Integrand size = 89, antiderivative size = 30 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=2 x-\left (x+e^x x-\log (x)\right )^2-4 x^2 \log ^2\left (x^2\right ) \] Output:
2*x-4*x^2*ln(x^2)^2-(x+exp(x)*x-ln(x))^2
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=2 \left (1+e^x\right ) x \log (x)-\log ^2(x)-x \left (-2+\left (1+e^x\right )^2 x+4 x \log ^2\left (x^2\right )\right ) \] Input:
Integrate[(4*x - 2*x^2 + E^x*(2*x - 4*x^2 - 2*x^3) + E^(2*x)*(-2*x^2 - 2*x ^3) + (-2 + 2*x + E^x*(2*x + 2*x^2))*Log[x] - 16*x^2*Log[x^2] - 8*x^2*Log[ x^2]^2)/x,x]
Output:
2*(1 + E^x)*x*Log[x] - Log[x]^2 - x*(-2 + (1 + E^x)^2*x + 4*x*Log[x^2]^2)
Time = 0.37 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {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.
\(\displaystyle \int \frac {-2 x^2-8 x^2 \log ^2\left (x^2\right )-16 x^2 \log \left (x^2\right )+\left (e^x \left (2 x^2+2 x\right )+2 x-2\right ) \log (x)+e^x \left (-2 x^3-4 x^2+2 x\right )+e^{2 x} \left (-2 x^3-2 x^2\right )+4 x}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {2 \left (x^2+4 x^2 \log ^2\left (x^2\right )+8 x^2 \log \left (x^2\right )-2 x-x \log (x)+\log (x)\right )}{x}-2 e^x \left (x^2+2 x-x \log (x)-\log (x)-1\right )-2 e^{2 x} x (x+1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^{2 x} x^2-x^2-4 x^2 \log ^2\left (x^2\right )-2 e^x \left (x^2-x \log (x)\right )+2 x-\log ^2(x)+2 x \log (x)\) |
Input:
Int[(4*x - 2*x^2 + E^x*(2*x - 4*x^2 - 2*x^3) + E^(2*x)*(-2*x^2 - 2*x^3) + (-2 + 2*x + E^x*(2*x + 2*x^2))*Log[x] - 16*x^2*Log[x^2] - 8*x^2*Log[x^2]^2 )/x,x]
Output:
2*x - x^2 - E^(2*x)*x^2 + 2*x*Log[x] - Log[x]^2 - 2*E^x*(x^2 - x*Log[x]) - 4*x^2*Log[x^2]^2
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 = 30.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83
method | result | size |
default | \(2 x -2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{x} \ln \left (x \right )-x^{2}-4 x^{2} \ln \left (x^{2}\right )^{2}+2 x \ln \left (x \right )-\ln \left (x \right )^{2}-{\mathrm e}^{2 x} x^{2}\) | \(55\) |
parallelrisch | \(2 x -2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{x} \ln \left (x \right )-x^{2}-4 x^{2} \ln \left (x^{2}\right )^{2}+2 x \ln \left (x \right )-\ln \left (x \right )^{2}-{\mathrm e}^{2 x} x^{2}\) | \(55\) |
parts | \(2 x -2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{x} \ln \left (x \right )-x^{2}-4 x^{2} \ln \left (x^{2}\right )^{2}+2 x \ln \left (x \right )-\ln \left (x \right )^{2}-{\mathrm e}^{2 x} x^{2}\) | \(55\) |
risch | \(\left (-16 x^{2}-1\right ) \ln \left (x \right )^{2}+\left (8 i x^{2} \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-16 i x^{2} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+8 i x^{2} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 \,{\mathrm e}^{x} x +2 x \right ) \ln \left (x \right )+x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+6 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-4 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+x^{2} \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}-{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{2}-x^{2}+2 x\) | \(217\) |
orering | \(\text {Expression too large to display}\) | \(4514\) |
Input:
int((-8*x^2*ln(x^2)^2-16*x^2*ln(x^2)+((2*x^2+2*x)*exp(x)+2*x-2)*ln(x)+(-2* x^3-2*x^2)*exp(x)^2+(-2*x^3-4*x^2+2*x)*exp(x)-2*x^2+4*x)/x,x,method=_RETUR NVERBOSE)
Output:
2*x-2*exp(x)*x^2+2*x*exp(x)*ln(x)-x^2-4*x^2*ln(x^2)^2+2*x*ln(x)-ln(x)^2-ex p(x)^2*x^2
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=-x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} - {\left (16 \, x^{2} + 1\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x e^{x} + x\right )} \log \left (x\right ) + 2 \, x \] Input:
integrate((-8*x^2*log(x^2)^2-16*x^2*log(x^2)+((2*x^2+2*x)*exp(x)+2*x-2)*lo g(x)+(-2*x^3-2*x^2)*exp(x)^2+(-2*x^3-4*x^2+2*x)*exp(x)-2*x^2+4*x)/x,x, alg orithm="fricas")
Output:
-x^2*e^(2*x) - 2*x^2*e^x - (16*x^2 + 1)*log(x)^2 - x^2 + 2*(x*e^x + x)*log (x) + 2*x
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=- x^{2} e^{2 x} - x^{2} + 2 x \log {\left (x \right )} + 2 x + \left (- 16 x^{2} - 1\right ) \log {\left (x \right )}^{2} + \left (- 2 x^{2} + 2 x \log {\left (x \right )}\right ) e^{x} \] Input:
integrate((-8*x**2*ln(x**2)**2-16*x**2*ln(x**2)+((2*x**2+2*x)*exp(x)+2*x-2 )*ln(x)+(-2*x**3-2*x**2)*exp(x)**2+(-2*x**3-4*x**2+2*x)*exp(x)-2*x**2+4*x) /x,x)
Output:
-x**2*exp(2*x) - x**2 + 2*x*log(x) + 2*x + (-16*x**2 - 1)*log(x)**2 + (-2* x**2 + 2*x*log(x))*exp(x)
\[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=\int { -\frac {2 \, {\left (4 \, x^{2} \log \left (x^{2}\right )^{2} + 8 \, x^{2} \log \left (x^{2}\right ) + x^{2} + {\left (x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + 2 \, x^{2} - x\right )} e^{x} - {\left ({\left (x^{2} + x\right )} e^{x} + x - 1\right )} \log \left (x\right ) - 2 \, x\right )}}{x} \,d x } \] Input:
integrate((-8*x^2*log(x^2)^2-16*x^2*log(x^2)+((2*x^2+2*x)*exp(x)+2*x-2)*lo g(x)+(-2*x^3-2*x^2)*exp(x)^2+(-2*x^3-4*x^2+2*x)*exp(x)-2*x^2+4*x)/x,x, alg orithm="maxima")
Output:
-16*x^2*log(x)^2 + 2*(x - 1)*e^x*log(x) - x^2 - 1/2*(2*x^2 - 2*x + 1)*e^(2 *x) - 1/2*(2*x - 1)*e^(2*x) - 2*(x^2 - 2*x + 2)*e^x - 4*(x - 1)*e^x + 2*x* log(x) + 2*e^x*log(x) - log(x)^2 + 2*x - 2*Ei(x) + 2*e^x - 2*integrate((x - 1)*e^x/x, x)
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=-16 \, x^{2} \log \left (x\right )^{2} - x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + 2 \, x e^{x} \log \left (x\right ) - x^{2} + 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, x \] Input:
integrate((-8*x^2*log(x^2)^2-16*x^2*log(x^2)+((2*x^2+2*x)*exp(x)+2*x-2)*lo g(x)+(-2*x^3-2*x^2)*exp(x)^2+(-2*x^3-4*x^2+2*x)*exp(x)-2*x^2+4*x)/x,x, alg orithm="giac")
Output:
-16*x^2*log(x)^2 - x^2*e^(2*x) - 2*x^2*e^x + 2*x*e^x*log(x) - x^2 + 2*x*lo g(x) - log(x)^2 + 2*x
Time = 3.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=2\,x-2\,x^2\,{\mathrm {e}}^x-{\ln \left (x\right )}^2-x^2\,{\mathrm {e}}^{2\,x}+2\,x\,\ln \left (x\right )-x^2-4\,x^2\,{\ln \left (x^2\right )}^2+2\,x\,{\mathrm {e}}^x\,\ln \left (x\right ) \] Input:
int(-(exp(2*x)*(2*x^2 + 2*x^3) - log(x)*(2*x + exp(x)*(2*x + 2*x^2) - 2) - 4*x + 16*x^2*log(x^2) + 2*x^2 + 8*x^2*log(x^2)^2 + exp(x)*(4*x^2 - 2*x + 2*x^3))/x,x)
Output:
2*x - 2*x^2*exp(x) - log(x)^2 - x^2*exp(2*x) + 2*x*log(x) - x^2 - 4*x^2*lo g(x^2)^2 + 2*x*exp(x)*log(x)
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \frac {4 x-2 x^2+e^x \left (2 x-4 x^2-2 x^3\right )+e^{2 x} \left (-2 x^2-2 x^3\right )+\left (-2+2 x+e^x \left (2 x+2 x^2\right )\right ) \log (x)-16 x^2 \log \left (x^2\right )-8 x^2 \log ^2\left (x^2\right )}{x} \, dx=-e^{2 x} x^{2}+2 e^{x} \mathrm {log}\left (x \right ) x -2 e^{x} x^{2}-4 \mathrm {log}\left (x^{2}\right )^{2} x^{2}-\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x -x^{2}+2 x \] Input:
int((-8*x^2*log(x^2)^2-16*x^2*log(x^2)+((2*x^2+2*x)*exp(x)+2*x-2)*log(x)+( -2*x^3-2*x^2)*exp(x)^2+(-2*x^3-4*x^2+2*x)*exp(x)-2*x^2+4*x)/x,x)
Output:
- e**(2*x)*x**2 + 2*e**x*log(x)*x - 2*e**x*x**2 - 4*log(x**2)**2*x**2 - l og(x)**2 + 2*log(x)*x - x**2 + 2*x