Integrand size = 31, antiderivative size = 19 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=\left (2-2 e^{x^4} x^2\right ) (-x+\log (4)) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.37, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2258, 2250, 2242, 2235, 2243} \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-2 e^{x^4} x^2 \log (4)-2 x \]
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Rule 2235
Rule 2242
Rule 2243
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = -2 x+\int e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right ) \, dx \\ & = -2 x+\int \left (6 e^{x^4} x^2+8 e^{x^4} x^6-4 e^{x^4} x \log (4)-8 e^{x^4} x^5 \log (4)\right ) \, dx \\ & = -2 x+6 \int e^{x^4} x^2 \, dx+8 \int e^{x^4} x^6 \, dx-(4 \log (4)) \int e^{x^4} x \, dx-(8 \log (4)) \int e^{x^4} x^5 \, dx \\ & = -2 x-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-2 e^{x^4} x^2 \log (4)-(2 \log (4)) \text {Subst}\left (\int e^{x^2} \, dx,x,x^2\right )+(4 \log (4)) \int e^{x^4} x \, dx \\ & = -2 x-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-2 e^{x^4} x^2 \log (4)-\sqrt {\pi } \text {erfi}\left (x^2\right ) \log (4)+(2 \log (4)) \text {Subst}\left (\int e^{x^2} \, dx,x,x^2\right ) \\ & = -2 x-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-2 e^{x^4} x^2 \log (4) \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.37 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=-2 x+\frac {3 \sqrt [4]{-x^4} \Gamma \left (\frac {3}{4},-x^4\right )}{2 x}-\frac {2 \sqrt [4]{-x^4} \Gamma \left (\frac {7}{4},-x^4\right )}{x}-2 e^{x^4} x^2 \log (4) \]
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Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\left (-4 x^{2} \ln \left (2\right )+2 x^{3}\right ) {\mathrm e}^{x^{4}}-2 x\) | \(23\) |
default | \(-2 x +2 x^{3} {\mathrm e}^{x^{4}}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{x^{4}}\) | \(25\) |
norman | \(-2 x +2 x^{3} {\mathrm e}^{x^{4}}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{x^{4}}\) | \(25\) |
parallelrisch | \(-2 x +2 x^{3} {\mathrm e}^{x^{4}}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{x^{4}}\) | \(25\) |
parts | \(-2 x +2 x^{3} {\mathrm e}^{x^{4}}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{x^{4}}\) | \(25\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=2 \, {\left (x^{3} - 2 \, x^{2} \log \left (2\right )\right )} e^{\left (x^{4}\right )} - 2 \, x \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=- 2 x + \left (2 x^{3} - 4 x^{2} \log {\left (2 \right )}\right ) e^{x^{4}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.21 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=-\frac {2 \, x^{7} \Gamma \left (\frac {7}{4}, -x^{4}\right )}{\left (-x^{4}\right )^{\frac {7}{4}}} - \frac {3 \, x^{3} \Gamma \left (\frac {3}{4}, -x^{4}\right )}{2 \, \left (-x^{4}\right )^{\frac {3}{4}}} + 2 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x^{2}\right ) \log \left (2\right ) - 2 \, {\left (2 \, x^{2} e^{\left (x^{4}\right )} + i \, \sqrt {\pi } \operatorname {erf}\left (i \, x^{2}\right )\right )} \log \left (2\right ) - 2 \, x \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=2 \, x^{3} e^{\left (x^{4}\right )} - 4 \, x^{2} e^{\left (x^{4}\right )} \log \left (2\right ) - 2 \, x \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \left (-2+e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right )\right ) \, dx=2\,x^3\,{\mathrm {e}}^{x^4}-2\,x-4\,x^2\,{\mathrm {e}}^{x^4}\,\ln \left (2\right ) \]
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