Integrand size = 58, antiderivative size = 24 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=x \left (-x+3 \left (-2-\frac {2 x^2}{e (-2+\log (x))}\right )\right ) \] Output:
x*(-6-6*x^2/exp(1)/(ln(x)-2)-x)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=-\frac {2 \left (\frac {1}{2} e x (6+x)+\frac {3 x^3}{-2+\log (x)}\right )}{e} \] Input:
Integrate[(E*(-24 - 8*x) + 42*x^2 + (-18*x^2 + E*(24 + 8*x))*Log[x] + E*(- 6 - 2*x)*Log[x]^2)/(4*E - 4*E*Log[x] + E*Log[x]^2),x]
Output:
(-2*((E*x*(6 + x))/2 + (3*x^3)/(-2 + Log[x])))/E
Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {7292, 27, 27, 7293, 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.
\(\displaystyle \int \frac {42 x^2+\left (e (8 x+24)-18 x^2\right ) \log (x)+e (-8 x-24)+e (-2 x-6) \log ^2(x)}{e \log ^2(x)-4 e \log (x)+4 e} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {42 x^2+\left (e (8 x+24)-18 x^2\right ) \log (x)+e (-8 x-24)+e (-2 x-6) \log ^2(x)}{e (2-\log (x))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {2 \left (21 x^2-e (x+3) \log ^2(x)-4 e (x+3)-\left (9 x^2-4 e (x+3)\right ) \log (x)\right )}{(2-\log (x))^2}dx}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \int \frac {21 x^2-e (x+3) \log ^2(x)-4 e (x+3)-\left (9 x^2-4 e (x+3)\right ) \log (x)}{(2-\log (x))^2}dx}{e}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2 \int \left (-\frac {9 x^2}{\log (x)-2}+\frac {3 x^2}{(\log (x)-2)^2}-e (x+3)\right )dx}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {3 x^3}{2-\log (x)}-\frac {1}{2} e (x+3)^2\right )}{e}\) |
Input:
Int[(E*(-24 - 8*x) + 42*x^2 + (-18*x^2 + E*(24 + 8*x))*Log[x] + E*(-6 - 2* x)*Log[x]^2)/(4*E - 4*E*Log[x] + E*Log[x]^2),x]
Output:
(2*(-1/2*(E*(3 + x)^2) + (3*x^3)/(2 - Log[x])))/E
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-x^{2}-6 x -\frac {6 x^{3} {\mathrm e}^{-1}}{\ln \left (x \right )-2}\) | \(23\) |
norman | \(\frac {12 x +2 x^{2}-6 x \ln \left (x \right )-x^{2} \ln \left (x \right )-6 x^{3} {\mathrm e}^{-1}}{\ln \left (x \right )-2}\) | \(38\) |
parallelrisch | \(\frac {\left (-{\mathrm e} x^{2} \ln \left (x \right )+2 x^{2} {\mathrm e}-6 x \,{\mathrm e} \ln \left (x \right )-6 x^{3}+12 x \,{\mathrm e}\right ) {\mathrm e}^{-1}}{\ln \left (x \right )-2}\) | \(46\) |
default | \(-2 \,{\mathrm e}^{-1} \left ({\mathrm e} \left (\frac {x^{2}}{2}-12 \,{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )+4\right )-\frac {4 x^{2}}{\ln \left (x \right )-2}\right )+\frac {3 x^{3}}{\ln \left (x \right )-2}+12 \,{\mathrm e} \left (-\frac {x}{\ln \left (x \right )-2}-{\mathrm e}^{2} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )+2\right )\right )+4 \,{\mathrm e} \left (-\frac {x^{2}}{\ln \left (x \right )-2}-2 \,{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )+4\right )\right )-12 \,{\mathrm e} \left (-3 \,{\mathrm e}^{2} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )+2\right )-\frac {2 x}{\ln \left (x \right )-2}\right )+3 \,{\mathrm e} \left (x -8 \,{\mathrm e}^{2} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )+2\right )-\frac {4 x}{\ln \left (x \right )-2}\right )-4 \,{\mathrm e} \left (-5 \,{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )+4\right )-\frac {2 x^{2}}{\ln \left (x \right )-2}\right )\right )\) | \(186\) |
Input:
int(((-2*x-6)*exp(1)*ln(x)^2+((8*x+24)*exp(1)-18*x^2)*ln(x)+(-8*x-24)*exp( 1)+42*x^2)/(exp(1)*ln(x)^2-4*exp(1)*ln(x)+4*exp(1)),x,method=_RETURNVERBOS E)
Output:
-x^2-6*x-6*x^3*exp(-1)/(ln(x)-2)
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=-\frac {6 \, x^{3} + {\left (x^{2} + 6 \, x\right )} e \log \left (x\right ) - 2 \, {\left (x^{2} + 6 \, x\right )} e}{e \log \left (x\right ) - 2 \, e} \] Input:
integrate(((-2*x-6)*exp(1)*log(x)^2+((8*x+24)*exp(1)-18*x^2)*log(x)+(-8*x- 24)*exp(1)+42*x^2)/(exp(1)*log(x)^2-4*exp(1)*log(x)+4*exp(1)),x, algorithm ="fricas")
Output:
-(6*x^3 + (x^2 + 6*x)*e*log(x) - 2*(x^2 + 6*x)*e)/(e*log(x) - 2*e)
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=- \frac {6 x^{3}}{e \log {\left (x \right )} - 2 e} - x^{2} - 6 x \] Input:
integrate(((-2*x-6)*exp(1)*ln(x)**2+((8*x+24)*exp(1)-18*x**2)*ln(x)+(-8*x- 24)*exp(1)+42*x**2)/(exp(1)*ln(x)**2-4*exp(1)*ln(x)+4*exp(1)),x)
Output:
-6*x**3/(E*log(x) - 2*E) - x**2 - 6*x
\[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=\int { -\frac {2 \, {\left ({\left (x + 3\right )} e \log \left (x\right )^{2} - 21 \, x^{2} + 4 \, {\left (x + 3\right )} e + {\left (9 \, x^{2} - 4 \, {\left (x + 3\right )} e\right )} \log \left (x\right )\right )}}{e \log \left (x\right )^{2} - 4 \, e \log \left (x\right ) + 4 \, e} \,d x } \] Input:
integrate(((-2*x-6)*exp(1)*log(x)^2+((8*x+24)*exp(1)-18*x^2)*log(x)+(-8*x- 24)*exp(1)+42*x^2)/(exp(1)*log(x)^2-4*exp(1)*log(x)+4*exp(1)),x, algorithm ="maxima")
Output:
24*e^2*exp_integral_e(2, -log(x) + 2)/(log(x) - 2) + 8*e^4*exp_integral_e( 2, -2*log(x) + 4)/(log(x) - 2) - 42*e^5*exp_integral_e(2, -3*log(x) + 6)/( log(x) - 2) + (36*x^3 - 6*x^2*e - 12*x*e - (x^2*e + 6*x*e)*log(x))/(e*log( x) - 2*e) - 2*integrate((63*x^2 - 8*x*e - 12*e)/(e*log(x) - 2*e), x)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=-\frac {x^{2} e \log \left (x\right )}{e \log \left (x\right ) - 2 \, e} - \frac {6 \, x^{3}}{e \log \left (x\right ) - 2 \, e} + \frac {2 \, x^{2} e}{e \log \left (x\right ) - 2 \, e} - \frac {6 \, x e \log \left (x\right )}{e \log \left (x\right ) - 2 \, e} + \frac {12 \, x e}{e \log \left (x\right ) - 2 \, e} \] Input:
integrate(((-2*x-6)*exp(1)*log(x)^2+((8*x+24)*exp(1)-18*x^2)*log(x)+(-8*x- 24)*exp(1)+42*x^2)/(exp(1)*log(x)^2-4*exp(1)*log(x)+4*exp(1)),x, algorithm ="giac")
Output:
-x^2*e*log(x)/(e*log(x) - 2*e) - 6*x^3/(e*log(x) - 2*e) + 2*x^2*e/(e*log(x ) - 2*e) - 6*x*e*log(x)/(e*log(x) - 2*e) + 12*x*e/(e*log(x) - 2*e)
Time = 3.70 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=\frac {6\,x^4}{2\,x\,\mathrm {e}-x\,\mathrm {e}\,\ln \left (x\right )}-\frac {{\mathrm {e}}^{-1}\,\left (\mathrm {e}\,x^3+6\,\mathrm {e}\,x^2\right )}{x} \] Input:
int(-(log(x)*(18*x^2 - exp(1)*(8*x + 24)) - 42*x^2 + exp(1)*(8*x + 24) + e xp(1)*log(x)^2*(2*x + 6))/(4*exp(1) + exp(1)*log(x)^2 - 4*exp(1)*log(x)),x )
Output:
(6*x^4)/(2*x*exp(1) - x*exp(1)*log(x)) - (exp(-1)*(6*x^2*exp(1) + x^3*exp( 1)))/x
Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {e (-24-8 x)+42 x^2+\left (-18 x^2+e (24+8 x)\right ) \log (x)+e (-6-2 x) \log ^2(x)}{4 e-4 e \log (x)+e \log ^2(x)} \, dx=\frac {x \left (-\mathrm {log}\left (x \right ) e x -6 \,\mathrm {log}\left (x \right ) e +2 e x +12 e -6 x^{2}\right )}{e \left (\mathrm {log}\left (x \right )-2\right )} \] Input:
int(((-2*x-6)*exp(1)*log(x)^2+((8*x+24)*exp(1)-18*x^2)*log(x)+(-8*x-24)*ex p(1)+42*x^2)/(exp(1)*log(x)^2-4*exp(1)*log(x)+4*exp(1)),x)
Output:
(x*( - log(x)*e*x - 6*log(x)*e + 2*e*x + 12*e - 6*x**2))/(e*(log(x) - 2))