Optimal. Leaf size=33 \[ x-e^x \left (\frac {e^5}{x^2}+3 x\right )+x (3-i \pi -\log (3))^2 \]
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Rubi [A] time = 0.18, antiderivative size = 45, normalized size of antiderivative = 1.36, number of steps used = 14, number of rules used = 7, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} -\frac {e^{x+5}}{x^2}-3 e^x x+x \left (10-\pi ^2+\log ^2(3)-2 i \pi (3-\log (3))-\log (729)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (e^5 (2-x)-3 x^3-3 x^4\right )+x^3 (i \pi +\log (3))^2+x^3 (10-6 (i \pi +\log (3)))}{x^3} \, dx\\ &=\int \frac {e^x \left (e^5 (2-x)-3 x^3-3 x^4\right )+x^3 \left (10-6 (i \pi +\log (3))+(i \pi +\log (3))^2\right )}{x^3} \, dx\\ &=\int \left (-\frac {e^x \left (-2 e^5+e^5 x+3 x^3+3 x^4\right )}{x^3}+10 \left (1+\frac {1}{10} \left (-\pi ^2+2 i \pi (-3+\log (3))+(-6+\log (3)) \log (3)\right )\right )\right ) \, dx\\ &=x \left (10-\pi ^2-2 i \pi (3-\log (3))+\log ^2(3)-\log (729)\right )-\int \frac {e^x \left (-2 e^5+e^5 x+3 x^3+3 x^4\right )}{x^3} \, dx\\ &=x \left (10-\pi ^2-2 i \pi (3-\log (3))+\log ^2(3)-\log (729)\right )-\int \left (3 e^x-\frac {2 e^{5+x}}{x^3}+\frac {e^{5+x}}{x^2}+3 e^x x\right ) \, dx\\ &=x \left (10-\pi ^2-2 i \pi (3-\log (3))+\log ^2(3)-\log (729)\right )+2 \int \frac {e^{5+x}}{x^3} \, dx-3 \int e^x \, dx-3 \int e^x x \, dx-\int \frac {e^{5+x}}{x^2} \, dx\\ &=-3 e^x-\frac {e^{5+x}}{x^2}+\frac {e^{5+x}}{x}-3 e^x x+x \left (10-\pi ^2-2 i \pi (3-\log (3))+\log ^2(3)-\log (729)\right )+3 \int e^x \, dx+\int \frac {e^{5+x}}{x^2} \, dx-\int \frac {e^{5+x}}{x} \, dx\\ &=-\frac {e^{5+x}}{x^2}-3 e^x x-e^5 \text {Ei}(x)+x \left (10-\pi ^2-2 i \pi (3-\log (3))+\log ^2(3)-\log (729)\right )+\int \frac {e^{5+x}}{x} \, dx\\ &=-\frac {e^{5+x}}{x^2}-3 e^x x+x \left (10-\pi ^2-2 i \pi (3-\log (3))+\log ^2(3)-\log (729)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 1.42 \begin {gather*} -\frac {e^{5+x}}{x^2}+10 x-3 e^x x-\pi ^2 x+2 i \pi x (-3+\log (3))-6 x \log (3)+x \log ^2(3) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 53, normalized size = 1.61 \begin {gather*} -\frac {2 \, {\left (-i \, \pi + 3\right )} x^{3} \log \relax (3) - x^{3} \log \relax (3)^{2} - {\left (-6 i \, \pi - \pi ^{2} + 10\right )} x^{3} + {\left (3 \, x^{3} + e^{5}\right )} e^{x}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 59, normalized size = 1.79 \begin {gather*} -\frac {\pi ^{2} x^{3} - 2 i \, \pi x^{3} \log \relax (3) - x^{3} \log \relax (3)^{2} + 6 i \, \pi x^{3} + 3 \, x^{3} e^{x} + 6 \, x^{3} \log \relax (3) - 10 \, x^{3} + e^{\left (x + 5\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 48, normalized size = 1.45
method | result | size |
norman | \(\frac {\left (2 i \ln \relax (3) \pi -\pi ^{2}-6 i \pi +\ln \relax (3)^{2}-6 \ln \relax (3)+10\right ) x^{3}-{\mathrm e}^{5} {\mathrm e}^{x}-3 \,{\mathrm e}^{x} x^{3}}{x^{2}}\) | \(48\) |
risch | \(2 i \ln \relax (3) \pi x -x \,\pi ^{2}-6 i x \pi +x \ln \relax (3)^{2}-6 x \ln \relax (3)+10 x -\frac {\left (3 x^{3}+{\mathrm e}^{5}\right ) {\mathrm e}^{x}}{x^{2}}\) | \(49\) |
default | \(10 x +x \ln \relax (3)^{2}-6 i x \pi +2 i \ln \relax (3) \pi x -x \,\pi ^{2}-3 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\expIntegralEi \left (1, -x \right )}{2}\right )-{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{x}}{x}-\expIntegralEi \left (1, -x \right )\right )-6 x \ln \relax (3)\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.37, size = 60, normalized size = 1.82 \begin {gather*} -\pi ^{2} x + 2 i \, \pi x \log \relax (3) + x \log \relax (3)^{2} - 6 i \, \pi x - 3 \, {\left (x - 1\right )} e^{x} - e^{5} \Gamma \left (-1, -x\right ) - 2 \, e^{5} \Gamma \left (-2, -x\right ) - 6 \, x \log \relax (3) + 10 \, x - 3 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 40, normalized size = 1.21 \begin {gather*} -x\,\left (\ln \left (729\right )+3\,{\mathrm {e}}^x+\Pi ^2-{\ln \relax (3)}^2-10+\Pi \,6{}\mathrm {i}-\Pi \,\ln \relax (3)\,2{}\mathrm {i}\right )-\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 46, normalized size = 1.39 \begin {gather*} x \left (- \pi ^{2} - 6 \log {\relax (3 )} + \log {\relax (3 )}^{2} + 10 - 6 i \pi + 2 i \pi \log {\relax (3 )}\right ) + \frac {\left (- 3 x^{3} - e^{5}\right ) e^{x}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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