3.32.23 \(\int \frac {-4-576 x-256 x^3+(81+x+72 x^2+16 x^4) \log (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16})}{(81+x+72 x^2+16 x^4) \log (43046721+2125764 x+153094374 x^2+5669028 x^3+238155553 x^4+6298848 x^5+211678272 x^6+3732544 x^7+117586944 x^8+1244160 x^9+41805312 x^{10}+221184 x^{11}+9289728 x^{12}+16384 x^{13}+1179648 x^{14}+65536 x^{16})} \, dx\)
Optimal. Leaf size=26 \[ -6+\log \left (\frac {e^x}{2 \log \left (\left (x+\left (9+4 x^2\right )^2\right )^4\right )}\right ) \]
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Rubi [A] time = 0.37, antiderivative size = 21, normalized size of antiderivative = 0.81,
number of steps used = 4, number of rules used = 3, integrand size = 194, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used
= {6688, 6742, 6684} \begin {gather*} x-\log \left (\log \left (\left (16 x^4+72 x^2+x+81\right )^4\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-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 + 22
1184*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 + 1175869
44*x^8 + 1244160*x^9 + 41805312*x^10 + 221184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16]),x
]
[Out]
x - Log[Log[(81 + x + 72*x^2 + 16*x^4)^4]]
Rule 6684
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !Fa
lseQ[q]]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-576 x-256 x^3+\left (81+x+72 x^2+16 x^4\right ) \log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )} \, dx\\ &=\int \left (1-\frac {4 \left (1+144 x+64 x^3\right )}{\left (81+x+72 x^2+16 x^4\right ) \log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )}\right ) \, dx\\ &=x-4 \int \frac {1+144 x+64 x^3}{\left (81+x+72 x^2+16 x^4\right ) \log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )} \, dx\\ &=x-\log \left (\log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 21, normalized size = 0.81 \begin {gather*} x-\log \left (\log \left (\left (81+x+72 x^2+16 x^4\right )^4\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-4 - 576*x - 256*x^3 + (81 + x + 72*x^2 + 16*x^4)*Log[43046721 + 2125764*x + 153094374*x^2 + 566902
8*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732544*x^7 + 117586944*x^8 + 1244160*x^9 + 41805312*x^1
0 + 221184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^16])/((81 + x + 72*x^2 + 16*x^4)*Log[4304
6721 + 2125764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732544*x^7 + 1
17586944*x^8 + 1244160*x^9 + 41805312*x^10 + 221184*x^11 + 9289728*x^12 + 16384*x^13 + 1179648*x^14 + 65536*x^
16]),x]
[Out]
x - Log[Log[(81 + x + 72*x^2 + 16*x^4)^4]]
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fricas [B] time = 0.94, size = 81, normalized size = 3.12 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((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+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+212576
4*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+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+
153094374*x^2+2125764*x+43046721),x, algorithm="fricas")
[Out]
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 + 211678272*x^6 + 6298848*x^5 + 238155553*x^4 + 5669028*x^3 + 153094374*x^2 + 21
25764*x + 43046721))
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giac [B] time = 0.29, size = 81, normalized size = 3.12 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((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+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+212576
4*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+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+
153094374*x^2+2125764*x+43046721),x, algorithm="giac")
[Out]
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 + 211678272*x^6 + 6298848*x^5 + 238155553*x^4 + 5669028*x^3 + 153094374*x^2 + 21
25764*x + 43046721))
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maple [B] time = 0.08, size = 82, normalized size = 3.15
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| method |
result |
size |
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| 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 |
\(x -\ln \left (\ln \left (65536 x^{16}+1179648 x^{14}+16384 x^{13}+9289728 x^{12}+221184 x^{11}+41805312 x^{10}+1244160 x^{9}+117586944 x^{8}+3732544 x^{7}+211678272 x^{6}+6298848 x^{5}+238155553 x^{4}+5669028 x^{3}+153094374 x^{2}+2125764 x +43046721\right )\right )\) |
\(82\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
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+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+2125764*x+430
46721)-256*x^3-576*x-4)/(16*x^4+72*x^2+x+81)/ln(65536*x^16+1179648*x^14+16384*x^13+9289728*x^12+221184*x^11+41
805312*x^10+1244160*x^9+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+15309437
4*x^2+2125764*x+43046721),x,method=_RETURNVERBOSE)
[Out]
x-ln(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+37
32544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+2125764*x+43046721))
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maxima [A] time = 0.78, size = 19, normalized size = 0.73 \begin {gather*} x - \log \left (\log \left (16 \, x^{4} + 72 \, x^{2} + x + 81\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((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+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+153094374*x^2+212576
4*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+117586944*x^8+3732544*x^7+211678272*x^6+6298848*x^5+238155553*x^4+5669028*x^3+
153094374*x^2+2125764*x+43046721),x, algorithm="maxima")
[Out]
x - log(log(16*x^4 + 72*x^2 + x + 81))
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mupad [B] time = 2.01, size = 81, normalized size = 3.12 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(576*x - log(2125764*x + 153094374*x^2 + 5669028*x^3 + 238155553*x^4 + 6298848*x^5 + 211678272*x^6 + 3732
544*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(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)),x
)
[Out]
x - log(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 + 655
36*x^16 + 43046721))
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sympy [B] time = 0.30, size = 80, normalized size = 3.08 \begin {gather*} 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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+3732544*x**7+211678272*x**6+6298848*x**5+238155553*x**4+5669028*x**3+153094
374*x**2+2125764*x+43046721)-256*x**3-576*x-4)/(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+211678272*x**6+6298848*x**
5+238155553*x**4+5669028*x**3+153094374*x**2+2125764*x+43046721),x)
[Out]
x - log(log(65536*x**16 + 1179648*x**14 + 16384*x**13 + 9289728*x**12 + 221184*x**11 + 41805312*x**10 + 124416
0*x**9 + 117586944*x**8 + 3732544*x**7 + 211678272*x**6 + 6298848*x**5 + 238155553*x**4 + 5669028*x**3 + 15309
4374*x**2 + 2125764*x + 43046721))
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