Optimal. Leaf size=30 \[ \frac {\left (\frac {e^4}{x}+\frac {4}{\log (x)}\right )^4}{(4+6 x) \log \left (x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 21.10, 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 {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{\left (8 x^5+24 x^6+18 x^7\right ) \log ^5(x) \log ^2\left (x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{x^5 \left (8+24 x+18 x^2\right ) \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{2 x^5 (2+3 x)^2 \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{x^5 (2+3 x)^2 \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (-16 x (2+3 x) \log \left (x^2\right )-4 x \log (x) \left (4+6 x+3 x \log \left (x^2\right )\right )-e^4 \log ^2(x) \left (4+6 x+(8+15 x) \log \left (x^2\right )\right )\right )}{x^5 (2+3 x)^2 \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {2 \left (4 x+e^4 \log (x)\right )^4}{x^5 (2+3 x) \log ^4(x) \log ^2\left (x^2\right )}-\frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^5 (2+3 x)^2 \log ^5(x) \log \left (x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^5 (2+3 x)^2 \log ^5(x) \log \left (x^2\right )} \, dx\right )-\int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^5 (2+3 x) \log ^4(x) \log ^2\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{4 x^5 \log ^5(x) \log \left (x^2\right )}-\frac {3 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{4 x^4 \log ^5(x) \log \left (x^2\right )}+\frac {27 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{16 x^3 \log ^5(x) \log \left (x^2\right )}-\frac {27 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{8 x^2 \log ^5(x) \log \left (x^2\right )}+\frac {405 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{64 x \log ^5(x) \log \left (x^2\right )}-\frac {243 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{32 (2+3 x)^2 \log ^5(x) \log \left (x^2\right )}-\frac {1215 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{64 (2+3 x) \log ^5(x) \log \left (x^2\right )}\right ) \, dx\right )-\int \left (\frac {\left (4 x+e^4 \log (x)\right )^4}{2 x^5 \log ^4(x) \log ^2\left (x^2\right )}-\frac {3 \left (4 x+e^4 \log (x)\right )^4}{4 x^4 \log ^4(x) \log ^2\left (x^2\right )}+\frac {9 \left (4 x+e^4 \log (x)\right )^4}{8 x^3 \log ^4(x) \log ^2\left (x^2\right )}-\frac {27 \left (4 x+e^4 \log (x)\right )^4}{16 x^2 \log ^4(x) \log ^2\left (x^2\right )}+\frac {81 \left (4 x+e^4 \log (x)\right )^4}{32 x \log ^4(x) \log ^2\left (x^2\right )}-\frac {243 \left (4 x+e^4 \log (x)\right )^4}{32 (2+3 x) \log ^4(x) \log ^2\left (x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^5 \log ^5(x) \log \left (x^2\right )} \, dx\right )+\frac {3}{8} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^4 \log ^5(x) \log \left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^5 \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {3}{4} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^4 \log ^4(x) \log ^2\left (x^2\right )} \, dx-\frac {27}{32} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^3 \log ^5(x) \log \left (x^2\right )} \, dx-\frac {9}{8} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^3 \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {27}{16} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^2 \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {27}{16} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^2 \log ^5(x) \log \left (x^2\right )} \, dx-\frac {81}{32} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x \log ^4(x) \log ^2\left (x^2\right )} \, dx-\frac {405}{128} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x \log ^5(x) \log \left (x^2\right )} \, dx+\frac {243}{64} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{(2+3 x)^2 \log ^5(x) \log \left (x^2\right )} \, dx+\frac {243}{32} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{(2+3 x) \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {1215}{128} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{(2+3 x) \log ^5(x) \log \left (x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 36, normalized size = 1.20 \begin {gather*} \frac {\left (4 x+e^4 \log (x)\right )^4}{2 x^4 (2+3 x) \log ^4(x) \log \left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 75.35, size = 114, normalized size = 3.80 \[\frac {{\mathrm e}^{16} \ln \left (x \right )^{4}+16 \,{\mathrm e}^{12} x \ln \left (x \right )^{3}+96 \,{\mathrm e}^{8} x^{2} \ln \left (x \right )^{2}+256 \,{\mathrm e}^{4} x^{3} \ln \left (x \right )+256 x^{4}}{\ln \left (x \right )^{4} \left (4 \ln \left (x \right )-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right ) x^{4} \left (3 x +2\right )}\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (30) = 60\).
time = 0.37, size = 61, normalized size = 2.03 \begin {gather*} \frac {256 \, x^{3} e^{4} \log \left (x\right ) + 96 \, x^{2} e^{8} \log \left (x\right )^{2} + 16 \, x e^{12} \log \left (x\right )^{3} + e^{16} \log \left (x\right )^{4} + 256 \, x^{4}}{4 \, {\left (3 \, x^{5} + 2 \, x^{4}\right )} \log \left (x\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (30) = 60\).
time = 0.38, size = 61, normalized size = 2.03 \begin {gather*} \frac {256 \, x^{3} e^{4} \log \left (x\right ) + 96 \, x^{2} e^{8} \log \left (x\right )^{2} + 16 \, x e^{12} \log \left (x\right )^{3} + e^{16} \log \left (x\right )^{4} + 256 \, x^{4}}{4 \, {\left (3 \, x^{5} + 2 \, x^{4}\right )} \log \left (x\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (20) = 40\).
time = 0.17, size = 65, normalized size = 2.17 \begin {gather*} \frac {256 x^{4} + 256 x^{3} e^{4} \log {\left (x \right )} + 96 x^{2} e^{8} \log {\left (x \right )}^{2} + 16 x e^{12} \log {\left (x \right )}^{3} + e^{16} \log {\left (x \right )}^{4}}{\left (12 x^{5} + 8 x^{4}\right ) \log {\left (x \right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (30) = 60\).
time = 0.46, size = 65, normalized size = 2.17 \begin {gather*} \frac {256 \, x^{3} e^{4} \log \left (x\right ) + 96 \, x^{2} e^{8} \log \left (x\right )^{2} + 16 \, x e^{12} \log \left (x\right )^{3} + e^{16} \log \left (x\right )^{4} + 256 \, x^{4}}{4 \, {\left (3 \, x^{5} \log \left (x\right )^{5} + 2 \, x^{4} \log \left (x\right )^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.45, size = 892, normalized size = 29.73 \begin {gather*} \frac {\frac {4\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+8\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^5+12288\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+16384\,x^4+24576\,x^5-8192\,x^3\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-12288\,x^4\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-128\,x\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-192\,x\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+6\,x\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+15\,x\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^5-4096\,x^3\,{\mathrm {e}}^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-12288\,x^4\,{\mathrm {e}}^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+1536\,x^2\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+1536\,x^2\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3+2304\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+3456\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-192\,x^2\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-384\,x^2\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}{4\,x^4\,{\left (3\,x+2\right )}^2\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}+\frac {\ln \left (x\right )\,\left (8\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+12288\,x^5-4096\,x^3\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-12288\,x^4\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-192\,x\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3+15\,x\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+1536\,x^2\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+3456\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-384\,x^2\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3\right )}{2\,x^4\,{\left (3\,x+2\right )}^2\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}}{\ln \left (x^2\right )}-\frac {8\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4-x\,\left (192\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-15\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4\right )+x^3\,\left (3456\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-4096\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )+x^2\,\left (1536\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-384\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3\right )+12288\,x^5-12288\,x^4\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}{16\,x^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+48\,x^5\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+36\,x^6\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}+\frac {128}{{\ln \left (x\right )}^4\,\left (3\,x+2\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}+\frac {8\,\left ({\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-128\,x^3+64\,x^2\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-12\,x\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2\right )}{x^3\,\ln \left (x\right )\,\left (3\,x+2\right )\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}+\frac {16\,\left (3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+32\,x^2-16\,x\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )}{x^2\,{\ln \left (x\right )}^2\,\left (3\,x+2\right )\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3}-\frac {128\,\left (2\,x-{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )}{x\,{\ln \left (x\right )}^3\,\left (3\,x+2\right )\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________