},\frac {5}{4};\frac {9}{4},\frac {9}{4};-x^4\right )-12 e^{-x^4}+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}+\frac {3 x \Gamma \left (\frac {5}{4}\right ) \log (x)}{\sqrt [4]{x^4}}-\frac {3 x \Gamma \left (\frac {1}{4}\right ) \log (x)}{4 \sqrt [4]{x^4}}+\frac {3 x \log (x) \Gamma \left (\frac {1}{4},x^4\right )}{4 \sqrt [4]{x^4}}+25 x^2+x^2 \log (x)-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}-\frac {3 x^5 \log (x) \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 2208
Rule 2209
Rule 2218
Rule 2554
Rule 6561
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4-78 e^{-x^4}+51 x+48 e^{-x^4} x^3+300 e^{-x^4} x^4+e^{-x^4} \left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx\\ &=4 x+\frac {51 x^2}{2}+48 \int e^{-x^4} x^3 \, dx-78 \int e^{-x^4} \, dx+300 \int e^{-x^4} x^4 \, dx+\int e^{-x^4} \left (-3+2 e^{x^4} x+12 x^4\right ) \log (x) \, dx\\ &=-12 e^{-x^4}+4 x+\frac {51 x^2}{2}+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\int \left (x+\frac {3 \Gamma \left (\frac {1}{4},x^4\right )}{4 \sqrt [4]{x^4}}-\frac {3 \Gamma \left (\frac {5}{4},x^4\right )}{\sqrt [4]{x^4}}\right ) \, dx\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\frac {3}{4} \int \frac {\Gamma \left (\frac {1}{4},x^4\right )}{\sqrt [4]{x^4}} \, dx+3 \int \frac {\Gamma \left (\frac {5}{4},x^4\right )}{\sqrt [4]{x^4}} \, dx\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\frac {(3 x) \int \frac {\Gamma \left (\frac {1}{4},x^4\right )}{x} \, dx}{4 \sqrt [4]{x^4}}+\frac {(3 x) \int \frac {\Gamma \left (\frac {5}{4},x^4\right )}{x} \, dx}{\sqrt [4]{x^4}}\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\frac {(3 x) \operatorname {Subst}\left (\int \frac {\Gamma \left (\frac {1}{4},x\right )}{x} \, dx,x,x^4\right )}{16 \sqrt [4]{x^4}}+\frac {(3 x) \operatorname {Subst}\left (\int \frac {\Gamma \left (\frac {5}{4},x\right )}{x} \, dx,x,x^4\right )}{4 \sqrt [4]{x^4}}\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+3 x \, _2F_2\left (\frac {1}{4},\frac {1}{4};\frac {5}{4},\frac {5}{4};-x^4\right )-\frac {12}{25} x^5 \, _2F_2\left (\frac {5}{4},\frac {5}{4};\frac {9}{4},\frac {9}{4};-x^4\right )+x^2 \log (x)-\frac {3 x \Gamma \left (\frac {1}{4}\right ) \log (x)}{4 \sqrt [4]{x^4}}+\frac {3 x \Gamma \left (\frac {5}{4}\right ) \log (x)}{\sqrt [4]{x^4}}+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.29, size = 26, normalized size = 1.13 \begin {gather*} e^{-x^4} \left (-3+e^{x^4} x\right ) (4+25 x+x \log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 41, normalized size = 1.78 \begin {gather*} {\left ({\left (25 \, x^{2} + 4 \, x\right )} e^{\left (x^{4}\right )} + {\left (x^{2} e^{\left (x^{4}\right )} - 3 \, x\right )} \log \relax (x) - 75 \, x - 12\right )} e^{\left (-x^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 43, normalized size = 1.87 \begin {gather*} x^{2} \log \relax (x) - 3 \, x e^{\left (-x^{4}\right )} \log \relax (x) + 25 \, x^{2} - 75 \, x e^{\left (-x^{4}\right )} + 4 \, x - 12 \, e^{\left (-x^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 1.43
method | result | size |
default | \(4 x +\left (-12-75 x -3 x \ln \relax (x )\right ) {\mathrm e}^{-x^{4}}+25 x^{2}+x^{2} \ln \relax (x )\) | \(33\) |
norman | \(\left (-12+x^{2} {\mathrm e}^{x^{4}} \ln \relax (x )-75 x +4 x \,{\mathrm e}^{x^{4}}-3 x \ln \relax (x )+25 x^{2} {\mathrm e}^{x^{4}}\right ) {\mathrm e}^{-x^{4}}\) | \(44\) |
risch | \(x \left (x \,{\mathrm e}^{x^{4}}-3\right ) {\mathrm e}^{-x^{4}} \ln \relax (x )+\left (25 x^{2} {\mathrm e}^{x^{4}}+4 x \,{\mathrm e}^{x^{4}}-75 x -12\right ) {\mathrm e}^{-x^{4}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {75 \, x^{5} \Gamma \left (\frac {5}{4}, x^{4}\right )}{{\left (x^{4}\right )}^{\frac {5}{4}}} + x^{2} \log \relax (x) - 3 \, x e^{\left (-x^{4}\right )} \log \relax (x) + 25 \, x^{2} + \frac {39 \, x \Gamma \left (\frac {1}{4}, x^{4}\right )}{2 \, {\left (x^{4}\right )}^{\frac {1}{4}}} + 4 \, x - 12 \, e^{\left (-x^{4}\right )} + 3 \, \int e^{\left (-x^{4}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\mathrm {e}}^{-x^4}\,\left ({\mathrm {e}}^{x^4}\,\left (51\,x+4\right )+48\,x^3+300\,x^4+\ln \relax (x)\,\left (2\,x\,{\mathrm {e}}^{x^4}+12\,x^4-3\right )-78\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 32, normalized size = 1.39 \begin {gather*} x^{2} \log {\relax (x )} + 25 x^{2} + 4 x + \left (- 3 x \log {\relax (x )} - 75 x - 12\right ) e^{- x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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