Integrand size = 80, antiderivative size = 26 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=\left (-4 e^{3/5}+e^{e^3}\right ) \left (-21+e^x-x\right ) \log \left (x^2\right ) \] Output:
ln(x^2)*(exp(x)-x-21)*(exp(exp(3))-4*exp(3/5))
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=\left (-4 e^{3/5}+e^{e^3}\right ) \left (-42 \log (x)+\left (e^x-x\right ) \log \left (x^2\right )\right ) \] Input:
Integrate[(-8*E^(3/5 + x) + E^E^3*(-42 + 2*E^x - 2*x) + E^(3/5)*(168 + 8*x ) + (4*E^(3/5)*x - 4*E^(3/5 + x)*x + E^E^3*(-x + E^x*x))*Log[x^2])/x,x]
Output:
(-4*E^(3/5) + E^E^3)*(-42*Log[x] + (E^x - x)*Log[x^2])
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).
Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, 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.
\(\displaystyle \int \frac {\left (-4 e^{x+\frac {3}{5}} x+4 e^{3/5} x+e^{e^3} \left (e^x x-x\right )\right ) \log \left (x^2\right )+e^{e^3} \left (-2 x+2 e^x-42\right )-8 e^{x+\frac {3}{5}}+e^{3/5} (8 x+168)}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {\left (e^{e^3}-4 e^{3/5}\right ) e^x \left (x \log \left (x^2\right )+2\right )}{x}-\frac {\left (e^{e^3}-4 e^{3/5}\right ) \left (x \log \left (x^2\right )+2 x+42\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \left (4 e^{3/5}-e^{e^3}\right ) x \log \left (x^2\right )-\left (4 e^{3/5}-e^{e^3}\right ) e^x \log \left (x^2\right )+42 \left (4 e^{3/5}-e^{e^3}\right ) \log (x)\) |
Input:
Int[(-8*E^(3/5 + x) + E^E^3*(-42 + 2*E^x - 2*x) + E^(3/5)*(168 + 8*x) + (4 *E^(3/5)*x - 4*E^(3/5 + x)*x + E^E^3*(-x + E^x*x))*Log[x^2])/x,x]
Output:
42*(4*E^(3/5) - E^E^3)*Log[x] - E^x*(4*E^(3/5) - E^E^3)*Log[x^2] + (4*E^(3 /5) - E^E^3)*x*Log[x^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]]
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(20)=40\).
Time = 0.68 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85
method | result | size |
norman | \(\left (-21 \,{\mathrm e}^{{\mathrm e}^{3}}+84 \,{\mathrm e}^{\frac {3}{5}}\right ) \ln \left (x^{2}\right )+\left (-{\mathrm e}^{{\mathrm e}^{3}}+4 \,{\mathrm e}^{\frac {3}{5}}\right ) x \ln \left (x^{2}\right )+\left ({\mathrm e}^{{\mathrm e}^{3}}-4 \,{\mathrm e}^{\frac {3}{5}}\right ) {\mathrm e}^{x} \ln \left (x^{2}\right )\) | \(48\) |
parallelrisch | \(-{\mathrm e}^{{\mathrm e}^{3}} \ln \left (x^{2}\right ) x +{\mathrm e}^{x} \ln \left (x^{2}\right ) {\mathrm e}^{{\mathrm e}^{3}}+4 \,{\mathrm e}^{\frac {3}{5}} \ln \left (x^{2}\right ) x -4 \,{\mathrm e}^{x} \ln \left (x^{2}\right ) {\mathrm e}^{\frac {3}{5}}-21 \ln \left (x^{2}\right ) {\mathrm e}^{{\mathrm e}^{3}}+84 \ln \left (x^{2}\right ) {\mathrm e}^{\frac {3}{5}}\) | \(58\) |
default | \(\left ({\mathrm e}^{{\mathrm e}^{3}} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )-4 \,{\mathrm e}^{\frac {3}{5}} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )\right ) {\mathrm e}^{x}+\left (2 \,{\mathrm e}^{{\mathrm e}^{3}}-8 \,{\mathrm e}^{\frac {3}{5}}\right ) {\mathrm e}^{x} \ln \left (x \right )+\left (-{\mathrm e}^{{\mathrm e}^{3}}+4 \,{\mathrm e}^{\frac {3}{5}}\right ) \left (x \ln \left (x^{2}\right )-2 x \right )-2 \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \,{\mathrm e}^{\frac {3}{5}}\right ) \left (x +21 \ln \left (x \right )\right )\) | \(84\) |
parts | \(\left ({\mathrm e}^{{\mathrm e}^{3}} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )-4 \,{\mathrm e}^{\frac {3}{5}} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )\right ) {\mathrm e}^{x}+\left (2 \,{\mathrm e}^{{\mathrm e}^{3}}-8 \,{\mathrm e}^{\frac {3}{5}}\right ) {\mathrm e}^{x} \ln \left (x \right )+\left (-{\mathrm e}^{{\mathrm e}^{3}}+4 \,{\mathrm e}^{\frac {3}{5}}\right ) \left (x \ln \left (x^{2}\right )-2 x \right )-2 \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \,{\mathrm e}^{\frac {3}{5}}\right ) \left (x +21 \ln \left (x \right )\right )\) | \(84\) |
risch | \(\left (-2 x \,{\mathrm e}^{{\mathrm e}^{3}}+2 \,{\mathrm e}^{{\mathrm e}^{3}+x}+8 x \,{\mathrm e}^{\frac {3}{5}}-8 \,{\mathrm e}^{x +\frac {3}{5}}\right ) \ln \left (x \right )+\frac {i \pi \,{\mathrm e}^{{\mathrm e}^{3}} x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}-i \pi \,{\mathrm e}^{{\mathrm e}^{3}} x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\frac {i \pi \,{\mathrm e}^{{\mathrm e}^{3}} x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-2 i \pi \,{\mathrm e}^{\frac {3}{5}} x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4 i \pi \,{\mathrm e}^{\frac {3}{5}} x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-2 i \pi \,{\mathrm e}^{\frac {3}{5}} x \operatorname {csgn}\left (i x^{2}\right )^{3}-42 \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{3}}+168 \ln \left (x \right ) {\mathrm e}^{\frac {3}{5}}-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{{\mathrm e}^{3}+x}}{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{{\mathrm e}^{3}+x}-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{{\mathrm e}^{3}+x}}{2}+2 i \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} \pi \,{\mathrm e}^{x +\frac {3}{5}}-4 i \operatorname {csgn}\left (i x^{2}\right )^{2} \pi \,\operatorname {csgn}\left (i x \right ) {\mathrm e}^{x +\frac {3}{5}}+2 i \operatorname {csgn}\left (i x^{2}\right )^{3} \pi \,{\mathrm e}^{x +\frac {3}{5}}\) | \(287\) |
orering | \(\text {Expression too large to display}\) | \(1036\) |
Input:
int((((exp(x)*x-x)*exp(exp(3))-4*x*exp(3/5)*exp(x)+4*x*exp(3/5))*ln(x^2)+( 2*exp(x)-2*x-42)*exp(exp(3))-8*exp(3/5)*exp(x)+(8*x+168)*exp(3/5))/x,x,met hod=_RETURNVERBOSE)
Output:
(-21*exp(exp(3))+84*exp(3/5))*ln(x^2)+(-exp(exp(3))+4*exp(3/5))*x*ln(x^2)+ (exp(exp(3))-4*exp(3/5))*exp(x)*ln(x^2)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx={\left (4 \, {\left (x + 21\right )} e^{\frac {6}{5}} - {\left ({\left (x + 21\right )} e^{\frac {3}{5}} - e^{\left (x + \frac {3}{5}\right )}\right )} e^{\left (e^{3}\right )} - 4 \, e^{\left (x + \frac {6}{5}\right )}\right )} e^{\left (-\frac {3}{5}\right )} \log \left (x^{2}\right ) \] Input:
integrate((((exp(x)*x-x)*exp(exp(3))-4*x*exp(3/5)*exp(x)+4*x*exp(3/5))*log (x^2)+(2*exp(x)-2*x-42)*exp(exp(3))-8*exp(3/5)*exp(x)+(8*x+168)*exp(3/5))/ x,x, algorithm="fricas")
Output:
(4*(x + 21)*e^(6/5) - ((x + 21)*e^(3/5) - e^(x + 3/5))*e^(e^3) - 4*e^(x + 6/5))*e^(-3/5)*log(x^2)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=\left (- x e^{e^{3}} + 4 x e^{\frac {3}{5}}\right ) \log {\left (x^{2} \right )} + \left (- 4 e^{\frac {3}{5}} \log {\left (x^{2} \right )} + e^{e^{3}} \log {\left (x^{2} \right )}\right ) e^{x} + \left (- 42 e^{e^{3}} + 168 e^{\frac {3}{5}}\right ) \log {\left (x \right )} \] Input:
integrate((((exp(x)*x-x)*exp(exp(3))-4*x*exp(3/5)*exp(x)+4*x*exp(3/5))*ln( x**2)+(2*exp(x)-2*x-42)*exp(exp(3))-8*exp(3/5)*exp(x)+(8*x+168)*exp(3/5))/ x,x)
Output:
(-x*exp(exp(3)) + 4*x*exp(3/5))*log(x**2) + (-4*exp(3/5)*log(x**2) + exp(e xp(3))*log(x**2))*exp(x) + (-42*exp(exp(3)) + 168*exp(3/5))*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=4 \, {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e^{\frac {3}{5}} + 8 \, x e^{\frac {3}{5}} - {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e^{\left (e^{3}\right )} - 2 \, x e^{\left (e^{3}\right )} + e^{\left (x + e^{3}\right )} \log \left (x^{2}\right ) - 4 \, e^{\left (x + \frac {3}{5}\right )} \log \left (x^{2}\right ) + 168 \, e^{\frac {3}{5}} \log \left (x\right ) - 42 \, e^{\left (e^{3}\right )} \log \left (x\right ) \] Input:
integrate((((exp(x)*x-x)*exp(exp(3))-4*x*exp(3/5)*exp(x)+4*x*exp(3/5))*log (x^2)+(2*exp(x)-2*x-42)*exp(exp(3))-8*exp(3/5)*exp(x)+(8*x+168)*exp(3/5))/ x,x, algorithm="maxima")
Output:
4*(x*log(x^2) - 2*x)*e^(3/5) + 8*x*e^(3/5) - (x*log(x^2) - 2*x)*e^(e^3) - 2*x*e^(e^3) + e^(x + e^3)*log(x^2) - 4*e^(x + 3/5)*log(x^2) + 168*e^(3/5)* log(x) - 42*e^(e^3)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=4 \, x e^{\frac {3}{5}} \log \left (x^{2}\right ) - x e^{\left (e^{3}\right )} \log \left (x^{2}\right ) + e^{\left (x + e^{3}\right )} \log \left (x^{2}\right ) - 4 \, e^{\left (x + \frac {3}{5}\right )} \log \left (x^{2}\right ) + 168 \, e^{\frac {3}{5}} \log \left (x\right ) - 42 \, e^{\left (e^{3}\right )} \log \left (x\right ) \] Input:
integrate((((exp(x)*x-x)*exp(exp(3))-4*x*exp(3/5)*exp(x)+4*x*exp(3/5))*log (x^2)+(2*exp(x)-2*x-42)*exp(exp(3))-8*exp(3/5)*exp(x)+(8*x+168)*exp(3/5))/ x,x, algorithm="giac")
Output:
4*x*e^(3/5)*log(x^2) - x*e^(e^3)*log(x^2) + e^(x + e^3)*log(x^2) - 4*e^(x + 3/5)*log(x^2) + 168*e^(3/5)*log(x) - 42*e^(e^3)*log(x)
Time = 2.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=\ln \left (x^2\right )\,\left (4\,{\mathrm {e}}^{3/5}-{\mathrm {e}}^{{\mathrm {e}}^3}\right )\,\left (x-{\mathrm {e}}^x+21\right ) \] Input:
int(-(log(x^2)*(exp(exp(3))*(x - x*exp(x)) - 4*x*exp(3/5) + 4*x*exp(3/5)*e xp(x)) + exp(exp(3))*(2*x - 2*exp(x) + 42) + 8*exp(3/5)*exp(x) - exp(3/5)* (8*x + 168))/x,x)
Output:
log(x^2)*(4*exp(3/5) - exp(exp(3)))*(x - exp(x) + 21)
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-8 e^{\frac {3}{5}+x}+e^{e^3} \left (-42+2 e^x-2 x\right )+e^{3/5} (168+8 x)+\left (4 e^{3/5} x-4 e^{\frac {3}{5}+x} x+e^{e^3} \left (-x+e^x x\right )\right ) \log \left (x^2\right )}{x} \, dx=\mathrm {log}\left (x^{2}\right ) \left (e^{e^{3}+x}-e^{e^{3}} x -21 e^{e^{3}}-4 e^{x +\frac {3}{5}}+4 e^{\frac {3}{5}} x +84 e^{\frac {3}{5}}\right ) \] Input:
int((((exp(x)*x-x)*exp(exp(3))-4*x*exp(3/5)*exp(x)+4*x*exp(3/5))*log(x^2)+ (2*exp(x)-2*x-42)*exp(exp(3))-8*exp(3/5)*exp(x)+(8*x+168)*exp(3/5))/x,x)
Output:
log(x**2)*(e**(e**3 + x) - e**(e**3)*x - 21*e**(e**3) - 4*e**((5*x + 3)/5) + 4*e**(3/5)*x + 84*e**(3/5))