Integrand size = 199, antiderivative size = 19 \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\frac {5}{3 (5+x (3-x+\log (2))+\log (x))^2} \]
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\frac {5}{3 \left (5-x^2+x (3+\log (2))+\log (x)\right )^2} \]
Integrate[(-10 - 30*x + 20*x^2 - 10*x*Log[2])/(375*x + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7 + (225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6)*Log[2] + (45*x^3 + 27*x^4 - 9*x^5)*Log[2]^2 + 3*x^4*Log[2]^3 + (22 5*x + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5 + (90*x^2 + 54*x^3 - 18*x^4)*Log[2] + 9*x^3*Log[2]^2)*Log[x] + (45*x + 27*x^2 - 9*x^3 + 9*x^2*Log[2])*Log[x]^ 2 + 3*x*Log[x]^3),x]
Time = 0.73 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6, 6, 7239, 27, 25, 7237}
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.
| \(\displaystyle \int \frac {20 x^2-30 x-10 x \log (2)-10}{-3 x^7+27 x^6-36 x^5-189 x^4+3 x^4 \log ^3(2)+180 x^3+675 x^2+\left (-9 x^3+27 x^2+9 x^2 \log (2)+45 x\right ) \log ^2(x)+\left (-9 x^5+27 x^4+45 x^3\right ) \log ^2(2)+\left (9 x^5-54 x^4-9 x^3+9 x^3 \log ^2(2)+270 x^2+\left (-18 x^4+54 x^3+90 x^2\right ) \log (2)+225 x\right ) \log (x)+\left (9 x^6-54 x^5-9 x^4+270 x^3+225 x^2\right ) \log (2)+375 x+3 x \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {20 x^2+x (-30-10 \log (2))-10}{-3 x^7+27 x^6-36 x^5-189 x^4+3 x^4 \log ^3(2)+180 x^3+675 x^2+\left (-9 x^3+27 x^2+9 x^2 \log (2)+45 x\right ) \log ^2(x)+\left (-9 x^5+27 x^4+45 x^3\right ) \log ^2(2)+\left (9 x^5-54 x^4-9 x^3+9 x^3 \log ^2(2)+270 x^2+\left (-18 x^4+54 x^3+90 x^2\right ) \log (2)+225 x\right ) \log (x)+\left (9 x^6-54 x^5-9 x^4+270 x^3+225 x^2\right ) \log (2)+375 x+3 x \log ^3(x)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {20 x^2+x (-30-10 \log (2))-10}{-3 x^7+27 x^6-36 x^5+x^4 \left (3 \log ^3(2)-189\right )+180 x^3+675 x^2+\left (-9 x^3+27 x^2+9 x^2 \log (2)+45 x\right ) \log ^2(x)+\left (-9 x^5+27 x^4+45 x^3\right ) \log ^2(2)+\left (9 x^5-54 x^4-9 x^3+9 x^3 \log ^2(2)+270 x^2+\left (-18 x^4+54 x^3+90 x^2\right ) \log (2)+225 x\right ) \log (x)+\left (9 x^6-54 x^5-9 x^4+270 x^3+225 x^2\right ) \log (2)+375 x+3 x \log ^3(x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {10 \left (2 x^2-x (3+\log (2))-1\right )}{3 x \left (-x^2+x (3+\log (2))+\log (x)+5\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{3} \int -\frac {-2 x^2+(3+\log (2)) x+1}{x \left (-x^2+(3+\log (2)) x+\log (x)+5\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {10}{3} \int \frac {-2 x^2+(3+\log (2)) x+1}{x \left (-x^2+(3+\log (2)) x+\log (x)+5\right )^3}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \frac {5}{3 \left (-x^2+x (3+\log (2))+\log (x)+5\right )^2}\) |
Int[(-10 - 30*x + 20*x^2 - 10*x*Log[2])/(375*x + 675*x^2 + 180*x^3 - 189*x ^4 - 36*x^5 + 27*x^6 - 3*x^7 + (225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6 )*Log[2] + (45*x^3 + 27*x^4 - 9*x^5)*Log[2]^2 + 3*x^4*Log[2]^3 + (225*x + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5 + (90*x^2 + 54*x^3 - 18*x^4)*Log[2] + 9*x ^3*Log[2]^2)*Log[x] + (45*x + 27*x^2 - 9*x^3 + 9*x^2*Log[2])*Log[x]^2 + 3* x*Log[x]^3),x]
3.5.66.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 8.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {5}{3 \left (x \ln \left (2\right )-x^{2}+\ln \left (x \right )+3 x +5\right )^{2}}\) | \(21\) |
| risch | \(\frac {5}{3 \left (x \ln \left (2\right )-x^{2}+\ln \left (x \right )+3 x +5\right )^{2}}\) | \(21\) |
| parallelrisch | \(\frac {5}{3 \left (x^{2} \ln \left (2\right )^{2}-2 x^{3} \ln \left (2\right )+x^{4}+6 x^{2} \ln \left (2\right )+2 x \ln \left (2\right ) \ln \left (x \right )-6 x^{3}-2 x^{2} \ln \left (x \right )+10 x \ln \left (2\right )-x^{2}+6 x \ln \left (x \right )+\ln \left (x \right )^{2}+30 x +10 \ln \left (x \right )+25\right )}\) | \(77\) |
int((-10*x*ln(2)+20*x^2-30*x-10)/(3*x*ln(x)^3+(9*x^2*ln(2)-9*x^3+27*x^2+45 *x)*ln(x)^2+(9*x^3*ln(2)^2+(-18*x^4+54*x^3+90*x^2)*ln(2)+9*x^5-54*x^4-9*x^ 3+270*x^2+225*x)*ln(x)+3*x^4*ln(2)^3+(-9*x^5+27*x^4+45*x^3)*ln(2)^2+(9*x^6 -54*x^5-9*x^4+270*x^3+225*x^2)*ln(2)-3*x^7+27*x^6-36*x^5-189*x^4+180*x^3+6 75*x^2+375*x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.53 \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\frac {5}{3 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 6 \, x^{3} - x^{2} - 2 \, {\left (x^{3} - 3 \, x^{2} - 5 \, x\right )} \log \left (2\right ) - 2 \, {\left (x^{2} - x \log \left (2\right ) - 3 \, x - 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 30 \, x + 25\right )}} \]
integrate((-10*x*log(2)+20*x^2-30*x-10)/(3*x*log(x)^3+(9*x^2*log(2)-9*x^3+ 27*x^2+45*x)*log(x)^2+(9*x^3*log(2)^2+(-18*x^4+54*x^3+90*x^2)*log(2)+9*x^5 -54*x^4-9*x^3+270*x^2+225*x)*log(x)+3*x^4*log(2)^3+(-9*x^5+27*x^4+45*x^3)* log(2)^2+(9*x^6-54*x^5-9*x^4+270*x^3+225*x^2)*log(2)-3*x^7+27*x^6-36*x^5-1 89*x^4+180*x^3+675*x^2+375*x),x, algorithm=\
5/3/(x^4 + x^2*log(2)^2 - 6*x^3 - x^2 - 2*(x^3 - 3*x^2 - 5*x)*log(2) - 2*( x^2 - x*log(2) - 3*x - 5)*log(x) + log(x)^2 + 30*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (17) = 34\).
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.32 \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\frac {5}{3 x^{4} - 18 x^{3} - 6 x^{3} \log {\left (2 \right )} - 3 x^{2} + 3 x^{2} \log {\left (2 \right )}^{2} + 18 x^{2} \log {\left (2 \right )} + 30 x \log {\left (2 \right )} + 90 x + \left (- 6 x^{2} + 6 x \log {\left (2 \right )} + 18 x + 30\right ) \log {\left (x \right )} + 3 \log {\left (x \right )}^{2} + 75} \]
integrate((-10*x*ln(2)+20*x**2-30*x-10)/(3*x*ln(x)**3+(9*x**2*ln(2)-9*x**3 +27*x**2+45*x)*ln(x)**2+(9*x**3*ln(2)**2+(-18*x**4+54*x**3+90*x**2)*ln(2)+ 9*x**5-54*x**4-9*x**3+270*x**2+225*x)*ln(x)+3*x**4*ln(2)**3+(-9*x**5+27*x* *4+45*x**3)*ln(2)**2+(9*x**6-54*x**5-9*x**4+270*x**3+225*x**2)*ln(2)-3*x** 7+27*x**6-36*x**5-189*x**4+180*x**3+675*x**2+375*x),x)
5/(3*x**4 - 18*x**3 - 6*x**3*log(2) - 3*x**2 + 3*x**2*log(2)**2 + 18*x**2* log(2) + 30*x*log(2) + 90*x + (-6*x**2 + 6*x*log(2) + 18*x + 30)*log(x) + 3*log(x)**2 + 75)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (19) = 38\).
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 3.11 \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\frac {5}{3 \, {\left (x^{4} - 2 \, x^{3} {\left (\log \left (2\right ) + 3\right )} + {\left (\log \left (2\right )^{2} + 6 \, \log \left (2\right ) - 1\right )} x^{2} + 10 \, x {\left (\log \left (2\right ) + 3\right )} - 2 \, {\left (x^{2} - x {\left (\log \left (2\right ) + 3\right )} - 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 25\right )}} \]
integrate((-10*x*log(2)+20*x^2-30*x-10)/(3*x*log(x)^3+(9*x^2*log(2)-9*x^3+ 27*x^2+45*x)*log(x)^2+(9*x^3*log(2)^2+(-18*x^4+54*x^3+90*x^2)*log(2)+9*x^5 -54*x^4-9*x^3+270*x^2+225*x)*log(x)+3*x^4*log(2)^3+(-9*x^5+27*x^4+45*x^3)* log(2)^2+(9*x^6-54*x^5-9*x^4+270*x^3+225*x^2)*log(2)-3*x^7+27*x^6-36*x^5-1 89*x^4+180*x^3+675*x^2+375*x),x, algorithm=\
5/3/(x^4 - 2*x^3*(log(2) + 3) + (log(2)^2 + 6*log(2) - 1)*x^2 + 10*x*(log( 2) + 3) - 2*(x^2 - x*(log(2) + 3) - 5)*log(x) + log(x)^2 + 25)
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 11.32 \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\frac {5 \, {\left (2 \, x^{2} - x \log \left (2\right ) - 3 \, x - 1\right )}}{3 \, {\left (2 \, x^{6} - 5 \, x^{5} \log \left (2\right ) + 4 \, x^{4} \log \left (2\right )^{2} - x^{3} \log \left (2\right )^{3} - 15 \, x^{5} + 24 \, x^{4} \log \left (2\right ) - 9 \, x^{3} \log \left (2\right )^{2} - 4 \, x^{4} \log \left (x\right ) + 6 \, x^{3} \log \left (2\right ) \log \left (x\right ) - 2 \, x^{2} \log \left (2\right )^{2} \log \left (x\right ) + 15 \, x^{4} + 5 \, x^{3} \log \left (2\right ) - 11 \, x^{2} \log \left (2\right )^{2} + 18 \, x^{3} \log \left (x\right ) - 12 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 2 \, x^{2} \log \left (x\right )^{2} - x \log \left (2\right ) \log \left (x\right )^{2} + 69 \, x^{3} - 66 \, x^{2} \log \left (2\right ) + 4 \, x^{2} \log \left (x\right ) - 12 \, x \log \left (2\right ) \log \left (x\right ) - 3 \, x \log \left (x\right )^{2} - 39 \, x^{2} - 35 \, x \log \left (2\right ) - 36 \, x \log \left (x\right ) - \log \left (x\right )^{2} - 105 \, x - 10 \, \log \left (x\right ) - 25\right )}} \]
integrate((-10*x*log(2)+20*x^2-30*x-10)/(3*x*log(x)^3+(9*x^2*log(2)-9*x^3+ 27*x^2+45*x)*log(x)^2+(9*x^3*log(2)^2+(-18*x^4+54*x^3+90*x^2)*log(2)+9*x^5 -54*x^4-9*x^3+270*x^2+225*x)*log(x)+3*x^4*log(2)^3+(-9*x^5+27*x^4+45*x^3)* log(2)^2+(9*x^6-54*x^5-9*x^4+270*x^3+225*x^2)*log(2)-3*x^7+27*x^6-36*x^5-1 89*x^4+180*x^3+675*x^2+375*x),x, algorithm=\
5/3*(2*x^2 - x*log(2) - 3*x - 1)/(2*x^6 - 5*x^5*log(2) + 4*x^4*log(2)^2 - x^3*log(2)^3 - 15*x^5 + 24*x^4*log(2) - 9*x^3*log(2)^2 - 4*x^4*log(x) + 6* x^3*log(2)*log(x) - 2*x^2*log(2)^2*log(x) + 15*x^4 + 5*x^3*log(2) - 11*x^2 *log(2)^2 + 18*x^3*log(x) - 12*x^2*log(2)*log(x) + 2*x^2*log(x)^2 - x*log( 2)*log(x)^2 + 69*x^3 - 66*x^2*log(2) + 4*x^2*log(x) - 12*x*log(2)*log(x) - 3*x*log(x)^2 - 39*x^2 - 35*x*log(2) - 36*x*log(x) - log(x)^2 - 105*x - 10 *log(x) - 25)
Timed out. \[ \int \frac {-10-30 x+20 x^2-10 x \log (2)}{375 x+675 x^2+180 x^3-189 x^4-36 x^5+27 x^6-3 x^7+\left (225 x^2+270 x^3-9 x^4-54 x^5+9 x^6\right ) \log (2)+\left (45 x^3+27 x^4-9 x^5\right ) \log ^2(2)+3 x^4 \log ^3(2)+\left (225 x+270 x^2-9 x^3-54 x^4+9 x^5+\left (90 x^2+54 x^3-18 x^4\right ) \log (2)+9 x^3 \log ^2(2)\right ) \log (x)+\left (45 x+27 x^2-9 x^3+9 x^2 \log (2)\right ) \log ^2(x)+3 x \log ^3(x)} \, dx=\int -\frac {30\,x+10\,x\,\ln \left (2\right )-20\,x^2+10}{375\,x+3\,x^4\,{\ln \left (2\right )}^3+3\,x\,{\ln \left (x\right )}^3+\ln \left (x\right )\,\left (225\,x+9\,x^3\,{\ln \left (2\right )}^2+\ln \left (2\right )\,\left (-18\,x^4+54\,x^3+90\,x^2\right )+270\,x^2-9\,x^3-54\,x^4+9\,x^5\right )+{\ln \left (2\right )}^2\,\left (-9\,x^5+27\,x^4+45\,x^3\right )+\ln \left (2\right )\,\left (9\,x^6-54\,x^5-9\,x^4+270\,x^3+225\,x^2\right )+675\,x^2+180\,x^3-189\,x^4-36\,x^5+27\,x^6-3\,x^7+{\ln \left (x\right )}^2\,\left (45\,x+9\,x^2\,\ln \left (2\right )+27\,x^2-9\,x^3\right )} \,d x \]
int(-(30*x + 10*x*log(2) - 20*x^2 + 10)/(375*x + 3*x^4*log(2)^3 + 3*x*log( x)^3 + log(x)*(225*x + 9*x^3*log(2)^2 + log(2)*(90*x^2 + 54*x^3 - 18*x^4) + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5) + log(2)^2*(45*x^3 + 27*x^4 - 9*x^5) + log(2)*(225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6) + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7 + log(x)^2*(45*x + 9*x^2*log(2) + 27*x^ 2 - 9*x^3)),x)
int(-(30*x + 10*x*log(2) - 20*x^2 + 10)/(375*x + 3*x^4*log(2)^3 + 3*x*log( x)^3 + log(x)*(225*x + 9*x^3*log(2)^2 + log(2)*(90*x^2 + 54*x^3 - 18*x^4) + 270*x^2 - 9*x^3 - 54*x^4 + 9*x^5) + log(2)^2*(45*x^3 + 27*x^4 - 9*x^5) + log(2)*(225*x^2 + 270*x^3 - 9*x^4 - 54*x^5 + 9*x^6) + 675*x^2 + 180*x^3 - 189*x^4 - 36*x^5 + 27*x^6 - 3*x^7 + log(x)^2*(45*x + 9*x^2*log(2) + 27*x^ 2 - 9*x^3)), x)