\(\int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+(2 x^3-4 x^5+2 x^7+16 x \log (3)) \log (\frac {1}{64} (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+(16 x^2-32 x^4+16 x^6) \log (3)+64 \log ^2(3)))}{x^2-2 x^4+x^6+8 \log (3)+(3 x^2-6 x^4+3 x^6+24 \log (3)) \log (\frac {1}{64} (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+(16 x^2-32 x^4+16 x^6) \log (3)+64 \log ^2(3)))+(3 x^2-6 x^4+3 x^6+24 \log (3)) \log ^2(\frac {1}{64} (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+(16 x^2-32 x^4+16 x^6) \log (3)+64 \log ^2(3)))+(x^2-2 x^4+x^6+8 \log (3)) \log ^3(\frac {1}{64} (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+(16 x^2-32 x^4+16 x^6) \log (3)+64 \log ^2(3)))} \, dx\) [1775]

Optimal result

Integrand size = 333, antiderivative size = 27 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx=\frac {x^2}{\left (1+\log \left (\left (\frac {1}{8} \left (x-x^3\right )^2+\log (3)\right )^2\right )\right )^2} \] Output:

x^2/(ln((ln(3)+1/8*(-x^3+x)^2)^2)+1)^2
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx=\frac {x^2}{\left (1+\log \left (\frac {1}{64} \left (x^2-2 x^4+x^6+8 \log (3)\right )^2\right )\right )^2} \] Input:

Integrate[(-6*x^3 + 28*x^5 - 22*x^7 + 16*x*Log[3] + (2*x^3 - 4*x^5 + 2*x^7 
 + 16*x*Log[3])*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16*x^2 - 32*x^ 
4 + 16*x^6)*Log[3] + 64*Log[3]^2)/64])/(x^2 - 2*x^4 + x^6 + 8*Log[3] + (3* 
x^2 - 6*x^4 + 3*x^6 + 24*Log[3])*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 
+ (16*x^2 - 32*x^4 + 16*x^6)*Log[3] + 64*Log[3]^2)/64] + (3*x^2 - 6*x^4 + 
3*x^6 + 24*Log[3])*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16*x^2 - 32 
*x^4 + 16*x^6)*Log[3] + 64*Log[3]^2)/64]^2 + (x^2 - 2*x^4 + x^6 + 8*Log[3] 
)*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16*x^2 - 32*x^4 + 16*x^6)*Lo 
g[3] + 64*Log[3]^2)/64]^3),x]
 

Output:

x^2/(1 + Log[(x^2 - 2*x^4 + x^6 + 8*Log[3])^2/64])^2
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-22 x^7+28 x^5-6 x^3+\left (2 x^7-4 x^5+2 x^3+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^{12}-4 x^{10}+6 x^8-4 x^6+x^4+\left (16 x^6-32 x^4+16 x^2\right ) \log (3)+64 \log ^2(3)\right )\right )+16 x \log (3)}{x^6-2 x^4+x^2+\left (3 x^6-6 x^4+3 x^2+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^{12}-4 x^{10}+6 x^8-4 x^6+x^4+\left (16 x^6-32 x^4+16 x^2\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^6-6 x^4+3 x^2+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^{12}-4 x^{10}+6 x^8-4 x^6+x^4+\left (16 x^6-32 x^4+16 x^2\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^6-2 x^4+x^2+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^{12}-4 x^{10}+6 x^8-4 x^6+x^4+\left (16 x^6-32 x^4+16 x^2\right ) \log (3)+64 \log ^2(3)\right )\right )+8 \log (3)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 x \left (-11 x^6+14 x^4-3 x^2+\left (x^6-2 x^4+x^2+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+8 \log (3)\right )}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {x \left (11 x^6-14 x^4+3 x^2-\left (x^6-2 x^4+x^2+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )-8 \log (3)\right )}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {x \left (11 x^6-14 x^4+3 x^2-\left (x^6-2 x^4+x^2+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )-8 \log (3)\right )}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx\)

\(\Big \downarrow \) 7266

\(\displaystyle -\int \frac {11 x^6-14 x^4+3 x^2-\left (x^6-2 x^4+x^2+8 \log (3)\right ) \log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )-8 \log (3)}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx^2\)

\(\Big \downarrow \) 7293

\(\displaystyle -\int \left (\frac {4 x^2 \left (3 x^4-4 x^2+1\right )}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}-\frac {1}{\left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^2}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -12 \int \frac {1}{\left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx^2+96 \log (3) \int \frac {1}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx^2+8 \int \frac {x^2}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx^2-8 \int \frac {x^4}{\left (x^6-2 x^4+x^2+8 \log (3)\right ) \left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^3}dx^2+\int \frac {1}{\left (\log \left (\frac {1}{64} \left (x^6-2 x^4+x^2+8 \log (3)\right )^2\right )+1\right )^2}dx^2\)

Input:

Int[(-6*x^3 + 28*x^5 - 22*x^7 + 16*x*Log[3] + (2*x^3 - 4*x^5 + 2*x^7 + 16* 
x*Log[3])*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16*x^2 - 32*x^4 + 16 
*x^6)*Log[3] + 64*Log[3]^2)/64])/(x^2 - 2*x^4 + x^6 + 8*Log[3] + (3*x^2 - 
6*x^4 + 3*x^6 + 24*Log[3])*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16* 
x^2 - 32*x^4 + 16*x^6)*Log[3] + 64*Log[3]^2)/64] + (3*x^2 - 6*x^4 + 3*x^6 
+ 24*Log[3])*Log[(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16*x^2 - 32*x^4 + 
 16*x^6)*Log[3] + 64*Log[3]^2)/64]^2 + (x^2 - 2*x^4 + x^6 + 8*Log[3])*Log[ 
(x^4 - 4*x^6 + 6*x^8 - 4*x^10 + x^12 + (16*x^2 - 32*x^4 + 16*x^6)*Log[3] + 
 64*Log[3]^2)/64]^3),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).

Time = 4.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22

method	result	size
risch	\(\frac {x^{2}}{{\left (\ln \left (\ln \left (3\right )^{2}+\frac {\left (16 x^{6}-32 x^{4}+16 x^{2}\right ) \ln \left (3\right )}{64}+\frac {x^{12}}{64}-\frac {x^{10}}{16}+\frac {3 x^{8}}{32}-\frac {x^{6}}{16}+\frac {x^{4}}{64}\right )+1\right )}^{2}}\)	\(60\)
parallelrisch	\(\frac {x^{2}}{\ln \left (\ln \left (3\right )^{2}+\frac {\left (16 x^{6}-32 x^{4}+16 x^{2}\right ) \ln \left (3\right )}{64}+\frac {x^{12}}{64}-\frac {x^{10}}{16}+\frac {3 x^{8}}{32}-\frac {x^{6}}{16}+\frac {x^{4}}{64}\right )^{2}+2 \ln \left (\ln \left (3\right )^{2}+\frac {\left (16 x^{6}-32 x^{4}+16 x^{2}\right ) \ln \left (3\right )}{64}+\frac {x^{12}}{64}-\frac {x^{10}}{16}+\frac {3 x^{8}}{32}-\frac {x^{6}}{16}+\frac {x^{4}}{64}\right )+1}\)	\(115\)

Input:

int(((16*x*ln(3)+2*x^7-4*x^5+2*x^3)*ln(ln(3)^2+1/64*(16*x^6-32*x^4+16*x^2) 
*ln(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)+16*x*ln(3)-22*x^7+2 
8*x^5-6*x^3)/((8*ln(3)+x^6-2*x^4+x^2)*ln(ln(3)^2+1/64*(16*x^6-32*x^4+16*x^ 
2)*ln(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^3+(24*ln(3)+3*x^6 
-6*x^4+3*x^2)*ln(ln(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*ln(3)+1/64*x^12-1/16* 
x^10+3/32*x^8-1/16*x^6+1/64*x^4)^2+(24*ln(3)+3*x^6-6*x^4+3*x^2)*ln(ln(3)^2 
+1/64*(16*x^6-32*x^4+16*x^2)*ln(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1 
/64*x^4)+8*ln(3)+x^6-2*x^4+x^2),x,method=_RETURNVERBOSE)
 

Output:

x^2/(ln(ln(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*ln(3)+1/64*x^12-1/16*x^10+3/32 
*x^8-1/16*x^6+1/64*x^4)+1)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.93 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx=\frac {x^{2}}{\log \left (\frac {1}{64} \, x^{12} - \frac {1}{16} \, x^{10} + \frac {3}{32} \, x^{8} - \frac {1}{16} \, x^{6} + \frac {1}{64} \, x^{4} + \frac {1}{4} \, {\left (x^{6} - 2 \, x^{4} + x^{2}\right )} \log \left (3\right ) + \log \left (3\right )^{2}\right )^{2} + 2 \, \log \left (\frac {1}{64} \, x^{12} - \frac {1}{16} \, x^{10} + \frac {3}{32} \, x^{8} - \frac {1}{16} \, x^{6} + \frac {1}{64} \, x^{4} + \frac {1}{4} \, {\left (x^{6} - 2 \, x^{4} + x^{2}\right )} \log \left (3\right ) + \log \left (3\right )^{2}\right ) + 1} \] Input:

integrate(((16*x*log(3)+2*x^7-4*x^5+2*x^3)*log(log(3)^2+1/64*(16*x^6-32*x^ 
4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)+16*x*log( 
3)-22*x^7+28*x^5-6*x^3)/((8*log(3)+x^6-2*x^4+x^2)*log(log(3)^2+1/64*(16*x^ 
6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^3+ 
(24*log(3)+3*x^6-6*x^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log 
(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^2+(24*log(3)+3*x^6-6*x 
^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x 
^10+3/32*x^8-1/16*x^6+1/64*x^4)+8*log(3)+x^6-2*x^4+x^2),x, algorithm="fric 
as")
 

Output:

x^2/(log(1/64*x^12 - 1/16*x^10 + 3/32*x^8 - 1/16*x^6 + 1/64*x^4 + 1/4*(x^6 
 - 2*x^4 + x^2)*log(3) + log(3)^2)^2 + 2*log(1/64*x^12 - 1/16*x^10 + 3/32* 
x^8 - 1/16*x^6 + 1/64*x^4 + 1/4*(x^6 - 2*x^4 + x^2)*log(3) + log(3)^2) + 1 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.15 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {x^{12}}{64} - \frac {x^{10}}{16} + \frac {3 x^{8}}{32} - \frac {x^{6}}{16} + \frac {x^{4}}{64} + \left (\frac {x^{6}}{4} - \frac {x^{4}}{2} + \frac {x^{2}}{4}\right ) \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )}^{2} + 2 \log {\left (\frac {x^{12}}{64} - \frac {x^{10}}{16} + \frac {3 x^{8}}{32} - \frac {x^{6}}{16} + \frac {x^{4}}{64} + \left (\frac {x^{6}}{4} - \frac {x^{4}}{2} + \frac {x^{2}}{4}\right ) \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} + 1} \] Input:

integrate(((16*x*ln(3)+2*x**7-4*x**5+2*x**3)*ln(ln(3)**2+1/64*(16*x**6-32* 
x**4+16*x**2)*ln(3)+1/64*x**12-1/16*x**10+3/32*x**8-1/16*x**6+1/64*x**4)+1 
6*x*ln(3)-22*x**7+28*x**5-6*x**3)/((8*ln(3)+x**6-2*x**4+x**2)*ln(ln(3)**2+ 
1/64*(16*x**6-32*x**4+16*x**2)*ln(3)+1/64*x**12-1/16*x**10+3/32*x**8-1/16* 
x**6+1/64*x**4)**3+(24*ln(3)+3*x**6-6*x**4+3*x**2)*ln(ln(3)**2+1/64*(16*x* 
*6-32*x**4+16*x**2)*ln(3)+1/64*x**12-1/16*x**10+3/32*x**8-1/16*x**6+1/64*x 
**4)**2+(24*ln(3)+3*x**6-6*x**4+3*x**2)*ln(ln(3)**2+1/64*(16*x**6-32*x**4+ 
16*x**2)*ln(3)+1/64*x**12-1/16*x**10+3/32*x**8-1/16*x**6+1/64*x**4)+8*ln(3 
)+x**6-2*x**4+x**2),x)
 

Output:

x**2/(log(x**12/64 - x**10/16 + 3*x**8/32 - x**6/16 + x**4/64 + (x**6/4 - 
x**4/2 + x**2/4)*log(3) + log(3)**2)**2 + 2*log(x**12/64 - x**10/16 + 3*x* 
*8/32 - x**6/16 + x**4/64 + (x**6/4 - x**4/2 + x**2/4)*log(3) + log(3)**2) 
 + 1)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27) = 54\).

Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx=\frac {x^{2}}{36 \, \log \left (2\right )^{2} - 4 \, {\left (6 \, \log \left (2\right ) - 1\right )} \log \left (x^{6} - 2 \, x^{4} + x^{2} + 8 \, \log \left (3\right )\right ) + 4 \, \log \left (x^{6} - 2 \, x^{4} + x^{2} + 8 \, \log \left (3\right )\right )^{2} - 12 \, \log \left (2\right ) + 1} \] Input:

integrate(((16*x*log(3)+2*x^7-4*x^5+2*x^3)*log(log(3)^2+1/64*(16*x^6-32*x^ 
4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)+16*x*log( 
3)-22*x^7+28*x^5-6*x^3)/((8*log(3)+x^6-2*x^4+x^2)*log(log(3)^2+1/64*(16*x^ 
6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^3+ 
(24*log(3)+3*x^6-6*x^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log 
(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^2+(24*log(3)+3*x^6-6*x 
^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x 
^10+3/32*x^8-1/16*x^6+1/64*x^4)+8*log(3)+x^6-2*x^4+x^2),x, algorithm="maxi 
ma")
 

Output:

x^2/(36*log(2)^2 - 4*(6*log(2) - 1)*log(x^6 - 2*x^4 + x^2 + 8*log(3)) + 4* 
log(x^6 - 2*x^4 + x^2 + 8*log(3))^2 - 12*log(2) + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (27) = 54\).

Time = 4.49 (sec) , antiderivative size = 573, normalized size of antiderivative = 21.22 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx =\text {Too large to display} \] Input:

integrate(((16*x*log(3)+2*x^7-4*x^5+2*x^3)*log(log(3)^2+1/64*(16*x^6-32*x^ 
4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)+16*x*log( 
3)-22*x^7+28*x^5-6*x^3)/((8*log(3)+x^6-2*x^4+x^2)*log(log(3)^2+1/64*(16*x^ 
6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^3+ 
(24*log(3)+3*x^6-6*x^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log 
(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^2+(24*log(3)+3*x^6-6*x 
^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x 
^10+3/32*x^8-1/16*x^6+1/64*x^4)+8*log(3)+x^6-2*x^4+x^2),x, algorithm="giac 
")
 

Output:

(3*x^7 - 4*x^5 + x^3)/(108*x^5*log(2)^2 - 36*x^5*log(2)*log(x^12 - 4*x^10 
+ 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32*x^4*log(3) + x^4 + 16*x^2*log(3) + 64 
*log(3)^2) + 3*x^5*log(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32* 
x^4*log(3) + x^4 + 16*x^2*log(3) + 64*log(3)^2)^2 - 36*x^5*log(2) + 6*x^5* 
log(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32*x^4*log(3) + x^4 + 
16*x^2*log(3) + 64*log(3)^2) + 3*x^5 - 144*x^3*log(2)^2 + 48*x^3*log(2)*lo 
g(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32*x^4*log(3) + x^4 + 16 
*x^2*log(3) + 64*log(3)^2) - 4*x^3*log(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log( 
3) - 4*x^6 - 32*x^4*log(3) + x^4 + 16*x^2*log(3) + 64*log(3)^2)^2 + 48*x^3 
*log(2) - 8*x^3*log(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32*x^4 
*log(3) + x^4 + 16*x^2*log(3) + 64*log(3)^2) - 4*x^3 + 36*x*log(2)^2 - 12* 
x*log(2)*log(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32*x^4*log(3) 
 + x^4 + 16*x^2*log(3) + 64*log(3)^2) + x*log(x^12 - 4*x^10 + 6*x^8 + 16*x 
^6*log(3) - 4*x^6 - 32*x^4*log(3) + x^4 + 16*x^2*log(3) + 64*log(3)^2)^2 - 
 12*x*log(2) + 2*x*log(x^12 - 4*x^10 + 6*x^8 + 16*x^6*log(3) - 4*x^6 - 32* 
x^4*log(3) + x^4 + 16*x^2*log(3) + 64*log(3)^2) + x)
 

Mupad [B] (verification not implemented)

Time = 4.62 (sec) , antiderivative size = 562, normalized size of antiderivative = 20.81 \[ \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=\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} \] Input:

int((log(log(3)^2 + (log(3)*(16*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/ 
16 + (3*x^8)/32 - x^10/16 + x^12/64)*(16*x*log(3) + 2*x^3 - 4*x^5 + 2*x^7) 
 + 16*x*log(3) - 6*x^3 + 28*x^5 - 22*x^7)/(8*log(3) + log(log(3)^2 + (log( 
3)*(16*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + (3*x^8)/32 - x^10/16 
 + x^12/64)^2*(24*log(3) + 3*x^2 - 6*x^4 + 3*x^6) + log(log(3)^2 + (log(3) 
*(16*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + (3*x^8)/32 - x^10/16 + 
 x^12/64)^3*(8*log(3) + x^2 - 2*x^4 + x^6) + log(log(3)^2 + (log(3)*(16*x^ 
2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + (3*x^8)/32 - x^10/16 + x^12/6 
4)*(24*log(3) + 3*x^2 - 6*x^4 + 3*x^6) + x^2 - 2*x^4 + x^6),x)
 

Output:

x^2/72 - ((8*log(3) - 3*x^2 + 14*x^4 - 11*x^6)/(4*(3*x^4 - 4*x^2 + 1)) + ( 
log(log(3)^2 + (log(3)*(16*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + 
(3*x^8)/32 - x^10/16 + x^12/64)*(8*log(3) + x^2 - 2*x^4 + x^6))/(4*(3*x^4 
- 4*x^2 + 1)))/(2*log(log(3)^2 + (log(3)*(16*x^2 - 32*x^4 + 16*x^6))/64 + 
x^4/64 - x^6/16 + (3*x^8)/32 - x^10/16 + x^12/64) + log(log(3)^2 + (log(3) 
*(16*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + (3*x^8)/32 - x^10/16 + 
 x^12/64)^2 + 1) + (9*log(3) - 432*x^2*log(3)^2 - x^8*(27*log(3) + 1/4) - 
x^4*(54*log(3) + 1/2) + x^6*(72*log(3) + 2/3) + 288*log(3)^2 + 1/12)/(513* 
x^4 - 108*x^2 - 1224*x^6 + 1539*x^8 - 972*x^10 + 243*x^12 + 9) + (((8*log( 
3) + x^2 - 2*x^4 + x^6)*(48*x^2*log(3) - 32*log(3) - 12*x^2 + 36*x^4 - 40* 
x^6 + 15*x^8 + 1))/(8*(3*x^4 - 4*x^2 + 1)^3) - (log(log(3)^2 + (log(3)*(16 
*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + (3*x^8)/32 - x^10/16 + x^1 
2/64)*(8*log(3) + x^2 - 2*x^4 + x^6)*(32*log(3) - 48*x^2*log(3) - 4*x^2 + 
8*x^4 - 8*x^6 + 3*x^8 + 1))/(8*(3*x^4 - 4*x^2 + 1)^3))/(log(log(3)^2 + (lo 
g(3)*(16*x^2 - 32*x^4 + 16*x^6))/64 + x^4/64 - x^6/16 + (3*x^8)/32 - x^10/ 
16 + x^12/64) + 1)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.30 \[ \int \frac {-6 x^3+28 x^5-22 x^7+16 x \log (3)+\left (2 x^3-4 x^5+2 x^7+16 x \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )}{x^2-2 x^4+x^6+8 \log (3)+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log \left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (3 x^2-6 x^4+3 x^6+24 \log (3)\right ) \log ^2\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )+\left (x^2-2 x^4+x^6+8 \log (3)\right ) \log ^3\left (\frac {1}{64} \left (x^4-4 x^6+6 x^8-4 x^{10}+x^{12}+\left (16 x^2-32 x^4+16 x^6\right ) \log (3)+64 \log ^2(3)\right )\right )} \, dx=\frac {x^{2}}{\mathrm {log}\left (\mathrm {log}\left (3\right )^{2}+\frac {\mathrm {log}\left (3\right ) x^{6}}{4}-\frac {\mathrm {log}\left (3\right ) x^{4}}{2}+\frac {\mathrm {log}\left (3\right ) x^{2}}{4}+\frac {x^{12}}{64}-\frac {x^{10}}{16}+\frac {3 x^{8}}{32}-\frac {x^{6}}{16}+\frac {x^{4}}{64}\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (3\right )^{2}+\frac {\mathrm {log}\left (3\right ) x^{6}}{4}-\frac {\mathrm {log}\left (3\right ) x^{4}}{2}+\frac {\mathrm {log}\left (3\right ) x^{2}}{4}+\frac {x^{12}}{64}-\frac {x^{10}}{16}+\frac {3 x^{8}}{32}-\frac {x^{6}}{16}+\frac {x^{4}}{64}\right )+1} \] Input:

int(((16*x*log(3)+2*x^7-4*x^5+2*x^3)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x 
^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)+16*x*log(3)-22* 
x^7+28*x^5-6*x^3)/((8*log(3)+x^6-2*x^4+x^2)*log(log(3)^2+1/64*(16*x^6-32*x 
^4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^3+(24*lo 
g(3)+3*x^6-6*x^4+3*x^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log(3)+1/ 
64*x^12-1/16*x^10+3/32*x^8-1/16*x^6+1/64*x^4)^2+(24*log(3)+3*x^6-6*x^4+3*x 
^2)*log(log(3)^2+1/64*(16*x^6-32*x^4+16*x^2)*log(3)+1/64*x^12-1/16*x^10+3/ 
32*x^8-1/16*x^6+1/64*x^4)+8*log(3)+x^6-2*x^4+x^2),x)
 

Output:

x**2/(log((64*log(3)**2 + 16*log(3)*x**6 - 32*log(3)*x**4 + 16*log(3)*x**2 
 + x**12 - 4*x**10 + 6*x**8 - 4*x**6 + x**4)/64)**2 + 2*log((64*log(3)**2 
+ 16*log(3)*x**6 - 32*log(3)*x**4 + 16*log(3)*x**2 + x**12 - 4*x**10 + 6*x 
**8 - 4*x**6 + x**4)/64) + 1)