Optimal. Leaf size=27 \[ \frac {x^2}{\left (1+\log \left (\left (\frac {1}{8} \left (x-x^3\right )^2+\log (3)\right )^2\right )\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.95, 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 {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (-3 x^2+14 x^4-11 x^6+8 \log (3)+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x^2-2 x^4+x^6+8 \log (3)\right )^2\right )\right )}{\left (x^2-2 x^4+x^6+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x^2-2 x^4+x^6+8 \log (3)\right )^2\right )\right )^3} \, dx\\ &=2 \int \frac {x \left (-3 x^2+14 x^4-11 x^6+8 \log (3)+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x^2-2 x^4+x^6+8 \log (3)\right )^2\right )\right )}{\left (x^2-2 x^4+x^6+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x^2-2 x^4+x^6+8 \log (3)\right )^2\right )\right )^3} \, dx\\ &=\text {Subst}\left (\int \frac {-3 x+14 x^2-11 x^3+8 \log (3)+\left (x-2 x^2+x^3+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )\\ &=\text {Subst}\left (\int \left (-\frac {4 x \left (1-4 x+3 x^2\right )}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3}+\frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^2}\right ) \, dx,x,x^2\right )\\ &=-\left (4 \text {Subst}\left (\int \frac {x \left (1-4 x+3 x^2\right )}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )\right )+\text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^2} \, dx,x,x^2\right )\\ &=-\left (4 \text {Subst}\left (\int \left (\frac {3}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3}+\frac {2 \left (-x+x^2-12 \log (3)\right )}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3}\right ) \, dx,x,x^2\right )\right )+\text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^2} \, dx,x,x^2\right )\\ &=-\left (8 \text {Subst}\left (\int \frac {-x+x^2-12 \log (3)}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )\right )-12 \text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^2} \, dx,x,x^2\right )\\ &=-\left (8 \text {Subst}\left (\int \left (-\frac {x}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3}+\frac {x^2}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3}-\frac {12 \log (3)}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3}\right ) \, dx,x,x^2\right )\right )-12 \text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^2} \, dx,x,x^2\right )\\ &=8 \text {Subst}\left (\int \frac {x}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )-8 \text {Subst}\left (\int \frac {x^2}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )-12 \text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )+(96 \log (3)) \text {Subst}\left (\int \frac {1}{\left (x-2 x^2+x^3+8 \log (3)\right ) \left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^3} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {1}{\left (1+\log \left (\frac {1}{64} \left (x-2 x^2+x^3+8 \log (3)\right )^2\right )\right )^2} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 31, normalized size = 1.15 \begin {gather*} \frac {x^2}{\left (1+\log \left (\frac {1}{64} \left (x^2-2 x^4+x^6+8 \log (3)\right )^2\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs.
\(2(25)=50\).
time = 25.23, size = 60, normalized size = 2.22
method | result | size |
risch | \(\frac {x^{2}}{\left (\ln \left (\ln \left (3\right )^{2}+\frac {\left (16 x^{6}-32 x^{4}+16 x^{2}\right ) \ln \left (3\right )}{64}+\frac {x^{12}}{64}-\frac {x^{10}}{16}+\frac {3 x^{8}}{32}-\frac {x^{6}}{16}+\frac {x^{4}}{64}\right )+1\right )^{2}}\) | \(60\) |
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. 64 vs.
\(2 (27) = 54\).
time = 0.57, size = 64, normalized size = 2.37 \begin {gather*} \frac {x^{2}}{36 \, \log \left (2\right )^{2} - 4 \, {\left (6 \, \log \left (2\right ) - 1\right )} \log \left (x^{6} - 2 \, x^{4} + x^{2} + 8 \, \log \left (3\right )\right ) + 4 \, \log \left (x^{6} - 2 \, x^{4} + x^{2} + 8 \, \log \left (3\right )\right )^{2} - 12 \, \log \left (2\right ) + 1} \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. 106 vs.
\(2 (27) = 54\).
time = 0.35, size = 106, normalized size = 3.93 \begin {gather*} \frac {x^{2}}{\log \left (\frac {1}{64} \, x^{12} - \frac {1}{16} \, x^{10} + \frac {3}{32} \, x^{8} - \frac {1}{16} \, x^{6} + \frac {1}{64} \, x^{4} + \frac {1}{4} \, {\left (x^{6} - 2 \, x^{4} + x^{2}\right )} \log \left (3\right ) + \log \left (3\right )^{2}\right )^{2} + 2 \, \log \left (\frac {1}{64} \, x^{12} - \frac {1}{16} \, x^{10} + \frac {3}{32} \, x^{8} - \frac {1}{16} \, x^{6} + \frac {1}{64} \, x^{4} + \frac {1}{4} \, {\left (x^{6} - 2 \, x^{4} + x^{2}\right )} \log \left (3\right ) + \log \left (3\right )^{2}\right ) + 1} \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. 112 vs.
\(2 (20) = 40\).
time = 0.14, size = 112, normalized size = 4.15 \begin {gather*} \frac {x^{2}}{\log {\left (\frac {x^{12}}{64} - \frac {x^{10}}{16} + \frac {3 x^{8}}{32} - \frac {x^{6}}{16} + \frac {x^{4}}{64} + \left (\frac {x^{6}}{4} - \frac {x^{4}}{2} + \frac {x^{2}}{4}\right ) \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )}^{2} + 2 \log {\left (\frac {x^{12}}{64} - \frac {x^{10}}{16} + \frac {3 x^{8}}{32} - \frac {x^{6}}{16} + \frac {x^{4}}{64} + \left (\frac {x^{6}}{4} - \frac {x^{4}}{2} + \frac {x^{2}}{4}\right ) \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} + 1} \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. 573 vs.
\(2 (27) = 54\).
time = 4.22, size = 573, normalized size = 21.22 \begin {gather*} \frac {3 \, x^{7} - 4 \, x^{5} + x^{3}}{108 \, x^{5} \log \left (2\right )^{2} - 36 \, x^{5} \log \left (2\right ) \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right ) + 3 \, x^{5} \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right )^{2} - 36 \, x^{5} \log \left (2\right ) + 6 \, x^{5} \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right ) + 3 \, x^{5} - 144 \, x^{3} \log \left (2\right )^{2} + 48 \, x^{3} \log \left (2\right ) \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right ) - 4 \, x^{3} \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right )^{2} + 48 \, x^{3} \log \left (2\right ) - 8 \, x^{3} \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right ) - 4 \, x^{3} + 36 \, x \log \left (2\right )^{2} - 12 \, x \log \left (2\right ) \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right ) + x \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right )^{2} - 12 \, x \log \left (2\right ) + 2 \, x \log \left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + 16 \, x^{6} \log \left (3\right ) - 4 \, x^{6} - 32 \, x^{4} \log \left (3\right ) + x^{4} + 16 \, x^{2} \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right ) + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.00, size = 562, normalized size = 20.81 \begin {gather*} \frac {x^2}{72}-\frac {\frac {-11\,x^6+14\,x^4-3\,x^2+8\,\ln \left (3\right )}{4\,\left (3\,x^4-4\,x^2+1\right )}+\frac {\ln \left ({\ln \left (3\right )}^2+\frac {\ln \left (3\right )\,\left (16\,x^6-32\,x^4+16\,x^2\right )}{64}+\frac {x^4}{64}-\frac {x^6}{16}+\frac {3\,x^8}{32}-\frac {x^{10}}{16}+\frac {x^{12}}{64}\right )\,\left (x^6-2\,x^4+x^2+8\,\ln \left (3\right )\right )}{4\,\left (3\,x^4-4\,x^2+1\right )}}{{\ln \left ({\ln \left (3\right )}^2+\frac {\ln \left (3\right )\,\left (16\,x^6-32\,x^4+16\,x^2\right )}{64}+\frac {x^4}{64}-\frac {x^6}{16}+\frac {3\,x^8}{32}-\frac {x^{10}}{16}+\frac {x^{12}}{64}\right )}^2+2\,\ln \left ({\ln \left (3\right )}^2+\frac {\ln \left (3\right )\,\left (16\,x^6-32\,x^4+16\,x^2\right )}{64}+\frac {x^4}{64}-\frac {x^6}{16}+\frac {3\,x^8}{32}-\frac {x^{10}}{16}+\frac {x^{12}}{64}\right )+1}+\frac {\left (-27\,\ln \left (3\right )-\frac {1}{4}\right )\,x^8+\left (72\,\ln \left (3\right )+\frac {2}{3}\right )\,x^6+\left (-54\,\ln \left (3\right )-\frac {1}{2}\right )\,x^4-432\,{\ln \left (3\right )}^2\,x^2+9\,\ln \left (3\right )+288\,{\ln \left (3\right )}^2+\frac {1}{12}}{243\,x^{12}-972\,x^{10}+1539\,x^8-1224\,x^6+513\,x^4-108\,x^2+9}+\frac {\frac {\left (x^6-2\,x^4+x^2+8\,\ln \left (3\right )\right )\,\left (48\,x^2\,\ln \left (3\right )-32\,\ln \left (3\right )-12\,x^2+36\,x^4-40\,x^6+15\,x^8+1\right )}{8\,{\left (3\,x^4-4\,x^2+1\right )}^3}-\frac {\ln \left ({\ln \left (3\right )}^2+\frac {\ln \left (3\right )\,\left (16\,x^6-32\,x^4+16\,x^2\right )}{64}+\frac {x^4}{64}-\frac {x^6}{16}+\frac {3\,x^8}{32}-\frac {x^{10}}{16}+\frac {x^{12}}{64}\right )\,\left (x^6-2\,x^4+x^2+8\,\ln \left (3\right )\right )\,\left (32\,\ln \left (3\right )-48\,x^2\,\ln \left (3\right )-4\,x^2+8\,x^4-8\,x^6+3\,x^8+1\right )}{8\,{\left (3\,x^4-4\,x^2+1\right )}^3}}{\ln \left ({\ln \left (3\right )}^2+\frac {\ln \left (3\right )\,\left (16\,x^6-32\,x^4+16\,x^2\right )}{64}+\frac {x^4}{64}-\frac {x^6}{16}+\frac {3\,x^8}{32}-\frac {x^{10}}{16}+\frac {x^{12}}{64}\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________