Optimal. Leaf size=34 \[ -4+\frac {e^5 x \left (\frac {\frac {x}{3}+x^3}{x}-x \log (5)\right )}{\log (5-x)} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.57, antiderivative size = 214, normalized size of antiderivative = 6.29, number of steps
used = 50, number of rules used = 15, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used =
{6820, 12, 6874, 2465, 2447, 2446, 2436, 2335, 2437, 2346, 2209, 2334, 2339, 30, 2464}
\begin {gather*} -20 e^5 \text {Ei}(2 \log (5-x))+2 e^5 (15-\log (5)) \text {Ei}(2 \log (5-x))-2 e^5 (5-\log (5)) \text {Ei}(2 \log (5-x))+25 e^5 \text {li}(5-x)+5 e^5 (5-\log (5)) \text {li}(5-x)-\frac {2}{3} e^5 (113-15 \log (5)) \text {li}(5-x)+\frac {1}{3} e^5 (76-15 \log (5)) \text {li}(5-x)-\frac {e^5 (5-x) x^2}{\log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}-\frac {e^5 (5-x) (76-15 \log (5))}{3 \log (5-x)}+\frac {5 e^5 (76-15 \log (5))}{3 \log (5-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2209
Rule 2334
Rule 2335
Rule 2339
Rule 2346
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2464
Rule 2465
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^5 \left (-x \left (-1-3 x^2+3 x \log (5)\right )-(-5+x) \left (1+9 x^2-6 x \log (5)\right ) \log (5-x)\right )}{3 (5-x) \log ^2(5-x)} \, dx\\ &=\frac {1}{3} e^5 \int \frac {-x \left (-1-3 x^2+3 x \log (5)\right )-(-5+x) \left (1+9 x^2-6 x \log (5)\right ) \log (5-x)}{(5-x) \log ^2(5-x)} \, dx\\ &=\frac {1}{3} e^5 \int \left (-\frac {x \left (1+3 x^2-3 x \log (5)\right )}{(-5+x) \log ^2(5-x)}+\frac {1+9 x^2-6 x \log (5)}{\log (5-x)}\right ) \, dx\\ &=-\left (\frac {1}{3} e^5 \int \frac {x \left (1+3 x^2-3 x \log (5)\right )}{(-5+x) \log ^2(5-x)} \, dx\right )+\frac {1}{3} e^5 \int \frac {1+9 x^2-6 x \log (5)}{\log (5-x)} \, dx\\ &=-\left (\frac {1}{3} e^5 \int \left (\frac {3 x^2}{\log ^2(5-x)}+\frac {76 \left (1-\frac {15 \log (5)}{76}\right )}{\log ^2(5-x)}-\frac {3 x (-5+\log (5))}{\log ^2(5-x)}-\frac {5 (-76+15 \log (5))}{(-5+x) \log ^2(5-x)}\right ) \, dx\right )+\frac {1}{3} e^5 \int \left (\frac {9 (5-x)^2}{\log (5-x)}+\frac {6 (5-x) (-15+\log (5))}{\log (5-x)}-\frac {2 (-113+15 \log (5))}{\log (5-x)}\right ) \, dx\\ &=-\left (e^5 \int \frac {x^2}{\log ^2(5-x)} \, dx\right )+\left (3 e^5\right ) \int \frac {(5-x)^2}{\log (5-x)} \, dx-\frac {1}{3} \left (e^5 (76-15 \log (5))\right ) \int \frac {1}{\log ^2(5-x)} \, dx-\frac {1}{3} \left (5 e^5 (76-15 \log (5))\right ) \int \frac {1}{(-5+x) \log ^2(5-x)} \, dx+\frac {1}{3} \left (2 e^5 (113-15 \log (5))\right ) \int \frac {1}{\log (5-x)} \, dx-\left (e^5 (5-\log (5))\right ) \int \frac {x}{\log ^2(5-x)} \, dx-\left (2 e^5 (15-\log (5))\right ) \int \frac {5-x}{\log (5-x)} \, dx\\ &=-\frac {e^5 (5-x) x^2}{\log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}-\left (3 e^5\right ) \int \frac {x^2}{\log (5-x)} \, dx-\left (3 e^5\right ) \text {Subst}\left (\int \frac {x^2}{\log (x)} \, dx,x,5-x\right )+\left (10 e^5\right ) \int \frac {x}{\log (5-x)} \, dx+\frac {1}{3} \left (e^5 (76-15 \log (5))\right ) \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,5-x\right )-\frac {1}{3} \left (5 e^5 (76-15 \log (5))\right ) \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,5-x\right )-\frac {1}{3} \left (2 e^5 (113-15 \log (5))\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )-\left (2 e^5 (5-\log (5))\right ) \int \frac {x}{\log (5-x)} \, dx+\left (5 e^5 (5-\log (5))\right ) \int \frac {1}{\log (5-x)} \, dx+\left (2 e^5 (15-\log (5))\right ) \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,5-x\right )\\ &=-\frac {e^5 (5-x) x^2}{\log (5-x)}-\frac {e^5 (5-x) (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}-\frac {2}{3} e^5 (113-15 \log (5)) \text {li}(5-x)-\left (3 e^5\right ) \int \left (\frac {25}{\log (5-x)}-\frac {10 (5-x)}{\log (5-x)}+\frac {(5-x)^2}{\log (5-x)}\right ) \, dx-\left (3 e^5\right ) \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (5-x)\right )+\left (10 e^5\right ) \int \left (\frac {5}{\log (5-x)}-\frac {5-x}{\log (5-x)}\right ) \, dx+\frac {1}{3} \left (e^5 (76-15 \log (5))\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )-\frac {1}{3} \left (5 e^5 (76-15 \log (5))\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (5-x)\right )-\left (2 e^5 (5-\log (5))\right ) \int \left (\frac {5}{\log (5-x)}-\frac {5-x}{\log (5-x)}\right ) \, dx-\left (5 e^5 (5-\log (5))\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )+\left (2 e^5 (15-\log (5))\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (5-x)\right )\\ &=-3 e^5 \text {Ei}(3 \log (5-x))+2 e^5 \text {Ei}(2 \log (5-x)) (15-\log (5))-\frac {e^5 (5-x) x^2}{\log (5-x)}+\frac {5 e^5 (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}+\frac {1}{3} e^5 (76-15 \log (5)) \text {li}(5-x)-\frac {2}{3} e^5 (113-15 \log (5)) \text {li}(5-x)-5 e^5 (5-\log (5)) \text {li}(5-x)-\left (3 e^5\right ) \int \frac {(5-x)^2}{\log (5-x)} \, dx-\left (10 e^5\right ) \int \frac {5-x}{\log (5-x)} \, dx+\left (30 e^5\right ) \int \frac {5-x}{\log (5-x)} \, dx+\left (50 e^5\right ) \int \frac {1}{\log (5-x)} \, dx-\left (75 e^5\right ) \int \frac {1}{\log (5-x)} \, dx+\left (2 e^5 (5-\log (5))\right ) \int \frac {5-x}{\log (5-x)} \, dx-\left (10 e^5 (5-\log (5))\right ) \int \frac {1}{\log (5-x)} \, dx\\ &=-3 e^5 \text {Ei}(3 \log (5-x))+2 e^5 \text {Ei}(2 \log (5-x)) (15-\log (5))-\frac {e^5 (5-x) x^2}{\log (5-x)}+\frac {5 e^5 (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}+\frac {1}{3} e^5 (76-15 \log (5)) \text {li}(5-x)-\frac {2}{3} e^5 (113-15 \log (5)) \text {li}(5-x)-5 e^5 (5-\log (5)) \text {li}(5-x)+\left (3 e^5\right ) \text {Subst}\left (\int \frac {x^2}{\log (x)} \, dx,x,5-x\right )+\left (10 e^5\right ) \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,5-x\right )-\left (30 e^5\right ) \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,5-x\right )-\left (50 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )+\left (75 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )-\left (2 e^5 (5-\log (5))\right ) \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,5-x\right )+\left (10 e^5 (5-\log (5))\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5-x\right )\\ &=-3 e^5 \text {Ei}(3 \log (5-x))+2 e^5 \text {Ei}(2 \log (5-x)) (15-\log (5))-\frac {e^5 (5-x) x^2}{\log (5-x)}+\frac {5 e^5 (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}+25 e^5 \text {li}(5-x)+\frac {1}{3} e^5 (76-15 \log (5)) \text {li}(5-x)-\frac {2}{3} e^5 (113-15 \log (5)) \text {li}(5-x)+5 e^5 (5-\log (5)) \text {li}(5-x)+\left (3 e^5\right ) \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (5-x)\right )+\left (10 e^5\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (5-x)\right )-\left (30 e^5\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (5-x)\right )-\left (2 e^5 (5-\log (5))\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (5-x)\right )\\ &=-20 e^5 \text {Ei}(2 \log (5-x))-2 e^5 \text {Ei}(2 \log (5-x)) (5-\log (5))+2 e^5 \text {Ei}(2 \log (5-x)) (15-\log (5))-\frac {e^5 (5-x) x^2}{\log (5-x)}+\frac {5 e^5 (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) (76-15 \log (5))}{3 \log (5-x)}-\frac {e^5 (5-x) x (5-\log (5))}{\log (5-x)}+25 e^5 \text {li}(5-x)+\frac {1}{3} e^5 (76-15 \log (5)) \text {li}(5-x)-\frac {2}{3} e^5 (113-15 \log (5)) \text {li}(5-x)+5 e^5 (5-\log (5)) \text {li}(5-x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 28, normalized size = 0.82 \begin {gather*} \frac {e^5 x \left (1+3 x^2-3 x \log (5)\right )}{3 \log (5-x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.27, size = 270, normalized size = 7.94
method | result | size |
risch | \(-\frac {x \,{\mathrm e}^{5} \left (3 x \ln \left (5\right )-3 x^{2}-1\right )}{3 \ln \left (5-x \right )}\) | \(26\) |
norman | \(\frac {x^{3} {\mathrm e}^{5}+\frac {x \,{\mathrm e}^{5}}{3}-x^{2} {\mathrm e}^{5} \ln \left (5\right )}{\ln \left (5-x \right )}\) | \(31\) |
derivativedivides | \(2 \,{\mathrm e}^{5} \ln \left (5\right ) \expIntegral \left (1, -2 \ln \left (5-x \right )\right )+3 \,{\mathrm e}^{5} \expIntegral \left (1, -3 \ln \left (5-x \right )\right )-10 \,{\mathrm e}^{5} \ln \left (5\right ) \expIntegral \left (1, -\ln \left (5-x \right )\right )+{\mathrm e}^{5} \ln \left (5\right ) \left (-\frac {\left (5-x \right )^{2}}{\ln \left (5-x \right )}-2 \expIntegral \left (1, -2 \ln \left (5-x \right )\right )\right )-30 \,{\mathrm e}^{5} \expIntegral \left (1, -2 \ln \left (5-x \right )\right )+{\mathrm e}^{5} \left (-\frac {\left (5-x \right )^{3}}{\ln \left (5-x \right )}-3 \expIntegral \left (1, -3 \ln \left (5-x \right )\right )\right )-10 \,{\mathrm e}^{5} \ln \left (5\right ) \left (-\frac {5-x}{\ln \left (5-x \right )}-\expIntegral \left (1, -\ln \left (5-x \right )\right )\right )+\frac {226 \,{\mathrm e}^{5} \expIntegral \left (1, -\ln \left (5-x \right )\right )}{3}-15 \,{\mathrm e}^{5} \left (-\frac {\left (5-x \right )^{2}}{\ln \left (5-x \right )}-2 \expIntegral \left (1, -2 \ln \left (5-x \right )\right )\right )-\frac {25 \,{\mathrm e}^{5} \ln \left (5\right )}{\ln \left (5-x \right )}+\frac {226 \,{\mathrm e}^{5} \left (-\frac {5-x}{\ln \left (5-x \right )}-\expIntegral \left (1, -\ln \left (5-x \right )\right )\right )}{3}+\frac {380 \,{\mathrm e}^{5}}{3 \ln \left (5-x \right )}\) | \(270\) |
default | \(2 \,{\mathrm e}^{5} \ln \left (5\right ) \expIntegral \left (1, -2 \ln \left (5-x \right )\right )+3 \,{\mathrm e}^{5} \expIntegral \left (1, -3 \ln \left (5-x \right )\right )-10 \,{\mathrm e}^{5} \ln \left (5\right ) \expIntegral \left (1, -\ln \left (5-x \right )\right )+{\mathrm e}^{5} \ln \left (5\right ) \left (-\frac {\left (5-x \right )^{2}}{\ln \left (5-x \right )}-2 \expIntegral \left (1, -2 \ln \left (5-x \right )\right )\right )-30 \,{\mathrm e}^{5} \expIntegral \left (1, -2 \ln \left (5-x \right )\right )+{\mathrm e}^{5} \left (-\frac {\left (5-x \right )^{3}}{\ln \left (5-x \right )}-3 \expIntegral \left (1, -3 \ln \left (5-x \right )\right )\right )-10 \,{\mathrm e}^{5} \ln \left (5\right ) \left (-\frac {5-x}{\ln \left (5-x \right )}-\expIntegral \left (1, -\ln \left (5-x \right )\right )\right )+\frac {226 \,{\mathrm e}^{5} \expIntegral \left (1, -\ln \left (5-x \right )\right )}{3}-15 \,{\mathrm e}^{5} \left (-\frac {\left (5-x \right )^{2}}{\ln \left (5-x \right )}-2 \expIntegral \left (1, -2 \ln \left (5-x \right )\right )\right )-\frac {25 \,{\mathrm e}^{5} \ln \left (5\right )}{\ln \left (5-x \right )}+\frac {226 \,{\mathrm e}^{5} \left (-\frac {5-x}{\ln \left (5-x \right )}-\expIntegral \left (1, -\ln \left (5-x \right )\right )\right )}{3}+\frac {380 \,{\mathrm e}^{5}}{3 \ln \left (5-x \right )}\) | \(270\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 36, normalized size = 1.06 \begin {gather*} \frac {3 \, x^{3} e^{5} - 3 \, x^{2} e^{5} \log \left (5\right ) + x e^{5}}{3 \, \log \left (-x + 5\right )} + \frac {5}{3} \, e^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 31, normalized size = 0.91 \begin {gather*} -\frac {3 \, x^{2} e^{5} \log \left (5\right ) - {\left (3 \, x^{3} + x\right )} e^{5}}{3 \, \log \left (-x + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 31, normalized size = 0.91 \begin {gather*} \frac {3 x^{3} e^{5} - 3 x^{2} e^{5} \log {\left (5 \right )} + x e^{5}}{3 \log {\left (5 - x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 66, normalized size = 1.94 \begin {gather*} \frac {3 \, {\left (x - 5\right )}^{3} e^{5} - 3 \, {\left (x - 5\right )}^{2} e^{5} \log \left (5\right ) + 45 \, {\left (x - 5\right )}^{2} e^{5} - 30 \, {\left (x - 5\right )} e^{5} \log \left (5\right ) + 226 \, {\left (x - 5\right )} e^{5} - 75 \, e^{5} \log \left (5\right ) + 380 \, e^{5}}{3 \, \log \left (-x + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 82, normalized size = 2.41 \begin {gather*} \frac {3\,{\mathrm {e}}^5\,x^5-3\,{\mathrm {e}}^5\,\left (\ln \left (5\right )+10\right )\,x^4+2\,{\mathrm {e}}^5\,\left (15\,\ln \left (5\right )+38\right )\,x^3-5\,{\mathrm {e}}^5\,\left (15\,\ln \left (5\right )+2\right )\,x^2+25\,{\mathrm {e}}^5\,x}{75\,\ln \left (5-x\right )-30\,x\,\ln \left (5-x\right )+3\,x^2\,\ln \left (5-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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