\(\int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} (-648 x-648 x^2-192 x^3-16 x^4) \log (\frac {27+18 x+2 x^2}{9+6 x+x^2})}{(81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7) \log (\frac {27+18 x+2 x^2}{9+6 x+x^2})} \, dx\) [5067]

Optimal result

Integrand size = 134, antiderivative size = 31 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^4}{1+x^2}}+\log \left (2 \log \left (3-\frac {x^2}{(3+x)^2}\right )\right ) \]

[Out]

ln(2*ln(3-x^2/(3+x)^2))+exp(4*exp(4)/(x^2+1))

Rubi [F]

\[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=\int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx \]

[In]

Int[(-6*x - 12*x^3 - 6*x^5 + E^(4 + (4*E^4)/(1 + x^2))*(-648*x - 648*x^2 - 192*x^3 - 16*x^4)*Log[(27 + 18*x +
2*x^2)/(9 + 6*x + x^2)])/((81 + 81*x + 186*x^2 + 164*x^3 + 129*x^4 + 85*x^5 + 24*x^6 + 2*x^7)*Log[(27 + 18*x +
2*x^2)/(9 + 6*x + x^2)]),x]

[Out]

E^((4*E^4)/(1 + x^2)) - 4*Sqrt[3]*Defer[Int][1/((-18 + 6*Sqrt[3] - 4*x)*Log[(27 + 18*x + 2*x^2)/(3 + x)^2]), x
] - 2*Defer[Int][1/((3 + x)*Log[(27 + 18*x + 2*x^2)/(3 + x)^2]), x] + 4*(1 - Sqrt[3])*Defer[Int][1/((18 - 6*Sq
rt[3] + 4*x)*Log[(27 + 18*x + 2*x^2)/(3 + x)^2]), x] - 4*Sqrt[3]*Defer[Int][1/((18 + 6*Sqrt[3] + 4*x)*Log[(27
+ 18*x + 2*x^2)/(3 + x)^2]), x] + 4*(1 + Sqrt[3])*Defer[Int][1/((18 + 6*Sqrt[3] + 4*x)*Log[(27 + 18*x + 2*x^2)
/(3 + x)^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 e^{4+\frac {4 e^4}{1+x^2}} x}{\left (1+x^2\right )^2}-\frac {6 x}{\left (81+81 x+24 x^2+2 x^3\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx \\ & = -\left (6 \int \frac {x}{\left (81+81 x+24 x^2+2 x^3\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx\right )-8 \int \frac {e^{4+\frac {4 e^4}{1+x^2}} x}{\left (1+x^2\right )^2} \, dx \\ & = e^{\frac {4 e^4}{1+x^2}}-6 \int \left (\frac {1}{3 (3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}+\frac {-9-2 x}{3 \left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx \\ & = e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx-2 \int \frac {-9-2 x}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx \\ & = e^{\frac {4 e^4}{1+x^2}}-2 \int \left (-\frac {9}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}-\frac {2 x}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx \\ & = e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+4 \int \frac {x}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+18 \int \frac {1}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx \\ & = e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+4 \int \left (\frac {1-\sqrt {3}}{\left (18-6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}+\frac {1+\sqrt {3}}{\left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx+18 \int \left (-\frac {2}{3 \sqrt {3} \left (-18+6 \sqrt {3}-4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}-\frac {2}{3 \sqrt {3} \left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx \\ & = e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx-\left (4 \sqrt {3}\right ) \int \frac {1}{\left (-18+6 \sqrt {3}-4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx-\left (4 \sqrt {3}\right ) \int \frac {1}{\left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+\left (4 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\left (18-6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+\left (4 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^4}{1+x^2}}+\log \left (\log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )\right ) \]

[In]

Integrate[(-6*x - 12*x^3 - 6*x^5 + E^(4 + (4*E^4)/(1 + x^2))*(-648*x - 648*x^2 - 192*x^3 - 16*x^4)*Log[(27 + 1
8*x + 2*x^2)/(9 + 6*x + x^2)])/((81 + 81*x + 186*x^2 + 164*x^3 + 129*x^4 + 85*x^5 + 24*x^6 + 2*x^7)*Log[(27 +
18*x + 2*x^2)/(9 + 6*x + x^2)]),x]

[Out]

E^((4*E^4)/(1 + x^2)) + Log[Log[(27 + 18*x + 2*x^2)/(3 + x)^2]]

Maple [A] (verified)

Time = 26.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03

method	result	size
default	\(\ln \left (\ln \left (2+\frac {6}{3+x}-\frac {9}{\left (3+x \right )^{2}}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\)	\(32\)
parts	\(\ln \left (\ln \left (2+\frac {6}{3+x}-\frac {9}{\left (3+x \right )^{2}}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\)	\(32\)
parallelrisch	\(\ln \left (\ln \left (\frac {2 x^{2}+18 x +27}{x^{2}+6 x +9}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\)	\(37\)
risch	\({\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}+\ln \left (\ln \left (x^{2}+9 x +\frac {27}{2}\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+9 x +\frac {27}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )}^{2}-\pi \operatorname {csgn}\left (i \left (3+x \right )\right )^{2} \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \left (3+x \right )\right ) \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )^{2}-\pi \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (i \left (x^{2}+9 x +\frac {27}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )}^{3}+2 i \ln \left (2\right )-4 i \ln \left (3+x \right )\right )}{2}\right )\)	\(223\)

[In]

int(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*ln((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5-12*x^3
-6*x)/(2*x^7+24*x^6+85*x^5+129*x^4+164*x^3+186*x^2+81*x+81)/ln((2*x^2+18*x+27)/(x^2+6*x+9)),x,method=_RETURNVE
RBOSE)

[Out]

ln(ln(2+6/(3+x)-9/(3+x)^2))+exp(4*exp(4)/(x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx={\left (e^{4} \log \left (\log \left (\frac {2 \, x^{2} + 18 \, x + 27}{x^{2} + 6 \, x + 9}\right )\right ) + e^{\left (\frac {4 \, {\left (x^{2} + e^{4} + 1\right )}}{x^{2} + 1}\right )}\right )} e^{\left (-4\right )} \]

[In]

integrate(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*log((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5
-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129*x^4+164*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x, algorit
hm="fricas")

[Out]

(e^4*log(log((2*x^2 + 18*x + 27)/(x^2 + 6*x + 9))) + e^(4*(x^2 + e^4 + 1)/(x^2 + 1)))*e^(-4)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^{4}}{x^{2} + 1}} + \log {\left (\log {\left (\frac {2 x^{2} + 18 x + 27}{x^{2} + 6 x + 9} \right )} \right )} \]

[In]

integrate(((-16*x**4-192*x**3-648*x**2-648*x)*exp(4)*exp(4*exp(4)/(x**2+1))*ln((2*x**2+18*x+27)/(x**2+6*x+9))-
6*x**5-12*x**3-6*x)/(2*x**7+24*x**6+85*x**5+129*x**4+164*x**3+186*x**2+81*x+81)/ln((2*x**2+18*x+27)/(x**2+6*x+
9)),x)

[Out]

exp(4*exp(4)/(x**2 + 1)) + log(log((2*x**2 + 18*x + 27)/(x**2 + 6*x + 9)))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\left (\frac {4 \, e^{4}}{x^{2} + 1}\right )} + \log \left (\log \left (2 \, x^{2} + 18 \, x + 27\right ) - 2 \, \log \left (x + 3\right )\right ) \]

[In]

integrate(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*log((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5
-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129*x^4+164*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x, algorit
hm="maxima")

[Out]

e^(4*e^4/(x^2 + 1)) + log(log(2*x^2 + 18*x + 27) - 2*log(x + 3))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (29) = 58\).

Time = 0.42 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\left (\frac {4 \, x^{2}}{x^{2} + 1} + \frac {4 \, e^{4}}{x^{2} + 1} + \frac {4}{x^{2} + 1} - 4\right )} + \log \left (\log \left (\frac {2 \, x^{2} + 18 \, x + 27}{x^{2} + 6 \, x + 9}\right )\right ) \]

[In]

integrate(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*log((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5
-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129*x^4+164*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x, algorit
hm="giac")

[Out]

e^(4*x^2/(x^2 + 1) + 4*e^4/(x^2 + 1) + 4/(x^2 + 1) - 4) + log(log((2*x^2 + 18*x + 27)/(x^2 + 6*x + 9)))

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=\ln \left (\ln \left (\frac {2\,x^2+18\,x+27}{x^2+6\,x+9}\right )\right )+{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^4}{x^2+1}} \]

[In]

int(-(6*x + 12*x^3 + 6*x^5 + exp(4)*exp((4*exp(4))/(x^2 + 1))*log((18*x + 2*x^2 + 27)/(6*x + x^2 + 9))*(648*x
+ 648*x^2 + 192*x^3 + 16*x^4))/(log((18*x + 2*x^2 + 27)/(6*x + x^2 + 9))*(81*x + 186*x^2 + 164*x^3 + 129*x^4 +
85*x^5 + 24*x^6 + 2*x^7 + 81)),x)

[Out]

log(log((18*x + 2*x^2 + 27)/(6*x + x^2 + 9))) + exp((4*exp(4))/(x^2 + 1))