Optimal. Leaf size=32 \[ \frac {4 \left (2+x+\frac {2}{x-\frac {5 (-e+\log (x))}{2-x}}\right )}{x+\log (x)} \]
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Rubi [F]
time = 7.31, 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 {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-25 e^2 (2+3 x)+10 e \left (-2-5 x-4 x^2+3 x^3\right )+x \left (12-26 x+10 x^2+6 x^3-3 x^4\right )+\left (40+32 x+25 e^2 x+48 x^2-28 x^3-4 x^4+x^5-10 e \left (-10-14 x-2 x^2+x^3\right )\right ) \log (x)-5 \left (10+(13+10 e) x+4 x^2-2 x^3\right ) \log ^2(x)+25 x \log ^3(x)\right )}{x (5 e-(-2+x) x-5 \log (x))^2 (x+\log (x))^2} \, dx\\ &=4 \int \frac {-25 e^2 (2+3 x)+10 e \left (-2-5 x-4 x^2+3 x^3\right )+x \left (12-26 x+10 x^2+6 x^3-3 x^4\right )+\left (40+32 x+25 e^2 x+48 x^2-28 x^3-4 x^4+x^5-10 e \left (-10-14 x-2 x^2+x^3\right )\right ) \log (x)-5 \left (10+(13+10 e) x+4 x^2-2 x^3\right ) \log ^2(x)+25 x \log ^3(x)}{x (5 e-(-2+x) x-5 \log (x))^2 (x+\log (x))^2} \, dx\\ &=4 \int \left (-\frac {10 \left (14+5 e-4 x+x^2\right )}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )}+\frac {(1+x) \left (-2 (2+5 e)-(12+5 e) x-5 x^2+x^3\right )}{x \left (5 e+7 x-x^2\right ) (x+\log (x))^2}+\frac {-28-10 e+25 e^2+2 (4+35 e) x+(47-10 e) x^2-14 x^3+x^4}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))}+\frac {10 \left (-10+9 x-6 x^2+2 x^3\right )}{x \left (-5 e-7 x+x^2\right ) \left (-5 e-2 x+x^2+5 \log (x)\right )^2}\right ) \, dx\\ &=4 \int \frac {(1+x) \left (-2 (2+5 e)-(12+5 e) x-5 x^2+x^3\right )}{x \left (5 e+7 x-x^2\right ) (x+\log (x))^2} \, dx+4 \int \frac {-28-10 e+25 e^2+2 (4+35 e) x+(47-10 e) x^2-14 x^3+x^4}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))} \, dx-40 \int \frac {14+5 e-4 x+x^2}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+40 \int \frac {-10+9 x-6 x^2+2 x^3}{x \left (-5 e-7 x+x^2\right ) \left (-5 e-2 x+x^2+5 \log (x)\right )^2} \, dx\\ &=4 \int \left (-\frac {3}{(x+\log (x))^2}-\frac {2 (2+5 e)}{5 e x (x+\log (x))^2}-\frac {x}{(x+\log (x))^2}+\frac {2 (14-5 e-(2-5 e) x)}{5 e \left (5 e+7 x-x^2\right ) (x+\log (x))^2}\right ) \, dx+4 \int \left (\frac {1}{x+\log (x)}-\frac {2 (14+10 e+3 x)}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))}+\frac {2}{\left (5 e+7 x-x^2\right ) (x+\log (x))}\right ) \, dx+40 \int \left (\frac {2}{\left (5 e+2 x-x^2-5 \log (x)\right )^2}+\frac {2}{e x \left (5 e+2 x-x^2-5 \log (x)\right )^2}+\frac {-14-9 e-10 e^2+2 (1-4 e) x}{e \left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}\right ) \, dx-40 \int \left (\frac {14+10 e+3 x}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )}-\frac {1}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {x}{(x+\log (x))^2} \, dx\right )+4 \int \frac {1}{x+\log (x)} \, dx-8 \int \frac {14+10 e+3 x}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))} \, dx+8 \int \frac {1}{\left (5 e+7 x-x^2\right ) (x+\log (x))} \, dx-12 \int \frac {1}{(x+\log (x))^2} \, dx-40 \int \frac {14+10 e+3 x}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+40 \int \frac {1}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+80 \int \frac {1}{\left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx+\frac {8 \int \frac {14-5 e-(2-5 e) x}{\left (5 e+7 x-x^2\right ) (x+\log (x))^2} \, dx}{5 e}+\frac {40 \int \frac {-14-9 e-10 e^2+2 (1-4 e) x}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}+\frac {80 \int \frac {1}{x \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}-\frac {(8 (2+5 e)) \int \frac {1}{x (x+\log (x))^2} \, dx}{5 e}\\ &=-\left (4 \int \frac {x}{(x+\log (x))^2} \, dx\right )+4 \int \frac {1}{x+\log (x)} \, dx+8 \int \left (\frac {2}{\sqrt {49+20 e} \left (7+\sqrt {49+20 e}-2 x\right ) (x+\log (x))}+\frac {2}{\sqrt {49+20 e} \left (-7+\sqrt {49+20 e}+2 x\right ) (x+\log (x))}\right ) \, dx-8 \int \left (\frac {14 \left (1+\frac {5 e}{7}\right )}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))}+\frac {3 x}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))}\right ) \, dx-12 \int \frac {1}{(x+\log (x))^2} \, dx+40 \int \left (\frac {2}{\sqrt {49+20 e} \left (7+\sqrt {49+20 e}-2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )}+\frac {2}{\sqrt {49+20 e} \left (-7+\sqrt {49+20 e}+2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )}\right ) \, dx-40 \int \left (\frac {14 \left (1+\frac {5 e}{7}\right )}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )}+\frac {3 x}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )}\right ) \, dx+80 \int \frac {1}{\left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx+\frac {8 \int \left (\frac {14 \left (1-\frac {5 e}{14}\right )}{\left (5 e+7 x-x^2\right ) (x+\log (x))^2}+\frac {(-2+5 e) x}{\left (5 e+7 x-x^2\right ) (x+\log (x))^2}\right ) \, dx}{5 e}+\frac {40 \int \left (-\frac {14 \left (1+\frac {1}{14} e (9+10 e)\right )}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}-\frac {2 (-1+4 e) x}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}\right ) \, dx}{e}+\frac {80 \int \frac {1}{x \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}-\frac {(8 (2+5 e)) \int \frac {1}{x (x+\log (x))^2} \, dx}{5 e}\\ &=-\left (4 \int \frac {x}{(x+\log (x))^2} \, dx\right )+4 \int \frac {1}{x+\log (x)} \, dx-12 \int \frac {1}{(x+\log (x))^2} \, dx-24 \int \frac {x}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))} \, dx+80 \int \frac {1}{\left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx-120 \int \frac {x}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+\frac {80 \int \frac {1}{x \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}+\frac {(8 (14-5 e)) \int \frac {1}{\left (5 e+7 x-x^2\right ) (x+\log (x))^2} \, dx}{5 e}+\frac {(80 (1-4 e)) \int \frac {x}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}+\frac {(8 (-2+5 e)) \int \frac {x}{\left (5 e+7 x-x^2\right ) (x+\log (x))^2} \, dx}{5 e}-\frac {(8 (2+5 e)) \int \frac {1}{x (x+\log (x))^2} \, dx}{5 e}-(16 (7+5 e)) \int \frac {1}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))} \, dx-(80 (7+5 e)) \int \frac {1}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+\frac {16 \int \frac {1}{\left (7+\sqrt {49+20 e}-2 x\right ) (x+\log (x))} \, dx}{\sqrt {49+20 e}}+\frac {16 \int \frac {1}{\left (-7+\sqrt {49+20 e}+2 x\right ) (x+\log (x))} \, dx}{\sqrt {49+20 e}}+\frac {80 \int \frac {1}{\left (7+\sqrt {49+20 e}-2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx}{\sqrt {49+20 e}}+\frac {80 \int \frac {1}{\left (-7+\sqrt {49+20 e}+2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx}{\sqrt {49+20 e}}-\frac {(40 (14+e (9+10 e))) \int \frac {1}{\left (5 e+7 x-x^2\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}\\ &=-\left (4 \int \frac {x}{(x+\log (x))^2} \, dx\right )+4 \int \frac {1}{x+\log (x)} \, dx-12 \int \frac {1}{(x+\log (x))^2} \, dx-24 \int \frac {x}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))} \, dx+80 \int \frac {1}{\left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx-120 \int \frac {x}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+\frac {80 \int \frac {1}{x \left (5 e+2 x-x^2-5 \log (x)\right )^2} \, dx}{e}+\frac {(8 (14-5 e)) \int \left (\frac {2}{\sqrt {49+20 e} \left (7+\sqrt {49+20 e}-2 x\right ) (x+\log (x))^2}+\frac {2}{\sqrt {49+20 e} \left (-7+\sqrt {49+20 e}+2 x\right ) (x+\log (x))^2}\right ) \, dx}{5 e}+\frac {(80 (1-4 e)) \int \left (\frac {1-\frac {7}{\sqrt {49+20 e}}}{\left (7-\sqrt {49+20 e}-2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}+\frac {1+\frac {7}{\sqrt {49+20 e}}}{\left (7+\sqrt {49+20 e}-2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}\right ) \, dx}{e}+\frac {(8 (-2+5 e)) \int \left (\frac {1-\frac {7}{\sqrt {49+20 e}}}{\left (7-\sqrt {49+20 e}-2 x\right ) (x+\log (x))^2}+\frac {1+\frac {7}{\sqrt {49+20 e}}}{\left (7+\sqrt {49+20 e}-2 x\right ) (x+\log (x))^2}\right ) \, dx}{5 e}-\frac {(8 (2+5 e)) \int \frac {1}{x (x+\log (x))^2} \, dx}{5 e}-(16 (7+5 e)) \int \frac {1}{\left (5 e+7 x-x^2\right )^2 (x+\log (x))} \, dx-(80 (7+5 e)) \int \frac {1}{\left (5 e+7 x-x^2\right )^2 \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx+\frac {16 \int \frac {1}{\left (7+\sqrt {49+20 e}-2 x\right ) (x+\log (x))} \, dx}{\sqrt {49+20 e}}+\frac {16 \int \frac {1}{\left (-7+\sqrt {49+20 e}+2 x\right ) (x+\log (x))} \, dx}{\sqrt {49+20 e}}+\frac {80 \int \frac {1}{\left (7+\sqrt {49+20 e}-2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx}{\sqrt {49+20 e}}+\frac {80 \int \frac {1}{\left (-7+\sqrt {49+20 e}+2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )} \, dx}{\sqrt {49+20 e}}-\frac {(40 (14+e (9+10 e))) \int \left (\frac {2}{\sqrt {49+20 e} \left (7+\sqrt {49+20 e}-2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}+\frac {2}{\sqrt {49+20 e} \left (-7+\sqrt {49+20 e}+2 x\right ) \left (5 e+2 x-x^2-5 \log (x)\right )^2}\right ) \, dx}{e}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 47, normalized size = 1.47 \begin {gather*} \frac {4 \left (4+2 x-x^3+5 e (2+x)-5 (2+x) \log (x)\right )}{(5 e-(-2+x) x-5 \log (x)) (x+\log (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs.
\(2(33)=66\).
time = 1.34, size = 89, normalized size = 2.78
method | result | size |
risch | \(\frac {-4 x^{3}+20 x \,{\mathrm e}-20 x \ln \left (x \right )+40 \,{\mathrm e}+8 x -40 \ln \left (x \right )+16}{-x^{3}-x^{2} \ln \left (x \right )+5 x \,{\mathrm e}+5 \,{\mathrm e} \ln \left (x \right )+2 x^{2}-3 x \ln \left (x \right )-5 \ln \left (x \right )^{2}}\) | \(73\) |
default | \(-\frac {4 \left (-x^{2} \ln \left (x \right )+5 \,{\mathrm e} \ln \left (x \right )-5 \ln \left (x \right )^{2}+2 x \ln \left (x \right )+2 x^{2}-10 \,{\mathrm e}+10 \ln \left (x \right )-2 x -4\right )}{-x^{3}-x^{2} \ln \left (x \right )+5 \,{\mathrm e}^{\ln \left (x \right )+1}+5 \,{\mathrm e} \ln \left (x \right )+2 x^{2}-3 x \ln \left (x \right )-5 \ln \left (x \right )^{2}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (31) = 62\).
time = 0.51, size = 63, normalized size = 1.97 \begin {gather*} \frac {4 \, {\left (x^{3} - x {\left (5 \, e + 2\right )} + 5 \, {\left (x + 2\right )} \log \left (x\right ) - 10 \, e - 4\right )}}{x^{3} - 2 \, x^{2} - 5 \, x e + {\left (x^{2} + 3 \, x - 5 \, e\right )} \log \left (x\right ) + 5 \, \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 60, normalized size = 1.88 \begin {gather*} \frac {4 \, {\left (x^{3} - 5 \, {\left (x + 2\right )} e + 5 \, {\left (x + 2\right )} \log \left (x\right ) - 2 \, x - 4\right )}}{x^{3} - 2 \, x^{2} - 5 \, x e + {\left (x^{2} + 3 \, x - 5 \, e\right )} \log \left (x\right ) + 5 \, \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (26) = 52\).
time = 0.19, size = 66, normalized size = 2.06 \begin {gather*} \frac {4 x^{3} - 20 e x - 8 x + \left (20 x + 40\right ) \log {\left (x \right )} - 40 e - 16}{x^{3} - 2 x^{2} - 5 e x + \left (x^{2} + 3 x - 5 e\right ) \log {\left (x \right )} + 5 \log {\left (x \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {48\,x+100\,x\,{\ln \left (x\right )}^3-\mathrm {e}\,\left (-120\,x^3+160\,x^2+200\,x+80\right )-{\ln \left (x\right )}^2\,\left (260\,x+200\,x\,\mathrm {e}+80\,x^2-40\,x^3+200\right )-104\,x^2+40\,x^3+24\,x^4-12\,x^5-{\mathrm {e}}^2\,\left (300\,x+200\right )+\ln \left (x\right )\,\left (128\,x+100\,x\,{\mathrm {e}}^2+\mathrm {e}\,\left (-40\,x^3+80\,x^2+560\,x+400\right )+192\,x^2-112\,x^3-16\,x^4+4\,x^5+160\right )}{{\ln \left (x\right )}^2\,\left (25\,x\,{\mathrm {e}}^2-\mathrm {e}\,\left (10\,x^3+80\,x^2\right )-11\,x^3+16\,x^4+x^5\right )+25\,x\,{\ln \left (x\right )}^4+\ln \left (x\right )\,\left (50\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (20\,x^4+10\,x^3\right )-12\,x^4+2\,x^5+2\,x^6\right )+\mathrm {e}\,\left (20\,x^4-10\,x^5\right )+25\,x^3\,{\mathrm {e}}^2+{\ln \left (x\right )}^3\,\left (10\,x^3+30\,x^2-50\,\mathrm {e}\,x\right )+4\,x^5-4\,x^6+x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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