Optimal. Leaf size=37 \[ \frac {5-x}{-2+x-\frac {\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{x}} \]
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Rubi [F] time = 7.69, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x \left (e^x (-5+x) \left (5+x^3\right )^2+3 \left (-25+30 x+10 x^3+8 x^4+x^7\right )\right )+(-5+2 x) \left (5+x^3\right )^2 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left ((-2+x) x-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=\int \left (-\frac {e^x (-5+x) x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8-125 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+50 x \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-50 x^3 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+20 x^4 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-5 x^6 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+2 x^7 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx\\ &=-\int \frac {e^x (-5+x) x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8-125 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+50 x \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-50 x^3 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+20 x^4 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-5 x^6 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+2 x^7 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=-\int \left (-\frac {5 e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+\int \frac {-3 x \left (-25+30 x+10 x^3+8 x^4+x^7\right )+(-5+2 x) \left (5+x^3\right )^2 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left ((-2+x) x-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \left (\frac {x \left (325-315 x+50 x^2+70 x^3-114 x^4+20 x^5+10 x^6-12 x^7+2 x^8\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {5-2 x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}\right ) \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \frac {x \left (325-315 x+50 x^2+70 x^3-114 x^4+20 x^5+10 x^6-12 x^7+2 x^8\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \frac {5-2 x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx\\ &=5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \left (\frac {10 x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}-\frac {12 x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {2 x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}-\frac {45 (-5+x) x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {6 (-5+x) x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+\int \left (\frac {5}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}-\frac {2 x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}\right ) \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \frac {(-5+x) x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \frac {(-5+x) x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \left (-\frac {5 x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {x^2}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \left (-\frac {5 x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \frac {x^2}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-30 \int \frac {x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+225 \int \frac {x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \left (\frac {1}{3 \left (-\sqrt [3]{-5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {1}{3 \left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {1}{3 \left ((-1)^{2/3} \sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-30 \int \left (-\frac {1}{3 \sqrt [3]{5} \left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}-\frac {(-1)^{2/3}}{3 \sqrt [3]{5} \left (\sqrt [3]{5}-\sqrt [3]{-1} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {\sqrt [3]{-\frac {1}{5}}}{3 \left (\sqrt [3]{5}+(-1)^{2/3} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx-45 \int \frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+225 \int \frac {x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ &=2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+2 \int \frac {1}{\left (-\sqrt [3]{-5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+2 \int \frac {1}{\left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+2 \int \frac {1}{\left ((-1)^{2/3} \sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+225 \int \frac {x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\left (2 (-5)^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{5}-\sqrt [3]{-1} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\left (2\ 5^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\left (2 \sqrt [3]{-1} 5^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{5}+(-1)^{2/3} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.20, size = 37, normalized size = 1.00 \begin {gather*} -\frac {(-5+x) x}{(-2+x) x-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 55, normalized size = 1.49 \begin {gather*} -\frac {x^{5} - 5 \, x^{4} + 5 \, x^{2} - 25 \, x}{x^{5} - 2 \, x^{4} + 5 \, x^{2} - {\left (x^{3} + 5\right )} e^{x} + {\left (x^{3} + 5\right )} \log \relax (3) - 13 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 59, normalized size = 1.59 \begin {gather*} -\frac {x^{5} - 5 \, x^{4} + 5 \, x^{2} - 25 \, x}{x^{5} - 2 \, x^{4} - x^{3} e^{x} + x^{3} \log \relax (3) + 5 \, x^{2} - 13 \, x - 5 \, e^{x} + 5 \, \log \relax (3)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.14, size = 48, normalized size = 1.30
method | result | size |
risch | \(-\frac {2 \left (x -5\right ) x}{2 x^{2}+2 \ln \relax (3)-4 x -2 \ln \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x^{3}+5 \,{\mathrm e}^{x}+3 x}{x^{3}+5}}\right )}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 57, normalized size = 1.54 \begin {gather*} -\frac {x^{5} - 5 \, x^{4} + 5 \, x^{2} - 25 \, x}{x^{5} - 2 \, x^{4} + x^{3} \log \relax (3) + 5 \, x^{2} - {\left (x^{3} + 5\right )} e^{x} - 13 \, x + 5 \, \log \relax (3)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 63, normalized size = 1.70 \begin {gather*} -\frac {-x^5+5\,x^4-5\,x^2+25\,x}{13\,x-5\,\ln \relax (3)+5\,{\mathrm {e}}^x+x^3\,{\mathrm {e}}^x-x^3\,\ln \relax (3)-5\,x^2+2\,x^4-x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.16, size = 53, normalized size = 1.43 \begin {gather*} \frac {x^{5} - 5 x^{4} + 5 x^{2} - 25 x}{- x^{5} + 2 x^{4} - x^{3} \log {\relax (3 )} - 5 x^{2} + 13 x + \left (x^{3} + 5\right ) e^{x} - 5 \log {\relax (3 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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