Integrand size = 69, antiderivative size = 29 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=x \left (-x+6 \left (-1+2 x^2-\log \left (5-\frac {4 x^2}{\log (4)}\right )\right )\right ) \] Output:
x*(12*x^2-6-6*ln(5-2*x^2/ln(2))-x)
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=2 \left (-3 x-\frac {x^2}{2}+6 x^3-3 x \log \left (5-\frac {4 x^2}{\log (4)}\right )\right ) \] Input:
Integrate[(72*x^2 + 8*x^3 - 144*x^4 + (-30 - 10*x + 180*x^2)*Log[4] + (24* x^2 - 30*Log[4])*Log[(-4*x^2 + 5*Log[4])/Log[4]])/(-4*x^2 + 5*Log[4]),x]
Output:
2*(-3*x - x^2/2 + 6*x^3 - 3*x*Log[5 - (4*x^2)/Log[4]])
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(29)=58\).
Time = 0.77 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {7276, 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.
\(\displaystyle \int \frac {-144 x^4+8 x^3+72 x^2+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {5 \log (4)-4 x^2}{\log (4)}\right )+\left (180 x^2-10 x-30\right ) \log (4)}{5 \log (4)-4 x^2} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 \left (72 x^4-4 x^3-18 x^2 (2+5 \log (4))+5 x \log (4)+15 \log (4)\right )}{4 x^2-5 \log (4)}-6 \log \left (5-\frac {4 x^2}{\log (4)}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -6 \sqrt {\log (1024)} \text {arctanh}\left (\frac {2 x}{\sqrt {5 \log (4)}}\right )+6 \sqrt {5 \log (4)} \text {arctanh}\left (\frac {2 x}{\sqrt {5 \log (4)}}\right )+12 x^3-x^2-6 x \log \left (5-\frac {4 x^2}{\log (4)}\right )-6 x\) |
Input:
Int[(72*x^2 + 8*x^3 - 144*x^4 + (-30 - 10*x + 180*x^2)*Log[4] + (24*x^2 - 30*Log[4])*Log[(-4*x^2 + 5*Log[4])/Log[4]])/(-4*x^2 + 5*Log[4]),x]
Output:
-6*x - x^2 + 12*x^3 + 6*ArcTanh[(2*x)/Sqrt[5*Log[4]]]*Sqrt[5*Log[4]] - 6*A rcTanh[(2*x)/Sqrt[5*Log[4]]]*Sqrt[Log[1024]] - 6*x*Log[5 - (4*x^2)/Log[4]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.75 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21
method | result | size |
default | \(12 x^{3}-x^{2}-6 x -6 x \ln \left (-2 x^{2}+5 \ln \left (2\right )\right )+6 x \ln \left (\ln \left (2\right )\right )\) | \(35\) |
norman | \(-6 x -x^{2}+12 x^{3}-6 x \ln \left (\frac {10 \ln \left (2\right )-4 x^{2}}{2 \ln \left (2\right )}\right )\) | \(35\) |
risch | \(-6 x -x^{2}+12 x^{3}-6 x \ln \left (\frac {10 \ln \left (2\right )-4 x^{2}}{2 \ln \left (2\right )}\right )\) | \(35\) |
parts | \(12 x^{3}-x^{2}-6 x -6 x \ln \left (-2 x^{2}+5 \ln \left (2\right )\right )+6 x \ln \left (\ln \left (2\right )\right )\) | \(35\) |
parallelrisch | \(12 x^{3}-x^{2}-6 x \ln \left (\frac {-2 x^{2}+5 \ln \left (2\right )}{\ln \left (2\right )}\right )-5 \ln \left (2\right )-6 x\) | \(38\) |
Input:
int(((-60*ln(2)+24*x^2)*ln(1/2*(10*ln(2)-4*x^2)/ln(2))+2*(180*x^2-10*x-30) *ln(2)-144*x^4+8*x^3+72*x^2)/(10*ln(2)-4*x^2),x,method=_RETURNVERBOSE)
Output:
12*x^3-x^2-6*x-6*x*ln(-2*x^2+5*ln(2))+6*x*ln(ln(2))
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=12 \, x^{3} - x^{2} - 6 \, x \log \left (-\frac {2 \, x^{2} - 5 \, \log \left (2\right )}{\log \left (2\right )}\right ) - 6 \, x \] Input:
integrate(((-60*log(2)+24*x^2)*log(1/2*(10*log(2)-4*x^2)/log(2))+2*(180*x^ 2-10*x-30)*log(2)-144*x^4+8*x^3+72*x^2)/(10*log(2)-4*x^2),x, algorithm="fr icas")
Output:
12*x^3 - x^2 - 6*x*log(-(2*x^2 - 5*log(2))/log(2)) - 6*x
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=12 x^{3} - x^{2} - 6 x \log {\left (\frac {- 2 x^{2} + 5 \log {\left (2 \right )}}{\log {\left (2 \right )}} \right )} - 6 x \] Input:
integrate(((-60*ln(2)+24*x**2)*ln(1/2*(10*ln(2)-4*x**2)/ln(2))+2*(180*x**2 -10*x-30)*ln(2)-144*x**4+8*x**3+72*x**2)/(10*ln(2)-4*x**2),x)
Output:
12*x**3 - x**2 - 6*x*log((-2*x**2 + 5*log(2))/log(2)) - 6*x
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.28 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=12 \, x^{3} + \frac {45}{2} \, \sqrt {10} \log \left (2\right )^{\frac {3}{2}} \log \left (\frac {2 \, x - \sqrt {10} \sqrt {\log \left (2\right )}}{2 \, x + \sqrt {10} \sqrt {\log \left (2\right )}}\right ) - x^{2} + 6 \, x {\left (\log \left (\log \left (2\right )\right ) + 2\right )} - \frac {45}{2} \, {\left (\sqrt {10} \sqrt {\log \left (2\right )} \log \left (\frac {2 \, x - \sqrt {10} \sqrt {\log \left (2\right )}}{2 \, x + \sqrt {10} \sqrt {\log \left (2\right )}}\right ) + 4 \, x\right )} \log \left (2\right ) + 90 \, x \log \left (2\right ) - 6 \, x \log \left (-2 \, x^{2} + 5 \, \log \left (2\right )\right ) - 18 \, x \] Input:
integrate(((-60*log(2)+24*x^2)*log(1/2*(10*log(2)-4*x^2)/log(2))+2*(180*x^ 2-10*x-30)*log(2)-144*x^4+8*x^3+72*x^2)/(10*log(2)-4*x^2),x, algorithm="ma xima")
Output:
12*x^3 + 45/2*sqrt(10)*log(2)^(3/2)*log((2*x - sqrt(10)*sqrt(log(2)))/(2*x + sqrt(10)*sqrt(log(2)))) - x^2 + 6*x*(log(log(2)) + 2) - 45/2*(sqrt(10)* sqrt(log(2))*log((2*x - sqrt(10)*sqrt(log(2)))/(2*x + sqrt(10)*sqrt(log(2) ))) + 4*x)*log(2) + 90*x*log(2) - 6*x*log(-2*x^2 + 5*log(2)) - 18*x
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=12 \, x^{3} - x^{2} + 6 \, x {\left (\log \left (\log \left (2\right )\right ) - 1\right )} - 6 \, x \log \left (-2 \, x^{2} + 5 \, \log \left (2\right )\right ) \] Input:
integrate(((-60*log(2)+24*x^2)*log(1/2*(10*log(2)-4*x^2)/log(2))+2*(180*x^ 2-10*x-30)*log(2)-144*x^4+8*x^3+72*x^2)/(10*log(2)-4*x^2),x, algorithm="gi ac")
Output:
12*x^3 - x^2 + 6*x*(log(log(2)) - 1) - 6*x*log(-2*x^2 + 5*log(2))
Time = 6.73 (sec) , antiderivative size = 251, normalized size of antiderivative = 8.66 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=2\,\mathrm {atan}\left (\frac {4\,x\,\sqrt {-\frac {45\,\ln \left (2\right )}{2}}}{30\,\ln \left (2\right )-450\,{\ln \left (2\right )}^2}-\frac {60\,x\,\ln \left (2\right )\,\sqrt {-\frac {45\,\ln \left (2\right )}{2}}}{30\,\ln \left (2\right )-450\,{\ln \left (2\right )}^2}\right )\,\sqrt {-\frac {45\,\ln \left (2\right )}{2}}-6\,x+\frac {5\,\ln \left (2\right )\,\ln \left (x^2-\frac {5\,\ln \left (2\right )}{2}\right )}{2}-\frac {\ln \left (32\right )\,\ln \left (x^2-\frac {5\,\ln \left (2\right )}{2}\right )}{2}-90\,x\,\ln \left (2\right )+18\,x\,\ln \left (32\right )-6\,x\,\ln \left (\ln \left (32\right )-2\,x^2\right )+6\,x\,\ln \left (\ln \left (2\right )\right )-x^2+12\,x^3-30\,\ln \left (2\right )\,\mathrm {atan}\left (\frac {4\,x\,\sqrt {-\frac {45\,\ln \left (2\right )}{2}}}{30\,\ln \left (2\right )-450\,{\ln \left (2\right )}^2}-\frac {60\,x\,\ln \left (2\right )\,\sqrt {-\frac {45\,\ln \left (2\right )}{2}}}{30\,\ln \left (2\right )-450\,{\ln \left (2\right )}^2}\right )\,\sqrt {-\frac {45\,\ln \left (2\right )}{2}}-6\,\sqrt {10}\,\mathrm {atanh}\left (\frac {\sqrt {10}\,x}{5\,\sqrt {\ln \left (2\right )}}\right )\,\sqrt {\ln \left (2\right )}+9\,\sqrt {2}\,\sqrt {5}\,\sqrt {\ln \left (2\right )}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {5}\,x}{5\,\sqrt {\ln \left (2\right )}}\right )-\frac {225\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{\sqrt {\ln \left (32\right )}}\right )\,{\ln \left (2\right )}^2}{\sqrt {\ln \left (32\right )}} \] Input:
int(-(2*log(2)*(10*x - 180*x^2 + 30) + log((5*log(2) - 2*x^2)/log(2))*(60* log(2) - 24*x^2) - 72*x^2 - 8*x^3 + 144*x^4)/(10*log(2) - 4*x^2),x)
Output:
2*atan((4*x*(-(45*log(2))/2)^(1/2))/(30*log(2) - 450*log(2)^2) - (60*x*log (2)*(-(45*log(2))/2)^(1/2))/(30*log(2) - 450*log(2)^2))*(-(45*log(2))/2)^( 1/2) - 6*x + (5*log(2)*log(x^2 - (5*log(2))/2))/2 - (log(32)*log(x^2 - (5* log(2))/2))/2 - 90*x*log(2) + 18*x*log(32) - 6*x*log(log(32) - 2*x^2) + 6* x*log(log(2)) - x^2 + 12*x^3 - 30*log(2)*atan((4*x*(-(45*log(2))/2)^(1/2)) /(30*log(2) - 450*log(2)^2) - (60*x*log(2)*(-(45*log(2))/2)^(1/2))/(30*log (2) - 450*log(2)^2))*(-(45*log(2))/2)^(1/2) - 6*10^(1/2)*atanh((10^(1/2)*x )/(5*log(2)^(1/2)))*log(2)^(1/2) + 9*2^(1/2)*5^(1/2)*log(2)^(1/2)*atanh((2 ^(1/2)*5^(1/2)*x)/(5*log(2)^(1/2))) - (225*2^(1/2)*atanh((2^(1/2)*x)/log(3 2)^(1/2))*log(2)^2)/log(32)^(1/2)
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21 \[ \int \frac {72 x^2+8 x^3-144 x^4+\left (-30-10 x+180 x^2\right ) \log (4)+\left (24 x^2-30 \log (4)\right ) \log \left (\frac {-4 x^2+5 \log (4)}{\log (4)}\right )}{-4 x^2+5 \log (4)} \, dx=-3 \sqrt {\mathrm {log}\left (2\right )}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {\mathrm {log}\left (2\right )}\, \sqrt {10}-2 x \right )-3 \sqrt {\mathrm {log}\left (2\right )}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {\mathrm {log}\left (2\right )}\, \sqrt {10}-2 x \right )+3 \sqrt {\mathrm {log}\left (2\right )}\, \sqrt {10}\, \mathrm {log}\left (\frac {5 \,\mathrm {log}\left (2\right )-2 x^{2}}{\mathrm {log}\left (2\right )}\right )-6 \,\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (2\right )-2 x^{2}}{\mathrm {log}\left (2\right )}\right ) x +12 x^{3}-x^{2}-6 x \] Input:
int(((-60*log(2)+24*x^2)*log(1/2*(10*log(2)-4*x^2)/log(2))+2*(180*x^2-10*x -30)*log(2)-144*x^4+8*x^3+72*x^2)/(10*log(2)-4*x^2),x)
Output:
- 3*sqrt(log(2))*sqrt(10)*log( - sqrt(log(2))*sqrt(10) - 2*x) - 3*sqrt(lo g(2))*sqrt(10)*log(sqrt(log(2))*sqrt(10) - 2*x) + 3*sqrt(log(2))*sqrt(10)* log((5*log(2) - 2*x**2)/log(2)) - 6*log((5*log(2) - 2*x**2)/log(2))*x + 12 *x**3 - x**2 - 6*x