Integrand size = 45, antiderivative size = 31 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx=4+\frac {3 x^2 \left (-x+5 \left (\frac {1}{x}+x\right )\right ) (3-\log (x))^2}{e^3}+\log (x) \] Output:
3/exp(3)*(4*x+5/x)*x^2*(3-ln(x))^2+ln(x)+4
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx=\frac {135 x+108 x^3+e^3 \log (x)-90 x \log (x)-72 x^3 \log (x)+15 x \log ^2(x)+12 x^3 \log ^2(x)}{e^3} \] Input:
Integrate[(E^3 + 45*x + 252*x^3 + (-60*x - 192*x^3)*Log[x] + (15*x + 36*x^ 3)*Log[x]^2)/(E^3*x),x]
Output:
(135*x + 108*x^3 + E^3*Log[x] - 90*x*Log[x] - 72*x^3*Log[x] + 15*x*Log[x]^ 2 + 12*x^3*Log[x]^2)/E^3
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 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.
\(\displaystyle \int \frac {252 x^3+\left (36 x^3+15 x\right ) \log ^2(x)+\left (-192 x^3-60 x\right ) \log (x)+45 x+e^3}{e^3 x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {252 x^3+45 x+3 \left (12 x^3+5 x\right ) \log ^2(x)-12 \left (16 x^3+5 x\right ) \log (x)+e^3}{x}dx}{e^3}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {\int \left (3 \left (12 x^2+5\right ) \log ^2(x)-12 \left (16 x^2+5\right ) \log (x)+\frac {252 x^3+45 x+e^3}{x}\right )dx}{e^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {108 x^3+12 x^3 \log ^2(x)-72 x^3 \log (x)+135 x+15 x \log ^2(x)-90 x \log (x)+e^3 \log (x)}{e^3}\) |
Input:
Int[(E^3 + 45*x + 252*x^3 + (-60*x - 192*x^3)*Log[x] + (15*x + 36*x^3)*Log [x]^2)/(E^3*x),x]
Output:
(135*x + 108*x^3 + E^3*Log[x] - 90*x*Log[x] - 72*x^3*Log[x] + 15*x*Log[x]^ 2 + 12*x^3*Log[x]^2)/E^3
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 = 0.47 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48
method | result | size |
risch | \({\mathrm e}^{-3} \left (12 x^{3}+15 x \right ) \ln \left (x \right )^{2}+{\mathrm e}^{-3} \left (-72 x^{3}-90 x \right ) \ln \left (x \right )+108 \,{\mathrm e}^{-3} x^{3}+135 x \,{\mathrm e}^{-3}+\ln \left (x \right )\) | \(46\) |
default | \({\mathrm e}^{-3} \left (12 x^{3} \ln \left (x \right )^{2}-72 x^{3} \ln \left (x \right )+108 x^{3}+15 x \ln \left (x \right )^{2}-90 x \ln \left (x \right )+135 x +{\mathrm e}^{3} \ln \left (x \right )\right )\) | \(48\) |
parallelrisch | \({\mathrm e}^{-3} \left (12 x^{3} \ln \left (x \right )^{2}-72 x^{3} \ln \left (x \right )+108 x^{3}+15 x \ln \left (x \right )^{2}-90 x \ln \left (x \right )+135 x +{\mathrm e}^{3} \ln \left (x \right )\right )\) | \(48\) |
norman | \(12 \,{\mathrm e}^{-3} \ln \left (x \right )^{2} x^{3}-72 \,{\mathrm e}^{-3} \ln \left (x \right ) x^{3}+15 \,{\mathrm e}^{-3} \ln \left (x \right )^{2} x +108 \,{\mathrm e}^{-3} x^{3}-90 \,{\mathrm e}^{-3} \ln \left (x \right ) x +135 x \,{\mathrm e}^{-3}+\ln \left (x \right )\) | \(64\) |
parts | \(84 \,{\mathrm e}^{-3} x^{3}+\ln \left (x \right )+45 x \,{\mathrm e}^{-3}+3 \,{\mathrm e}^{-3} \left (4 x^{3} \ln \left (x \right )^{2}-\frac {8 x^{3} \ln \left (x \right )}{3}+\frac {8 x^{3}}{9}+5 x \ln \left (x \right )^{2}-10 x \ln \left (x \right )+10 x \right )-12 \,{\mathrm e}^{-3} \left (\frac {16 x^{3} \ln \left (x \right )}{3}-\frac {16 x^{3}}{9}+5 x \ln \left (x \right )-5 x \right )\) | \(90\) |
Input:
int(((36*x^3+15*x)*ln(x)^2+(-192*x^3-60*x)*ln(x)+exp(3)+252*x^3+45*x)/x/ex p(3),x,method=_RETURNVERBOSE)
Output:
exp(-3)*(12*x^3+15*x)*ln(x)^2+exp(-3)*(-72*x^3-90*x)*ln(x)+108*exp(-3)*x^3 +135*x*exp(-3)+ln(x)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx={\left (108 \, x^{3} + 3 \, {\left (4 \, x^{3} + 5 \, x\right )} \log \left (x\right )^{2} - {\left (72 \, x^{3} + 90 \, x - e^{3}\right )} \log \left (x\right ) + 135 \, x\right )} e^{\left (-3\right )} \] Input:
integrate(((36*x^3+15*x)*log(x)^2+(-192*x^3-60*x)*log(x)+exp(3)+252*x^3+45 *x)/x/exp(3),x, algorithm="fricas")
Output:
(108*x^3 + 3*(4*x^3 + 5*x)*log(x)^2 - (72*x^3 + 90*x - e^3)*log(x) + 135*x )*e^(-3)
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx=\frac {\left (- 72 x^{3} - 90 x\right ) \log {\left (x \right )}}{e^{3}} + \frac {\left (12 x^{3} + 15 x\right ) \log {\left (x \right )}^{2}}{e^{3}} + \frac {108 x^{3} + 135 x + e^{3} \log {\left (x \right )}}{e^{3}} \] Input:
integrate(((36*x**3+15*x)*ln(x)**2+(-192*x**3-60*x)*ln(x)+exp(3)+252*x**3+ 45*x)/x/exp(3),x)
Output:
(-72*x**3 - 90*x)*exp(-3)*log(x) + (12*x**3 + 15*x)*exp(-3)*log(x)**2 + (1 08*x**3 + 135*x + exp(3)*log(x))*exp(-3)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx=\frac {1}{3} \, {\left (4 \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} - 192 \, x^{3} \log \left (x\right ) + 316 \, x^{3} + 45 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - 180 \, x \log \left (x\right ) + 3 \, e^{3} \log \left (x\right ) + 315 \, x\right )} e^{\left (-3\right )} \] Input:
integrate(((36*x^3+15*x)*log(x)^2+(-192*x^3-60*x)*log(x)+exp(3)+252*x^3+45 *x)/x/exp(3),x, algorithm="maxima")
Output:
1/3*(4*(9*log(x)^2 - 6*log(x) + 2)*x^3 - 192*x^3*log(x) + 316*x^3 + 45*(lo g(x)^2 - 2*log(x) + 2)*x - 180*x*log(x) + 3*e^3*log(x) + 315*x)*e^(-3)
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx={\left (12 \, x^{3} \log \left (x\right )^{2} - 72 \, x^{3} \log \left (x\right ) + 108 \, x^{3} + 15 \, x \log \left (x\right )^{2} - 90 \, x \log \left (x\right ) + e^{3} \log \left (x\right ) + 135 \, x\right )} e^{\left (-3\right )} \] Input:
integrate(((36*x^3+15*x)*log(x)^2+(-192*x^3-60*x)*log(x)+exp(3)+252*x^3+45 *x)/x/exp(3),x, algorithm="giac")
Output:
(12*x^3*log(x)^2 - 72*x^3*log(x) + 108*x^3 + 15*x*log(x)^2 - 90*x*log(x) + e^3*log(x) + 135*x)*e^(-3)
Time = 0.65 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx=12\,{\mathrm {e}}^{-3}\,x^3\,{\ln \left (x\right )}^2-72\,{\mathrm {e}}^{-3}\,x^3\,\ln \left (x\right )+108\,{\mathrm {e}}^{-3}\,x^3+15\,{\mathrm {e}}^{-3}\,x\,{\ln \left (x\right )}^2-90\,{\mathrm {e}}^{-3}\,x\,\ln \left (x\right )+135\,{\mathrm {e}}^{-3}\,x+\ln \left (x\right ) \] Input:
int((exp(-3)*(45*x + exp(3) + log(x)^2*(15*x + 36*x^3) - log(x)*(60*x + 19 2*x^3) + 252*x^3))/x,x)
Output:
log(x) + 135*x*exp(-3) + 108*x^3*exp(-3) - 90*x*exp(-3)*log(x) + 15*x*exp( -3)*log(x)^2 - 72*x^3*exp(-3)*log(x) + 12*x^3*exp(-3)*log(x)^2
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^3+45 x+252 x^3+\left (-60 x-192 x^3\right ) \log (x)+\left (15 x+36 x^3\right ) \log ^2(x)}{e^3 x} \, dx=\frac {12 \mathrm {log}\left (x \right )^{2} x^{3}+15 \mathrm {log}\left (x \right )^{2} x +\mathrm {log}\left (x \right ) e^{3}-72 \,\mathrm {log}\left (x \right ) x^{3}-90 \,\mathrm {log}\left (x \right ) x +108 x^{3}+135 x}{e^{3}} \] Input:
int(((36*x^3+15*x)*log(x)^2+(-192*x^3-60*x)*log(x)+exp(3)+252*x^3+45*x)/x/ exp(3),x)
Output:
(12*log(x)**2*x**3 + 15*log(x)**2*x + log(x)*e**3 - 72*log(x)*x**3 - 90*lo g(x)*x + 108*x**3 + 135*x)/e**3