Integrand size = 85, antiderivative size = 23 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=9 \left (16-\frac {5}{x}+\left (x+\frac {9}{\log ^2\left (2+x^4\right )}\right )^2\right ) \] Output:
9*(x+9/ln(x^4+2)^2)^2+144-45/x
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=9 \left (-\frac {5}{x}+x^2+\frac {81}{\log ^4\left (2+x^4\right )}+\frac {18 x}{\log ^2\left (2+x^4\right )}\right ) \] Input:
Integrate[(-11664*x^5 - 1296*x^6*Log[2 + x^4]^2 + (324*x^2 + 162*x^6)*Log[ 2 + x^4]^3 + (90 + 36*x^3 + 45*x^4 + 18*x^7)*Log[2 + x^4]^5)/((2*x^2 + x^6 )*Log[2 + x^4]^5),x]
Output:
9*(-5/x + x^2 + 81/Log[2 + x^4]^4 + (18*x)/Log[2 + x^4]^2)
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.
\(\displaystyle \int \frac {-11664 x^5-1296 x^6 \log ^2\left (x^4+2\right )+\left (18 x^7+45 x^4+36 x^3+90\right ) \log ^5\left (x^4+2\right )+\left (162 x^6+324 x^2\right ) \log ^3\left (x^4+2\right )}{\left (x^6+2 x^2\right ) \log ^5\left (x^4+2\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-11664 x^5-1296 x^6 \log ^2\left (x^4+2\right )+\left (18 x^7+45 x^4+36 x^3+90\right ) \log ^5\left (x^4+2\right )+\left (162 x^6+324 x^2\right ) \log ^3\left (x^4+2\right )}{x^2 \left (x^4+2\right ) \log ^5\left (x^4+2\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {1296 x^4}{\left (x^4+2\right ) \log ^3\left (x^4+2\right )}+\frac {162}{\log ^2\left (x^4+2\right )}-\frac {11664 x^3}{\left (x^4+2\right ) \log ^5\left (x^4+2\right )}+\frac {9 \left (2 x^3+5\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -1296 \int \frac {x^4}{\left (x^4+2\right ) \log ^3\left (x^4+2\right )}dx+162 \int \frac {1}{\log ^2\left (x^4+2\right )}dx+\frac {729}{\log ^4\left (x^4+2\right )}+9 x^2-\frac {45}{x}\) |
Input:
Int[(-11664*x^5 - 1296*x^6*Log[2 + x^4]^2 + (324*x^2 + 162*x^6)*Log[2 + x^ 4]^3 + (90 + 36*x^3 + 45*x^4 + 18*x^7)*Log[2 + x^4]^5)/((2*x^2 + x^6)*Log[ 2 + x^4]^5),x]
Output:
$Aborted
Time = 9.69 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\frac {9 x^{3}-45}{x}+\frac {162 x \ln \left (x^{4}+2\right )^{2}+729}{\ln \left (x^{4}+2\right )^{4}}\) | \(35\) |
parallelrisch | \(-\frac {-36 \ln \left (x^{4}+2\right )^{4} x^{3}-648 \ln \left (x^{4}+2\right )^{2} x^{2}+180 \ln \left (x^{4}+2\right )^{4}-2916 x}{4 \ln \left (x^{4}+2\right )^{4} x}\) | \(54\) |
Input:
int(((18*x^7+45*x^4+36*x^3+90)*ln(x^4+2)^5+(162*x^6+324*x^2)*ln(x^4+2)^3-1 296*x^6*ln(x^4+2)^2-11664*x^5)/(x^6+2*x^2)/ln(x^4+2)^5,x,method=_RETURNVER BOSE)
Output:
9*(x^3-5)/x+81*(2*x*ln(x^4+2)^2+9)/ln(x^4+2)^4
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=\frac {9 \, {\left ({\left (x^{3} - 5\right )} \log \left (x^{4} + 2\right )^{4} + 18 \, x^{2} \log \left (x^{4} + 2\right )^{2} + 81 \, x\right )}}{x \log \left (x^{4} + 2\right )^{4}} \] Input:
integrate(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^ 4+2)^3-1296*x^6*log(x^4+2)^2-11664*x^5)/(x^6+2*x^2)/log(x^4+2)^5,x, algori thm="fricas")
Output:
9*((x^3 - 5)*log(x^4 + 2)^4 + 18*x^2*log(x^4 + 2)^2 + 81*x)/(x*log(x^4 + 2 )^4)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=9 x^{2} + \frac {162 x \log {\left (x^{4} + 2 \right )}^{2} + 729}{\log {\left (x^{4} + 2 \right )}^{4}} - \frac {45}{x} \] Input:
integrate(((18*x**7+45*x**4+36*x**3+90)*ln(x**4+2)**5+(162*x**6+324*x**2)* ln(x**4+2)**3-1296*x**6*ln(x**4+2)**2-11664*x**5)/(x**6+2*x**2)/ln(x**4+2) **5,x)
Output:
9*x**2 + (162*x*log(x**4 + 2)**2 + 729)/log(x**4 + 2)**4 - 45/x
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=\frac {9 \, {\left ({\left (x^{3} - 5\right )} \log \left (x^{4} + 2\right )^{2} + 18 \, x^{2}\right )}}{x \log \left (x^{4} + 2\right )^{2}} + \frac {729}{\log \left (x^{4} + 2\right )^{4}} \] Input:
integrate(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^ 4+2)^3-1296*x^6*log(x^4+2)^2-11664*x^5)/(x^6+2*x^2)/log(x^4+2)^5,x, algori thm="maxima")
Output:
9*((x^3 - 5)*log(x^4 + 2)^2 + 18*x^2)/(x*log(x^4 + 2)^2) + 729/log(x^4 + 2 )^4
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (23) = 46\).
Time = 0.57 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.39 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=9 \, x^{2} + \frac {81 \, {\left (2 \, x^{11} \log \left (x^{4} + 2\right )^{2} + 9 \, x^{10} + 4 \, x^{7} \log \left (x^{4} + 2\right )^{2} + 18 \, x^{6}\right )}}{x^{10} \log \left (x^{4} + 2\right )^{4} + 2 \, x^{6} \log \left (x^{4} + 2\right )^{4}} - \frac {45}{x} \] Input:
integrate(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^ 4+2)^3-1296*x^6*log(x^4+2)^2-11664*x^5)/(x^6+2*x^2)/log(x^4+2)^5,x, algori thm="giac")
Output:
9*x^2 + 81*(2*x^11*log(x^4 + 2)^2 + 9*x^10 + 4*x^7*log(x^4 + 2)^2 + 18*x^6 )/(x^10*log(x^4 + 2)^4 + 2*x^6*log(x^4 + 2)^4) - 45/x
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=\frac {729}{{\ln \left (x^4+2\right )}^4}-\frac {45}{x}+9\,x^2+\frac {162\,x}{{\ln \left (x^4+2\right )}^2} \] Input:
int((log(x^4 + 2)^3*(324*x^2 + 162*x^6) - 1296*x^6*log(x^4 + 2)^2 + log(x^ 4 + 2)^5*(36*x^3 + 45*x^4 + 18*x^7 + 90) - 11664*x^5)/(log(x^4 + 2)^5*(2*x ^2 + x^6)),x)
Output:
729/log(x^4 + 2)^4 - 45/x + 9*x^2 + (162*x)/log(x^4 + 2)^2
Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26 \[ \int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx=\frac {9 \mathrm {log}\left (x^{4}+2\right )^{4} x^{3}-45 \mathrm {log}\left (x^{4}+2\right )^{4}+162 \mathrm {log}\left (x^{4}+2\right )^{2} x^{2}+729 x}{\mathrm {log}\left (x^{4}+2\right )^{4} x} \] Input:
int(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^4+2)^3 -1296*x^6*log(x^4+2)^2-11664*x^5)/(x^6+2*x^2)/log(x^4+2)^5,x)
Output:
(9*(log(x**4 + 2)**4*x**3 - 5*log(x**4 + 2)**4 + 18*log(x**4 + 2)**2*x**2 + 81*x))/(log(x**4 + 2)**4*x)