Integrand size = 194, antiderivative size = 26 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=-6+\log \left (\frac {e^x}{2 \log \left (\left (x+\left (9+4 x^2\right )^2\right )^4\right )}\right ) \]
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=x-\log \left (\log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )\right ) \]
Integrate[(-4 - 576*x - 256*x^3 + (81 + x + 72*x^2 + 16*x^4)*Log[43046721 + 2125764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732544*x^7 + 117586944*x^8 + 1244160*x^9 + 41805312*x^10 + 221184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16])/(( 81 + x + 72*x^2 + 16*x^4)*Log[43046721 + 2125764*x + 153094374*x^2 + 56690 28*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732544*x^7 + 11758 6944*x^8 + 1244160*x^9 + 41805312*x^10 + 221184*x^11 + 9289728*x^12 + 1638 4*x^13 + 1179648*x^14 + 65536*x^16]),x]
Time = 0.62 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {7239, 7293, 2009}
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.
| \(\displaystyle \int \frac {-256 x^3-576 x+\left (16 x^4+72 x^2+x+81\right ) \log \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^9+117586944 x^8+3732544 x^7+211678272 x^6+6298848 x^5+238155553 x^4+5669028 x^3+153094374 x^2+2125764 x+43046721\right )-4}{\left (16 x^4+72 x^2+x+81\right ) \log \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^9+117586944 x^8+3732544 x^7+211678272 x^6+6298848 x^5+238155553 x^4+5669028 x^3+153094374 x^2+2125764 x+43046721\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-256 x^3+\left (16 x^4+72 x^2+x+81\right ) \log \left (\left (16 x^4+72 x^2+x+81\right )^4\right )-576 x-4}{\left (16 x^4+72 x^2+x+81\right ) \log \left (\left (16 x^4+72 x^2+x+81\right )^4\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (1-\frac {4 \left (64 x^3+144 x+1\right )}{\left (16 x^4+72 x^2+x+81\right ) \log \left (\left (16 x^4+72 x^2+x+81\right )^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x-\log \left (\log \left (\left (16 x^4+72 x^2+x+81\right )^4\right )\right )\) |
Int[(-4 - 576*x - 256*x^3 + (81 + x + 72*x^2 + 16*x^4)*Log[43046721 + 2125 764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 211678 272*x^6 + 3732544*x^7 + 117586944*x^8 + 1244160*x^9 + 41805312*x^10 + 2211 84*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16])/((81 + x + 72*x^2 + 16*x^4)*Log[43046721 + 2125764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732544*x^7 + 117586944*x ^8 + 1244160*x^9 + 41805312*x^10 + 221184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16]),x]
3.2.23.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(23)=46\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15
| method | result | size |
| default | \(x -\ln \left (\ln \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^{9}+117586944 x^{8}+3732544 x^{7}+211678272 x^{6}+6298848 x^{5}+238155553 x^{4}+5669028 x^{3}+153094374 x^{2}+2125764 x +43046721\right )\right )\) | \(82\) |
| norman | \(x -\ln \left (\ln \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^{9}+117586944 x^{8}+3732544 x^{7}+211678272 x^{6}+6298848 x^{5}+238155553 x^{4}+5669028 x^{3}+153094374 x^{2}+2125764 x +43046721\right )\right )\) | \(82\) |
| risch | \(x -\ln \left (\ln \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^{9}+117586944 x^{8}+3732544 x^{7}+211678272 x^{6}+6298848 x^{5}+238155553 x^{4}+5669028 x^{3}+153094374 x^{2}+2125764 x +43046721\right )\right )\) | \(82\) |
| parallelrisch | \(x -\ln \left (\ln \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^{9}+117586944 x^{8}+3732544 x^{7}+211678272 x^{6}+6298848 x^{5}+238155553 x^{4}+5669028 x^{3}+153094374 x^{2}+2125764 x +43046721\right )\right )\) | \(82\) |
| parts | \(x -\ln \left (\ln \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^{9}+117586944 x^{8}+3732544 x^{7}+211678272 x^{6}+6298848 x^{5}+238155553 x^{4}+5669028 x^{3}+153094374 x^{2}+2125764 x +43046721\right )\right )\) | \(82\) |
int(((16*x^4+72*x^2+x+81)*ln(65536*x^16+1179648*x^14+16384*x^13+9289728*x^ 12+221184*x^11+41805312*x^10+1244160*x^9+117586944*x^8+3732544*x^7+2116782 72*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+2125764*x+43046 721)-256*x^3-576*x-4)/(16*x^4+72*x^2+x+81)/ln(65536*x^16+1179648*x^14+1638 4*x^13+9289728*x^12+221184*x^11+41805312*x^10+1244160*x^9+117586944*x^8+37 32544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^ 2+2125764*x+43046721),x,method=_RETURNVERBOSE)
x-ln(ln(65536*x^16+1179648*x^14+16384*x^13+9289728*x^12+221184*x^11+418053 12*x^10+1244160*x^9+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+23 8155553*x^4+5669028*x^3+153094374*x^2+2125764*x+43046721))
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=x - \log \left (\log \left (65536 \, x^{16} + 1179648 \, x^{14} + 16384 \, x^{13} + 9289728 \, x^{12} + 221184 \, x^{11} + 41805312 \, x^{10} + 1244160 \, x^{9} + 117586944 \, x^{8} + 3732544 \, x^{7} + 211678272 \, x^{6} + 6298848 \, x^{5} + 238155553 \, x^{4} + 5669028 \, x^{3} + 153094374 \, x^{2} + 2125764 \, x + 43046721\right )\right ) \]
integrate(((16*x^4+72*x^2+x+81)*log(65536*x^16+1179648*x^14+16384*x^13+928 9728*x^12+221184*x^11+41805312*x^10+1244160*x^9+117586944*x^8+3732544*x^7+ 211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+2125764* x+43046721)-256*x^3-576*x-4)/(16*x^4+72*x^2+x+81)/log(65536*x^16+1179648*x ^14+16384*x^13+9289728*x^12+221184*x^11+41805312*x^10+1244160*x^9+11758694 4*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+1530 94374*x^2+2125764*x+43046721),x, algorithm=\
x - log(log(65536*x^16 + 1179648*x^14 + 16384*x^13 + 9289728*x^12 + 221184 *x^11 + 41805312*x^10 + 1244160*x^9 + 117586944*x^8 + 3732544*x^7 + 211678 272*x^6 + 6298848*x^5 + 238155553*x^4 + 5669028*x^3 + 153094374*x^2 + 2125 764*x + 43046721))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=x - \log {\left (\log {\left (65536 x^{16} + 1179648 x^{14} + 16384 x^{13} + 9289728 x^{12} + 221184 x^{11} + 41805312 x^{10} + 1244160 x^{9} + 117586944 x^{8} + 3732544 x^{7} + 211678272 x^{6} + 6298848 x^{5} + 238155553 x^{4} + 5669028 x^{3} + 153094374 x^{2} + 2125764 x + 43046721 \right )} \right )} \]
integrate(((16*x**4+72*x**2+x+81)*ln(65536*x**16+1179648*x**14+16384*x**13 +9289728*x**12+221184*x**11+41805312*x**10+1244160*x**9+117586944*x**8+373 2544*x**7+211678272*x**6+6298848*x**5+238155553*x**4+5669028*x**3+15309437 4*x**2+2125764*x+43046721)-256*x**3-576*x-4)/(16*x**4+72*x**2+x+81)/ln(655 36*x**16+1179648*x**14+16384*x**13+9289728*x**12+221184*x**11+41805312*x** 10+1244160*x**9+117586944*x**8+3732544*x**7+211678272*x**6+6298848*x**5+23 8155553*x**4+5669028*x**3+153094374*x**2+2125764*x+43046721),x)
x - log(log(65536*x**16 + 1179648*x**14 + 16384*x**13 + 9289728*x**12 + 22 1184*x**11 + 41805312*x**10 + 1244160*x**9 + 117586944*x**8 + 3732544*x**7 + 211678272*x**6 + 6298848*x**5 + 238155553*x**4 + 5669028*x**3 + 1530943 74*x**2 + 2125764*x + 43046721))
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=x - \log \left (\log \left (16 \, x^{4} + 72 \, x^{2} + x + 81\right )\right ) \]
integrate(((16*x^4+72*x^2+x+81)*log(65536*x^16+1179648*x^14+16384*x^13+928 9728*x^12+221184*x^11+41805312*x^10+1244160*x^9+117586944*x^8+3732544*x^7+ 211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+2125764* x+43046721)-256*x^3-576*x-4)/(16*x^4+72*x^2+x+81)/log(65536*x^16+1179648*x ^14+16384*x^13+9289728*x^12+221184*x^11+41805312*x^10+1244160*x^9+11758694 4*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+1530 94374*x^2+2125764*x+43046721),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=x - \log \left (\log \left (65536 \, x^{16} + 1179648 \, x^{14} + 16384 \, x^{13} + 9289728 \, x^{12} + 221184 \, x^{11} + 41805312 \, x^{10} + 1244160 \, x^{9} + 117586944 \, x^{8} + 3732544 \, x^{7} + 211678272 \, x^{6} + 6298848 \, x^{5} + 238155553 \, x^{4} + 5669028 \, x^{3} + 153094374 \, x^{2} + 2125764 \, x + 43046721\right )\right ) \]
integrate(((16*x^4+72*x^2+x+81)*log(65536*x^16+1179648*x^14+16384*x^13+928 9728*x^12+221184*x^11+41805312*x^10+1244160*x^9+117586944*x^8+3732544*x^7+ 211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+2125764* x+43046721)-256*x^3-576*x-4)/(16*x^4+72*x^2+x+81)/log(65536*x^16+1179648*x ^14+16384*x^13+9289728*x^12+221184*x^11+41805312*x^10+1244160*x^9+11758694 4*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+1530 94374*x^2+2125764*x+43046721),x, algorithm=\
x - log(log(65536*x^16 + 1179648*x^14 + 16384*x^13 + 9289728*x^12 + 221184 *x^11 + 41805312*x^10 + 1244160*x^9 + 117586944*x^8 + 3732544*x^7 + 211678 272*x^6 + 6298848*x^5 + 238155553*x^4 + 5669028*x^3 + 153094374*x^2 + 2125 764*x + 43046721))
Time = 9.78 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16}\right )} \, dx=x-\ln \left (\ln \left (65536\,x^{16}+1179648\,x^{14}+16384\,x^{13}+9289728\,x^{12}+221184\,x^{11}+41805312\,x^{10}+1244160\,x^9+117586944\,x^8+3732544\,x^7+211678272\,x^6+6298848\,x^5+238155553\,x^4+5669028\,x^3+153094374\,x^2+2125764\,x+43046721\right )\right ) \]
int(-(576*x - log(2125764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732544*x^7 + 117586944*x^8 + 1244160*x^9 + 41805312*x^10 + 221184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16 + 43046721)*(x + 72*x^2 + 16*x^4 + 81) + 256*x^3 + 4)/(log(212 5764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 21167 8272*x^6 + 3732544*x^7 + 117586944*x^8 + 1244160*x^9 + 41805312*x^10 + 221 184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16 + 4304672 1)*(x + 72*x^2 + 16*x^4 + 81)),x)