Optimal. Leaf size=33 \[ \frac {e^x x^2}{\frac {2}{-2+\frac {-e^4+x}{x^2}}-\frac {3}{\log (4)}} \]
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Rubi [C] time = 9.37, antiderivative size = 1873, normalized size of antiderivative = 56.76, number of steps used = 33, number of rules used = 9, integrand size = 158, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6, 6688, 12, 6742, 2176, 2194, 2270, 2178, 2177}
result too large to display
Warning: Unable to verify antiderivative.
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Rule 6
Rule 12
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2270
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+x^4 \left (36+4 \log ^2(4)\right )} \, dx\\ &=\int \frac {e^x x \log (4) \left (-3 e^8 (2+x)-2 e^4 x \left (-6+9 x+x^2 (6+\log (4))\right )-x^2 \left (6+6 x^2 (2+\log (4))+4 x^3 (3+\log (4))-x (21+\log (16))\right )\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx\\ &=\log (4) \int \frac {e^x x \left (-3 e^8 (2+x)-2 e^4 x \left (-6+9 x+x^2 (6+\log (4))\right )-x^2 \left (6+6 x^2 (2+\log (4))+4 x^3 (3+\log (4))-x (21+\log (16))\right )\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx\\ &=\log (4) \int \left (-\frac {e^x x^2}{3+\log (4)}-\frac {3 e^x x (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \left (-2 e^4 \log (4) (3+\log (4))+\log (4) \log (16)+\log (262144)\right )}{4 (3+\log (4))^3}+\frac {3 e^x \left (4 e^8 \log (4) (3+\log (4))-3 \log (16)-2 x \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )+2 e^4 \left (8 \log ^2(4)-\log ^2(16)+\log (262144)\right )\right )}{8 (3+\log (4))^3 \left (3 e^4-3 x+2 x^2 (3+\log (4))\right )}+\frac {3 e^x \left (-x \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )+e^4 \left (3 \log (4096)-2 e^4 (3+\log (4)) \log (68719476736)\right )\right )}{8 (3+\log (4))^3 \left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2}\right ) \, dx\\ &=\frac {(3 \log (4)) \int \frac {e^x \left (4 e^8 \log (4) (3+\log (4))-3 \log (16)-2 x \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )+2 e^4 \left (8 \log ^2(4)-\log ^2(16)+\log (262144)\right )\right )}{3 e^4-3 x+2 x^2 (3+\log (4))} \, dx}{8 (3+\log (4))^3}+\frac {(3 \log (4)) \int \frac {e^x \left (-x \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )+e^4 \left (3 \log (4096)-2 e^4 (3+\log (4)) \log (68719476736)\right )\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx}{8 (3+\log (4))^3}-\frac {\log (4) \int e^x x^2 \, dx}{3+\log (4)}-\frac {(3 \log (4) (4+\log (4))) \int e^x x \, dx}{2 (3+\log (4))^2}+\frac {\left (\log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )\right ) \int e^x \, dx}{4 (3+\log (4))^3}\\ &=-\frac {e^x x^2 \log (4)}{3+\log (4)}-\frac {3 e^x x \log (4) (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )}{4 (3+\log (4))^3}+\frac {(3 \log (4)) \int \left (\frac {e^x \left (-2 \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )-\frac {2 i \left (72 e^8 \log (4)-12 \log ^2(4)+96 e^4 \log ^2(4)+48 e^8 \log ^2(4)+32 e^4 \log ^3(4)+8 e^8 \log ^3(4)-12 e^4 \log ^2(16)-4 e^4 \log (4) \log ^2(16)+12 e^4 \log (4096)+4 e^4 \log (4) \log (4096)-3 \log (262144)\right )}{\sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}\right )}{-3+4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}+\frac {e^x \left (-2 \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )+\frac {2 i \left (72 e^8 \log (4)-12 \log ^2(4)+96 e^4 \log ^2(4)+48 e^8 \log ^2(4)+32 e^4 \log ^3(4)+8 e^8 \log ^3(4)-12 e^4 \log ^2(16)-4 e^4 \log (4) \log ^2(16)+12 e^4 \log (4096)+4 e^4 \log (4) \log (4096)-3 \log (262144)\right )}{\sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}\right )}{-3+4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}\right ) \, dx}{8 (3+\log (4))^3}+\frac {(3 \log (4)) \int \left (\frac {e^x x \left (-16 e^8 \log (4) (3+\log (4))^2-9 \log (16)+12 e^4 (3+\log (4)) \log (256)\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2}+\frac {e^{4+x} \left (3 \log (4096)-2 e^4 (3+\log (4)) \log (68719476736)\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2}\right ) \, dx}{8 (3+\log (4))^3}+\frac {(2 \log (4)) \int e^x x \, dx}{3+\log (4)}+\frac {(3 \log (4) (4+\log (4))) \int e^x \, dx}{2 (3+\log (4))^2}\\ &=\frac {2 e^x x \log (4)}{3+\log (4)}-\frac {e^x x^2 \log (4)}{3+\log (4)}+\frac {3 e^x \log (4) (4+\log (4))}{2 (3+\log (4))^2}-\frac {3 e^x x \log (4) (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )}{4 (3+\log (4))^3}-\frac {(2 \log (4)) \int e^x \, dx}{3+\log (4)}-\frac {\left (3 \log (4) \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )\right ) \int \frac {e^x x}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx}{8 (3+\log (4))^3}+\frac {\left (9 \log (4) \left (1-2 e^4 (3+\log (4))\right ) \log (4096)\right ) \int \frac {e^{4+x}}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx}{8 (3+\log (4))^3}-\frac {\left (3 \log (4) \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)-\frac {i \left (8 e^8 \log (4) (3+\log (4))^2+4 e^4 (3+\log (4)) \left (8 \log ^2(4)-\log ^2(16)+\log (4096)\right )-3 \left (4 \log ^2(4)+\log (262144)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )\right ) \int \frac {e^x}{-3+4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}} \, dx}{4 (3+\log (4))^3}-\frac {\left (3 \log (4) \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)+\frac {i \left (8 e^8 \log (4) (3+\log (4))^2+4 e^4 (3+\log (4)) \left (8 \log ^2(4)-\log ^2(16)+\log (4096)\right )-3 \left (4 \log ^2(4)+\log (262144)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )\right ) \int \frac {e^x}{-3+4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}} \, dx}{4 (3+\log (4))^3}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F] time = 2.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.60, size = 45, normalized size = 1.36 \begin {gather*} -\frac {2 \, {\left (2 \, x^{4} - x^{3} + x^{2} e^{4}\right )} e^{x} \log \relax (2)}{4 \, x^{2} \log \relax (2) + 6 \, x^{2} - 3 \, x + 3 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.21, size = 288, normalized size = 8.73 \begin {gather*} -\frac {32 \, x^{4} e^{x} \log \relax (2)^{4} + 144 \, x^{4} e^{x} \log \relax (2)^{3} - 16 \, x^{3} e^{x} \log \relax (2)^{4} + 216 \, x^{4} e^{x} \log \relax (2)^{2} - 72 \, x^{3} e^{x} \log \relax (2)^{3} + 16 \, x^{2} e^{\left (x + 4\right )} \log \relax (2)^{4} + 108 \, x^{4} e^{x} \log \relax (2) - 108 \, x^{3} e^{x} \log \relax (2)^{2} + 72 \, x^{2} e^{\left (x + 4\right )} \log \relax (2)^{3} - 54 \, x^{3} e^{x} \log \relax (2) + 108 \, x^{2} e^{\left (x + 4\right )} \log \relax (2)^{2} + 24 \, x e^{\left (x + 4\right )} \log \relax (2)^{3} + 54 \, x^{2} e^{\left (x + 4\right )} \log \relax (2) + 36 \, x e^{\left (x + 4\right )} \log \relax (2)^{2} - 9 \, x e^{x} \log \relax (2)^{2} - 12 \, e^{\left (x + 8\right )} \log \relax (2)^{3} - 18 \, e^{\left (x + 8\right )} \log \relax (2)^{2} + 9 \, e^{\left (x + 4\right )} \log \relax (2)^{2}}{32 \, x^{2} \log \relax (2)^{4} + 192 \, x^{2} \log \relax (2)^{3} + 432 \, x^{2} \log \relax (2)^{2} - 24 \, x \log \relax (2)^{3} + 24 \, e^{4} \log \relax (2)^{3} + 432 \, x^{2} \log \relax (2) - 108 \, x \log \relax (2)^{2} + 108 \, e^{4} \log \relax (2)^{2} + 162 \, x^{2} - 162 \, x \log \relax (2) + 162 \, e^{4} \log \relax (2) - 81 \, x + 81 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 43, normalized size = 1.30
method | result | size |
gosper | \(-\frac {2 x^{2} \left (2 x^{2}+{\mathrm e}^{4}-x \right ) \ln \relax (2) {\mathrm e}^{x}}{4 x^{2} \ln \relax (2)+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\) | \(43\) |
risch | \(-\frac {2 x^{2} \left (2 x^{2}+{\mathrm e}^{4}-x \right ) \ln \relax (2) {\mathrm e}^{x}}{4 x^{2} \ln \relax (2)+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\) | \(43\) |
norman | \(\frac {2 x^{3} \ln \relax (2) {\mathrm e}^{x}-4 x^{4} \ln \relax (2) {\mathrm e}^{x}-2 x^{2} {\mathrm e}^{4} \ln \relax (2) {\mathrm e}^{x}}{4 x^{2} \ln \relax (2)+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\) | \(54\) |
default | \(\text {Expression too large to display}\) | \(45072\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 48, normalized size = 1.45 \begin {gather*} -\frac {2 \, {\left (2 \, x^{4} \log \relax (2) - x^{3} \log \relax (2) + x^{2} e^{4} \log \relax (2)\right )} e^{x}}{2 \, x^{2} {\left (2 \, \log \relax (2) + 3\right )} - 3 \, x + 3 \, e^{4}} \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 = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left ({\mathrm {e}}^8\,\left (3\,x^2+6\,x\right )+{\mathrm {e}}^4\,\left (12\,x^4+18\,x^3-12\,x^2\right )+6\,x^3-21\,x^4+12\,x^5+12\,x^6\right )+4\,{\ln \relax (2)}^2\,\left (2\,x^4\,{\mathrm {e}}^4-2\,x^4+6\,x^5+4\,x^6\right )\right )}{9\,{\mathrm {e}}^8+16\,x^4\,{\ln \relax (2)}^2-{\mathrm {e}}^4\,\left (18\,x-36\,x^2\right )+2\,\ln \relax (2)\,\left (24\,x^4-12\,x^3+12\,{\mathrm {e}}^4\,x^2\right )+9\,x^2-36\,x^3+36\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 53, normalized size = 1.61 \begin {gather*} \frac {\left (- 4 x^{4} \log {\relax (2 )} + 2 x^{3} \log {\relax (2 )} - 2 x^{2} e^{4} \log {\relax (2 )}\right ) e^{x}}{4 x^{2} \log {\relax (2 )} + 6 x^{2} - 3 x + 3 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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