Optimal. Leaf size=28 \[ \frac {1}{3} \left (3+e^5+\frac {\left (-7+\frac {5}{x}-x\right )^4}{\log ^8(4)}\right )^2 \]
________________________________________________________________________________________
Rubi [B] time = 0.22, antiderivative size = 303, normalized size of antiderivative = 10.82, number of steps used = 3, number of rules used = 2, integrand size = 177, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {12, 14} \begin {gather*} \frac {x^8}{3 \log ^{16}(4)}+\frac {390625}{3 x^8 \log ^{16}(4)}+\frac {56 x^7}{3 \log ^{16}(4)}-\frac {4375000}{3 x^7 \log ^{16}(4)}+\frac {444 x^6}{\log ^{16}(4)}+\frac {6937500}{x^6 \log ^{16}(4)}+\frac {17248 x^5}{3 \log ^{16}(4)}-\frac {53900000}{3 x^5 \log ^{16}(4)}+\frac {2 x^4 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}+\frac {1250 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 x^4 \log ^{16}(4)}+\frac {56 x^3 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}-\frac {7000 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 x^3 \log ^{16}(4)}+\frac {4 x^2 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}+\frac {100 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{3 x^2 \log ^{16}(4)}-\frac {56 x \left (53081-102 \log ^8(4)-34 e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}+\frac {280 \left (53081-102 \log ^8(4)-34 e^5 \log ^8(4)\right )}{3 x \log ^{16}(4)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-3125000+30625000 x-124875000 x^2+269500000 x^3-319025000 x^4+183897000 x^5-22013600 x^6-14862680 x^7-2972536 x^9+880544 x^{10}+1471176 x^{11}+510440 x^{12}+86240 x^{13}+7992 x^{14}+392 x^{15}+8 x^{16}+\left (-15000 x^4+63000 x^5-82200 x^6+28560 x^7+5712 x^9+3288 x^{10}+504 x^{11}+24 x^{12}+e^5 \left (-5000 x^4+21000 x^5-27400 x^6+9520 x^7+1904 x^9+1096 x^{10}+168 x^{11}+8 x^{12}\right )\right ) \log ^8(4)}{x^9} \, dx}{3 \log ^{16}(4)}\\ &=\frac {\int \left (-\frac {3125000}{x^9}+\frac {30625000}{x^8}-\frac {124875000}{x^7}+\frac {269500000}{x^6}+86240 x^4+7992 x^5+392 x^6+8 x^7+\frac {21000 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{x^4}+168 x^2 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )-\frac {5000 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{x^5}+8 x^3 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )+56 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )+\frac {280 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )}{x^2}-\frac {200 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{x^3}+8 x \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )\right ) \, dx}{3 \log ^{16}(4)}\\ &=\frac {390625}{3 x^8 \log ^{16}(4)}-\frac {4375000}{3 x^7 \log ^{16}(4)}+\frac {6937500}{x^6 \log ^{16}(4)}-\frac {53900000}{3 x^5 \log ^{16}(4)}+\frac {17248 x^5}{3 \log ^{16}(4)}+\frac {444 x^6}{\log ^{16}(4)}+\frac {56 x^7}{3 \log ^{16}(4)}+\frac {x^8}{3 \log ^{16}(4)}+\frac {280 \left (53081-102 \log ^8(4)-34 e^5 \log ^8(4)\right )}{3 x \log ^{16}(4)}-\frac {56 x \left (53081-102 \log ^8(4)-34 e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}-\frac {7000 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 x^3 \log ^{16}(4)}+\frac {56 x^3 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}+\frac {1250 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 x^4 \log ^{16}(4)}+\frac {2 x^4 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}+\frac {100 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{3 x^2 \log ^{16}(4)}+\frac {4 x^2 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{3 \log ^{16}(4)}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.06, size = 216, normalized size = 7.71 \begin {gather*} \frac {390625-4375000 x+20812500 x^2-53900000 x^3+17248 x^{13}+1332 x^{14}+56 x^{15}+x^{16}-7000 x^5 \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )+56 x^{11} \left (8757+3 \log ^8(4)+e^5 \log ^8(4)\right )+1250 x^4 \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )+2 x^{12} \left (63805+3 \log ^8(4)+e^5 \log ^8(4)\right )-280 x^7 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )+56 x^9 \left (-53081+102 \log ^8(4)+34 e^5 \log ^8(4)\right )+100 x^6 \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )+4 x^{10} \left (110068+411 \log ^8(4)+137 e^5 \log ^8(4)\right )}{3 x^8 \log ^{16}(4)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 171, normalized size = 6.11 \begin {gather*} \frac {x^{16} + 56 \, x^{15} + 1332 \, x^{14} + 17248 \, x^{13} + 127610 \, x^{12} + 490392 \, x^{11} + 440272 \, x^{10} - 2972536 \, x^{9} + 512 \, {\left (3 \, x^{12} + 84 \, x^{11} + 822 \, x^{10} + 2856 \, x^{9} - 14280 \, x^{7} + 20550 \, x^{6} - 10500 \, x^{5} + 1875 \, x^{4} + {\left (x^{12} + 28 \, x^{11} + 274 \, x^{10} + 952 \, x^{9} - 4760 \, x^{7} + 6850 \, x^{6} - 3500 \, x^{5} + 625 \, x^{4}\right )} e^{5}\right )} \log \relax (2)^{8} + 14862680 \, x^{7} + 11006800 \, x^{6} - 61299000 \, x^{5} + 79756250 \, x^{4} - 53900000 \, x^{3} + 20812500 \, x^{2} - 4375000 \, x + 390625}{196608 \, x^{8} \log \relax (2)^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 239, normalized size = 8.54 \begin {gather*} \frac {512 \, x^{4} e^{5} \log \relax (2)^{8} + 1536 \, x^{4} \log \relax (2)^{8} + 14336 \, x^{3} e^{5} \log \relax (2)^{8} + 43008 \, x^{3} \log \relax (2)^{8} + 140288 \, x^{2} e^{5} \log \relax (2)^{8} + 420864 \, x^{2} \log \relax (2)^{8} + 487424 \, x e^{5} \log \relax (2)^{8} + 1462272 \, x \log \relax (2)^{8} + x^{8} + 56 \, x^{7} + 1332 \, x^{6} + 17248 \, x^{5} + 127610 \, x^{4} + 490392 \, x^{3} + 440272 \, x^{2} - 2972536 \, x - \frac {5 \, {\left (487424 \, x^{7} e^{5} \log \relax (2)^{8} + 1462272 \, x^{7} \log \relax (2)^{8} - 701440 \, x^{6} e^{5} \log \relax (2)^{8} - 2104320 \, x^{6} \log \relax (2)^{8} + 358400 \, x^{5} e^{5} \log \relax (2)^{8} + 1075200 \, x^{5} \log \relax (2)^{8} - 64000 \, x^{4} e^{5} \log \relax (2)^{8} - 192000 \, x^{4} \log \relax (2)^{8} - 2972536 \, x^{7} - 2201360 \, x^{6} + 12259800 \, x^{5} - 15951250 \, x^{4} + 10780000 \, x^{3} - 4162500 \, x^{2} + 875000 \, x - 78125\right )}}{x^{8}}}{196608 \, \log \relax (2)^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.18, size = 226, normalized size = 8.07
method | result | size |
default | \(\frac {64 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{4}+1792 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{3}+192 \ln \relax (2)^{8} x^{4}+17536 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{2}+5376 \ln \relax (2)^{8} x^{3}+60928 \,{\mathrm e}^{5} \ln \relax (2)^{8} x +52608 \ln \relax (2)^{8} x^{2}+182784 x \ln \relax (2)^{8}+\frac {x^{8}}{8}+7 x^{7}+\frac {333 x^{6}}{2}+2156 x^{5}+\frac {63805 x^{4}}{4}+61299 x^{3}+55034 x^{2}-371567 x +\frac {390625}{8 x^{8}}-\frac {-160000 \,{\mathrm e}^{5} \ln \relax (2)^{8}-480000 \ln \relax (2)^{8}-39878125}{4 x^{4}}-\frac {6737500}{x^{5}}+\frac {5203125}{2 x^{6}}-\frac {-876800 \,{\mathrm e}^{5} \ln \relax (2)^{8}-2630400 \ln \relax (2)^{8}-2751700}{2 x^{2}}-\frac {304640 \,{\mathrm e}^{5} \ln \relax (2)^{8}+913920 \ln \relax (2)^{8}-1857835}{x}-\frac {672000 \,{\mathrm e}^{5} \ln \relax (2)^{8}+2016000 \ln \relax (2)^{8}+22987125}{3 x^{3}}-\frac {546875}{x^{7}}}{24576 \ln \relax (2)^{16}}\) | \(226\) |
gosper | \(\frac {390625-4375000 x +1462272 x^{9} \ln \relax (2)^{8}+490392 x^{11}+127610 x^{12}+17248 x^{13}+1332 x^{14}+x^{16}+56 x^{15}+14862680 x^{7}+440272 x^{10}-2972536 x^{9}+11006800 x^{6}-61299000 x^{5}+79756250 x^{4}-53900000 x^{3}+20812500 x^{2}-7311360 x^{7} \ln \relax (2)^{8}+420864 x^{10} \ln \relax (2)^{8}+512 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{12}+14336 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{11}+10521600 \ln \relax (2)^{8} x^{6}-5376000 \ln \relax (2)^{8} x^{5}+960000 \ln \relax (2)^{8} x^{4}-1792000 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{5}+3507200 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{6}+320000 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{4}+140288 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{10}+487424 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{9}-2437120 \,{\mathrm e}^{5} \ln \relax (2)^{8} x^{7}+1536 \ln \relax (2)^{8} x^{12}+43008 \ln \relax (2)^{8} x^{11}}{196608 \ln \relax (2)^{16} x^{8}}\) | \(243\) |
risch | \(\frac {{\mathrm e}^{5} x^{4}}{384 \ln \relax (2)^{8}}+\frac {7 \,{\mathrm e}^{5} x^{3}}{96 \ln \relax (2)^{8}}+\frac {x^{4}}{128 \ln \relax (2)^{8}}+\frac {137 \,{\mathrm e}^{5} x^{2}}{192 \ln \relax (2)^{8}}+\frac {7 x^{3}}{32 \ln \relax (2)^{8}}+\frac {119 \,{\mathrm e}^{5} x}{48 \ln \relax (2)^{8}}+\frac {137 x^{2}}{64 \ln \relax (2)^{8}}+\frac {119 x}{16 \ln \relax (2)^{8}}+\frac {x^{8}}{196608 \ln \relax (2)^{16}}+\frac {7 x^{7}}{24576 \ln \relax (2)^{16}}+\frac {111 x^{6}}{16384 \ln \relax (2)^{16}}+\frac {539 x^{5}}{6144 \ln \relax (2)^{16}}+\frac {63805 x^{4}}{98304 \ln \relax (2)^{16}}+\frac {20433 x^{3}}{8192 \ln \relax (2)^{16}}+\frac {27517 x^{2}}{12288 \ln \relax (2)^{16}}-\frac {371567 x}{24576 \ln \relax (2)^{16}}+\frac {\left (-2437120 \,{\mathrm e}^{5} \ln \relax (2)^{8}-7311360 \ln \relax (2)^{8}+14862680\right ) x^{7}+\left (3507200 \,{\mathrm e}^{5} \ln \relax (2)^{8}+10521600 \ln \relax (2)^{8}+11006800\right ) x^{6}+\left (-1792000 \,{\mathrm e}^{5} \ln \relax (2)^{8}-5376000 \ln \relax (2)^{8}-61299000\right ) x^{5}+\left (320000 \,{\mathrm e}^{5} \ln \relax (2)^{8}+960000 \ln \relax (2)^{8}+79756250\right ) x^{4}-53900000 x^{3}+20812500 x^{2}-4375000 x +390625}{196608 \ln \relax (2)^{16} x^{8}}\) | \(252\) |
norman | \(\frac {\frac {390625}{196608 \ln \relax (2)}-\frac {546875 x}{24576 \ln \relax (2)}+\frac {1734375 x^{2}}{16384 \ln \relax (2)}-\frac {1684375 x^{3}}{6144 \ln \relax (2)}+\frac {539 x^{13}}{6144 \ln \relax (2)}+\frac {111 x^{14}}{16384 \ln \relax (2)}+\frac {7 x^{15}}{24576 \ln \relax (2)}+\frac {x^{16}}{196608 \ln \relax (2)}-\frac {875 \left (256 \,{\mathrm e}^{5} \ln \relax (2)^{8}+768 \ln \relax (2)^{8}+8757\right ) x^{5}}{24576 \ln \relax (2)}+\frac {7 \left (256 \,{\mathrm e}^{5} \ln \relax (2)^{8}+768 \ln \relax (2)^{8}+8757\right ) x^{11}}{24576 \ln \relax (2)}+\frac {625 \left (256 \,{\mathrm e}^{5} \ln \relax (2)^{8}+768 \ln \relax (2)^{8}+63805\right ) x^{4}}{98304 \ln \relax (2)}+\frac {\left (256 \,{\mathrm e}^{5} \ln \relax (2)^{8}+768 \ln \relax (2)^{8}+63805\right ) x^{12}}{98304 \ln \relax (2)}-\frac {35 \left (8704 \,{\mathrm e}^{5} \ln \relax (2)^{8}+26112 \ln \relax (2)^{8}-53081\right ) x^{7}}{24576 \ln \relax (2)}+\frac {7 \left (8704 \,{\mathrm e}^{5} \ln \relax (2)^{8}+26112 \ln \relax (2)^{8}-53081\right ) x^{9}}{24576 \ln \relax (2)}+\frac {25 \left (8768 \,{\mathrm e}^{5} \ln \relax (2)^{8}+26304 \ln \relax (2)^{8}+27517\right ) x^{6}}{12288 \ln \relax (2)}+\frac {\left (8768 \,{\mathrm e}^{5} \ln \relax (2)^{8}+26304 \ln \relax (2)^{8}+27517\right ) x^{10}}{12288 \ln \relax (2)}}{x^{8} \ln \relax (2)^{15}}\) | \(277\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.37, size = 179, normalized size = 6.39 \begin {gather*} \frac {x^{8} + 56 \, x^{7} + 1332 \, x^{6} + 2 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} + 63805\right )} x^{4} + 17248 \, x^{5} + 56 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} + 8757\right )} x^{3} + 16 \, {\left (8768 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} + 27517\right )} x^{2} + 56 \, {\left (8704 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} - 53081\right )} x - \frac {5 \, {\left (56 \, {\left (8704 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} - 53081\right )} x^{7} - 80 \, {\left (8768 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} + 27517\right )} x^{6} + 1400 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} + 8757\right )} x^{5} - 250 \, {\left (256 \, {\left (e^{5} + 3\right )} \log \relax (2)^{8} + 63805\right )} x^{4} + 10780000 \, x^{3} - 4162500 \, x^{2} + 875000 \, x - 78125\right )}}{x^{8}}}{196608 \, \log \relax (2)^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.25, size = 241, normalized size = 8.61 \begin {gather*} \frac {539\,x^5}{6144\,{\ln \relax (2)}^{16}}+\frac {111\,x^6}{16384\,{\ln \relax (2)}^{16}}+\frac {7\,x^7}{24576\,{\ln \relax (2)}^{16}}+\frac {x^8}{196608\,{\ln \relax (2)}^{16}}+\frac {x^4\,\left (256\,{\mathrm {e}}^5\,{\ln \relax (2)}^8+768\,{\ln \relax (2)}^8+63805\right )}{98304\,{\ln \relax (2)}^{16}}+\frac {x^3\,\left (5376\,{\mathrm {e}}^5\,{\ln \relax (2)}^8+16128\,{\ln \relax (2)}^8+183897\right )}{73728\,{\ln \relax (2)}^{16}}+\frac {x^2\,\left (35072\,{\mathrm {e}}^5\,{\ln \relax (2)}^8+105216\,{\ln \relax (2)}^8+110068\right )}{49152\,{\ln \relax (2)}^{16}}+\frac {x\,\left (60928\,{\mathrm {e}}^5\,{\ln \relax (2)}^8+182784\,{\ln \relax (2)}^8-371567\right )}{24576\,{\ln \relax (2)}^{16}}-\frac {\left (304640\,{\mathrm {e}}^5\,{\ln \relax (2)}^8+913920\,{\ln \relax (2)}^8-1857835\right )\,x^7+\left (-438400\,{\mathrm {e}}^5\,{\ln \relax (2)}^8-1315200\,{\ln \relax (2)}^8-1375850\right )\,x^6+\left (224000\,{\mathrm {e}}^5\,{\ln \relax (2)}^8+672000\,{\ln \relax (2)}^8+7662375\right )\,x^5+\left (-40000\,{\mathrm {e}}^5\,{\ln \relax (2)}^8-120000\,{\ln \relax (2)}^8-\frac {39878125}{4}\right )\,x^4+6737500\,x^3-\frac {5203125\,x^2}{2}+546875\,x-\frac {390625}{8}}{24576\,x^8\,{\ln \relax (2)}^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 15.76, size = 226, normalized size = 8.07 \begin {gather*} \frac {\frac {x^{8}}{8} + 7 x^{7} + \frac {333 x^{6}}{2} + 2156 x^{5} + x^{4} \left (192 \log {\relax (2 )}^{8} + 64 e^{5} \log {\relax (2 )}^{8} + \frac {63805}{4}\right ) + x^{3} \left (5376 \log {\relax (2 )}^{8} + 1792 e^{5} \log {\relax (2 )}^{8} + 61299\right ) + x^{2} \left (52608 \log {\relax (2 )}^{8} + 55034 + 17536 e^{5} \log {\relax (2 )}^{8}\right ) + x \left (-371567 + 182784 \log {\relax (2 )}^{8} + 60928 e^{5} \log {\relax (2 )}^{8}\right ) + \frac {x^{7} \left (- 2437120 e^{5} \log {\relax (2 )}^{8} - 7311360 \log {\relax (2 )}^{8} + 14862680\right ) + x^{6} \left (10521600 \log {\relax (2 )}^{8} + 11006800 + 3507200 e^{5} \log {\relax (2 )}^{8}\right ) + x^{5} \left (-61299000 - 1792000 e^{5} \log {\relax (2 )}^{8} - 5376000 \log {\relax (2 )}^{8}\right ) + x^{4} \left (960000 \log {\relax (2 )}^{8} + 320000 e^{5} \log {\relax (2 )}^{8} + 79756250\right ) - 53900000 x^{3} + 20812500 x^{2} - 4375000 x + 390625}{8 x^{8}}}{24576 \log {\relax (2 )}^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________