Integrand size = 56, antiderivative size = 21 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=\frac {\left (4+x+\frac {5}{3} \log ^2\left (\frac {14 x}{3}\right )\right )^2}{x} \] Output:
(x+4+5/3*ln(14/3*x)^2)^2/x
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(21)=42\).
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=\frac {16}{x}+x+\frac {10}{3} \log ^2\left (\frac {14 x}{3}\right )+\frac {40 \log ^2\left (\frac {14 x}{3}\right )}{3 x}+\frac {25 \log ^4\left (\frac {14 x}{3}\right )}{9 x} \] Input:
Integrate[(-144 + 9*x^2 + (240 + 60*x)*Log[(14*x)/3] - 120*Log[(14*x)/3]^2 + 100*Log[(14*x)/3]^3 - 25*Log[(14*x)/3]^4)/(9*x^2),x]
Output:
16/x + x + (10*Log[(14*x)/3]^2)/3 + (40*Log[(14*x)/3]^2)/(3*x) + (25*Log[( 14*x)/3]^4)/(9*x)
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(21)=42\).
Time = 0.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.62, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 25, 2010, 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.
\(\displaystyle \int \frac {9 x^2-25 \log ^4\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+(60 x+240) \log \left (\frac {14 x}{3}\right )-144}{9 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -\frac {25 \log ^4\left (\frac {14 x}{3}\right )-100 \log ^3\left (\frac {14 x}{3}\right )+120 \log ^2\left (\frac {14 x}{3}\right )-60 (x+4) \log \left (\frac {14 x}{3}\right )-9 x^2+144}{x^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{9} \int \frac {25 \log ^4\left (\frac {14 x}{3}\right )-100 \log ^3\left (\frac {14 x}{3}\right )+120 \log ^2\left (\frac {14 x}{3}\right )-60 (x+4) \log \left (\frac {14 x}{3}\right )-9 x^2+144}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {1}{9} \int \left (\frac {25 \log ^4\left (\frac {14 x}{3}\right )}{x^2}-\frac {100 \log ^3\left (\frac {14 x}{3}\right )}{x^2}+\frac {120 \log ^2\left (\frac {14 x}{3}\right )}{x^2}-\frac {60 (x+4) \log \left (\frac {14 x}{3}\right )}{x^2}-\frac {9 \left (x^2-16\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{9} \left (9 x+\frac {144}{x}+\frac {25 \log ^4\left (\frac {14 x}{3}\right )}{x}+\frac {120 \log ^2\left (\frac {14 x}{3}\right )}{x}-30 \log ^2(x)+60 \log (x) \log \left (\frac {14 x}{3}\right )\right )\) |
Input:
Int[(-144 + 9*x^2 + (240 + 60*x)*Log[(14*x)/3] - 120*Log[(14*x)/3]^2 + 100 *Log[(14*x)/3]^3 - 25*Log[(14*x)/3]^4)/(9*x^2),x]
Output:
(144/x + 9*x - 30*Log[x]^2 + 60*Log[x]*Log[(14*x)/3] + (120*Log[(14*x)/3]^ 2)/x + (25*Log[(14*x)/3]^4)/x)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 1.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67
method | result | size |
norman | \(\frac {16+x^{2}+\frac {40 \ln \left (\frac {14 x}{3}\right )^{2}}{3}+\frac {25 \ln \left (\frac {14 x}{3}\right )^{4}}{9}+\frac {10 x \ln \left (\frac {14 x}{3}\right )^{2}}{3}}{x}\) | \(35\) |
risch | \(\frac {25 \ln \left (\frac {14 x}{3}\right )^{4}}{9 x}+\frac {10 \left (4+x \right ) \ln \left (\frac {14 x}{3}\right )^{2}}{3 x}+\frac {x^{2}+16}{x}\) | \(36\) |
derivativedivides | \(\frac {25 \ln \left (\frac {14 x}{3}\right )^{4}}{9 x}+\frac {40 \ln \left (\frac {14 x}{3}\right )^{2}}{3 x}+\frac {16}{x}+\frac {10 \ln \left (\frac {14 x}{3}\right )^{2}}{3}+x\) | \(38\) |
default | \(\frac {25 \ln \left (\frac {14 x}{3}\right )^{4}}{9 x}+\frac {40 \ln \left (\frac {14 x}{3}\right )^{2}}{3 x}+\frac {16}{x}+\frac {10 \ln \left (\frac {14 x}{3}\right )^{2}}{3}+x\) | \(38\) |
parallelrisch | \(\frac {25 \ln \left (\frac {14 x}{3}\right )^{4}+144+30 x \ln \left (\frac {14 x}{3}\right )^{2}+9 x^{2}+120 \ln \left (\frac {14 x}{3}\right )^{2}}{9 x}\) | \(38\) |
parts | \(\frac {25 \ln \left (\frac {14 x}{3}\right )^{4}}{9 x}+\frac {40 \ln \left (\frac {14 x}{3}\right )^{2}}{3 x}+\frac {16}{x}+\frac {10 \ln \left (\frac {14 x}{3}\right )^{2}}{3}+x\) | \(38\) |
orering | \(\text {Expression too large to display}\) | \(997\) |
Input:
int(1/9*(-25*ln(14/3*x)^4+100*ln(14/3*x)^3-120*ln(14/3*x)^2+(60*x+240)*ln( 14/3*x)+9*x^2-144)/x^2,x,method=_RETURNVERBOSE)
Output:
(16+x^2+40/3*ln(14/3*x)^2+25/9*ln(14/3*x)^4+10/3*x*ln(14/3*x)^2)/x
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=\frac {25 \, \log \left (\frac {14}{3} \, x\right )^{4} + 30 \, {\left (x + 4\right )} \log \left (\frac {14}{3} \, x\right )^{2} + 9 \, x^{2} + 144}{9 \, x} \] Input:
integrate(1/9*(-25*log(14/3*x)^4+100*log(14/3*x)^3-120*log(14/3*x)^2+(60*x +240)*log(14/3*x)+9*x^2-144)/x^2,x, algorithm="fricas")
Output:
1/9*(25*log(14/3*x)^4 + 30*(x + 4)*log(14/3*x)^2 + 9*x^2 + 144)/x
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=x + \frac {\left (10 x + 40\right ) \log {\left (\frac {14 x}{3} \right )}^{2}}{3 x} + \frac {25 \log {\left (\frac {14 x}{3} \right )}^{4}}{9 x} + \frac {16}{x} \] Input:
integrate(1/9*(-25*ln(14/3*x)**4+100*ln(14/3*x)**3-120*ln(14/3*x)**2+(60*x +240)*ln(14/3*x)+9*x**2-144)/x**2,x)
Output:
x + (10*x + 40)*log(14*x/3)**2/(3*x) + 25*log(14*x/3)**4/(9*x) + 16/x
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.00 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=\frac {10}{3} \, \log \left (\frac {14}{3} \, x\right )^{2} + x + \frac {25 \, {\left (\log \left (\frac {14}{3} \, x\right )^{4} + 4 \, \log \left (\frac {14}{3} \, x\right )^{3} + 12 \, \log \left (\frac {14}{3} \, x\right )^{2} + 24 \, \log \left (\frac {14}{3} \, x\right ) + 24\right )}}{9 \, x} - \frac {100 \, {\left (\log \left (\frac {14}{3} \, x\right )^{3} + 3 \, \log \left (\frac {14}{3} \, x\right )^{2} + 6 \, \log \left (\frac {14}{3} \, x\right ) + 6\right )}}{9 \, x} + \frac {40 \, {\left (\log \left (\frac {14}{3} \, x\right )^{2} + 2 \, \log \left (\frac {14}{3} \, x\right ) + 2\right )}}{3 \, x} - \frac {80 \, \log \left (\frac {14}{3} \, x\right )}{3 \, x} - \frac {32}{3 \, x} \] Input:
integrate(1/9*(-25*log(14/3*x)^4+100*log(14/3*x)^3-120*log(14/3*x)^2+(60*x +240)*log(14/3*x)+9*x^2-144)/x^2,x, algorithm="maxima")
Output:
10/3*log(14/3*x)^2 + x + 25/9*(log(14/3*x)^4 + 4*log(14/3*x)^3 + 12*log(14 /3*x)^2 + 24*log(14/3*x) + 24)/x - 100/9*(log(14/3*x)^3 + 3*log(14/3*x)^2 + 6*log(14/3*x) + 6)/x + 40/3*(log(14/3*x)^2 + 2*log(14/3*x) + 2)/x - 80/3 *log(14/3*x)/x - 32/3/x
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=\frac {10}{3} \, {\left (\frac {4}{x} + 1\right )} \log \left (\frac {14}{3} \, x\right )^{2} + \frac {25 \, \log \left (\frac {14}{3} \, x\right )^{4}}{9 \, x} + x + \frac {16}{x} \] Input:
integrate(1/9*(-25*log(14/3*x)^4+100*log(14/3*x)^3-120*log(14/3*x)^2+(60*x +240)*log(14/3*x)+9*x^2-144)/x^2,x, algorithm="giac")
Output:
10/3*(4/x + 1)*log(14/3*x)^2 + 25/9*log(14/3*x)^4/x + x + 16/x
Time = 3.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=x+\frac {\frac {25\,{\ln \left (\frac {14\,x}{3}\right )}^4}{9}+\frac {40\,{\ln \left (\frac {14\,x}{3}\right )}^2}{3}+16}{x}+\frac {10\,{\ln \left (\frac {14\,x}{3}\right )}^2}{3} \] Input:
int(-((40*log((14*x)/3)^2)/3 - (100*log((14*x)/3)^3)/9 + (25*log((14*x)/3) ^4)/9 - x^2 - (log((14*x)/3)*(60*x + 240))/9 + 16)/x^2,x)
Output:
x + ((40*log((14*x)/3)^2)/3 + (25*log((14*x)/3)^4)/9 + 16)/x + (10*log((14 *x)/3)^2)/3
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-144+9 x^2+(240+60 x) \log \left (\frac {14 x}{3}\right )-120 \log ^2\left (\frac {14 x}{3}\right )+100 \log ^3\left (\frac {14 x}{3}\right )-25 \log ^4\left (\frac {14 x}{3}\right )}{9 x^2} \, dx=\frac {25 \mathrm {log}\left (\frac {14 x}{3}\right )^{4}+30 \mathrm {log}\left (\frac {14 x}{3}\right )^{2} x +120 \mathrm {log}\left (\frac {14 x}{3}\right )^{2}+9 x^{2}+144}{9 x} \] Input:
int(1/9*(-25*log(14/3*x)^4+100*log(14/3*x)^3-120*log(14/3*x)^2+(60*x+240)* log(14/3*x)+9*x^2-144)/x^2,x)
Output:
(25*log((14*x)/3)**4 + 30*log((14*x)/3)**2*x + 120*log((14*x)/3)**2 + 9*x* *2 + 144)/(9*x)