Integrand size = 74, antiderivative size = 22 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=\left (7+e^{-3 x}+3 x-\log \left (\frac {x^2}{4}\right )\right )^2 \] Output:
(7-ln(1/4*x^2)+3*x+1/exp(x)^3)^2
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(22)=44\).
Time = 5.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=e^{-6 x}+42 x+9 x^2+2 e^{-3 x} (7+3 x)-28 \log (x)+2 \left (-e^{-3 x}-3 x\right ) \log \left (\frac {x^2}{4}\right )+\log ^2\left (\frac {x^2}{4}\right ) \] Input:
Integrate[(-6*x + E^(3*x)*(-4 - 36*x - 18*x^2) + E^(6*x)*(-28 + 30*x + 18* x^2) + (E^(6*x)*(4 - 6*x) + 6*E^(3*x)*x)*Log[x^2/4])/(E^(6*x)*x),x]
Output:
E^(-6*x) + 42*x + 9*x^2 + (2*(7 + 3*x))/E^(3*x) - 28*Log[x] + 2*(-E^(-3*x) - 3*x)*Log[x^2/4] + Log[x^2/4]^2
Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(22)=44\).
Time = 1.43 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {7292, 27, 25, 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.
\(\displaystyle \int \frac {e^{-6 x} \left (e^{3 x} \left (-18 x^2-36 x-4\right )+e^{6 x} \left (18 x^2+30 x-28\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )-6 x\right )}{x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 e^{-6 x} \left (-3 e^{3 x} x+3 x+2 e^{3 x}\right ) \left (e^{3 x} \log \left (\frac {x^2}{4}\right )-3 e^{3 x} x-7 e^{3 x}-1\right )}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {e^{-6 x} \left (-3 e^{3 x} x+3 x+2 e^{3 x}\right ) \left (3 e^{3 x} x+7 e^{3 x}-e^{3 x} \log \left (\frac {x^2}{4}\right )+1\right )}{x}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {e^{-6 x} \left (-3 e^{3 x} x+3 x+2 e^{3 x}\right ) \left (3 e^{3 x} x+7 e^{3 x}-e^{3 x} \log \left (\frac {x^2}{4}\right )+1\right )}{x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {e^{-3 x} \left (9 x^2-3 \log \left (\frac {x^2}{4}\right ) x+18 x+2\right )}{x}+3 e^{-6 x}+\frac {(2-3 x) \left (3 x-\log \left (x^2\right )+7 \left (1+\frac {2 \log (2)}{7}\right )\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (-\frac {1}{2} \left (-\log \left (x^2\right )+3 x+7+\log (4)\right )^2+e^{-3 x} \log \left (x^2\right )-\frac {e^{-6 x}}{2}-7 e^{-3 x}-3 e^{-3 x} x-\frac {1}{3} e^{-3 x} \log (64)\right )\) |
Input:
Int[(-6*x + E^(3*x)*(-4 - 36*x - 18*x^2) + E^(6*x)*(-28 + 30*x + 18*x^2) + (E^(6*x)*(4 - 6*x) + 6*E^(3*x)*x)*Log[x^2/4])/(E^(6*x)*x),x]
Output:
-2*(-1/2*1/E^(6*x) - 7/E^(3*x) - (3*x)/E^(3*x) - Log[64]/(3*E^(3*x)) - (7 + 3*x + Log[4] - Log[x^2])^2/2 + Log[x^2]/E^(3*x))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(19)=38\).
Time = 3.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00
method | result | size |
default | \(2 \left (7+2 \ln \left (2\right )-\ln \left (x^{2}\right )+2 \ln \left (x \right )\right ) {\mathrm e}^{-3 x}+6 x \,{\mathrm e}^{-3 x}-4 \ln \left (x \right ) {\mathrm e}^{-3 x}+{\mathrm e}^{-6 x}+9 x^{2}+42 x -28 \ln \left (x \right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}-6 x \ln \left (x^{2}\right )+4 \ln \left (2\right ) \left (-2 \ln \left (x \right )+3 x \right )\) | \(88\) |
parts | \(2 \left (7+2 \ln \left (2\right )-\ln \left (x^{2}\right )+2 \ln \left (x \right )\right ) {\mathrm e}^{-3 x}+6 x \,{\mathrm e}^{-3 x}-4 \ln \left (x \right ) {\mathrm e}^{-3 x}+{\mathrm e}^{-6 x}+9 x^{2}+42 x -28 \ln \left (x \right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}-6 x \ln \left (x^{2}\right )+4 \ln \left (2\right ) \left (-2 \ln \left (x \right )+3 x \right )\) | \(88\) |
parallelrisch | \(\frac {\left (2+18 x^{2} {\mathrm e}^{6 x}-12 \,{\mathrm e}^{6 x} \ln \left (\frac {x^{2}}{4}\right ) x +2 \ln \left (\frac {x^{2}}{4}\right )^{2} {\mathrm e}^{6 x}+84 x \,{\mathrm e}^{6 x}-28 \,{\mathrm e}^{6 x} \ln \left (\frac {x^{2}}{4}\right )+12 x \,{\mathrm e}^{3 x}-4 \ln \left (\frac {x^{2}}{4}\right ) {\mathrm e}^{3 x}+28 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{-6 x}}{2}\) | \(89\) |
risch | \(3 i \operatorname {csgn}\left (i x^{2}\right )^{3} x \pi -6 i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} x \pi -2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) {\mathrm e}^{-3 x}+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{-3 x}-2 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{2}\right )^{3}+3 i \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) x \pi -2 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} {\mathrm e}^{-3 x}+4 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-8 \ln \left (2\right ) \ln \left (x \right )+12 x \ln \left (2\right )+9 x^{2}-12 x \ln \left (x \right )+4 \ln \left (2\right ) {\mathrm e}^{-3 x}-28 \ln \left (x \right )+42 x +6 x \,{\mathrm e}^{-3 x}+14 \,{\mathrm e}^{-3 x}-4 \ln \left (x \right ) {\mathrm e}^{-3 x}+4 \ln \left (x \right )^{2}+{\mathrm e}^{-6 x}\) | \(237\) |
Input:
int((((-6*x+4)*exp(x)^6+6*x*exp(x)^3)*ln(1/4*x^2)+(18*x^2+30*x-28)*exp(x)^ 6+(-18*x^2-36*x-4)*exp(x)^3-6*x)/x/exp(x)^6,x,method=_RETURNVERBOSE)
Output:
2*(7+2*ln(2)-ln(x^2)+2*ln(x))*exp(-3*x)+6*x*exp(-3*x)-4*ln(x)*exp(-3*x)+1/ exp(x)^6+9*x^2+42*x-28*ln(x)+4*ln(x)*ln(x^2)-4*ln(x)^2-6*x*ln(x^2)+4*ln(2) *(-2*ln(x)+3*x)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx={\left (e^{\left (6 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right )^{2} + 3 \, {\left (3 \, x^{2} + 14 \, x\right )} e^{\left (6 \, x\right )} + 2 \, {\left (3 \, x + 7\right )} e^{\left (3 \, x\right )} - 2 \, {\left ({\left (3 \, x + 7\right )} e^{\left (6 \, x\right )} + e^{\left (3 \, x\right )}\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + 1\right )} e^{\left (-6 \, x\right )} \] Input:
integrate((((-6*x+4)*exp(x)^6+6*x*exp(x)^3)*log(1/4*x^2)+(18*x^2+30*x-28)* exp(x)^6+(-18*x^2-36*x-4)*exp(x)^3-6*x)/x/exp(x)^6,x, algorithm="fricas")
Output:
(e^(6*x)*log(1/4*x^2)^2 + 3*(3*x^2 + 14*x)*e^(6*x) + 2*(3*x + 7)*e^(3*x) - 2*((3*x + 7)*e^(6*x) + e^(3*x))*log(1/4*x^2) + 1)*e^(-6*x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=9 x^{2} - 6 x \log {\left (\frac {x^{2}}{4} \right )} + 42 x + \left (6 x - 2 \log {\left (\frac {x^{2}}{4} \right )} + 14\right ) e^{- 3 x} - 28 \log {\left (x \right )} + \log {\left (\frac {x^{2}}{4} \right )}^{2} + e^{- 6 x} \] Input:
integrate((((-6*x+4)*exp(x)**6+6*x*exp(x)**3)*ln(1/4*x**2)+(18*x**2+30*x-2 8)*exp(x)**6+(-18*x**2-36*x-4)*exp(x)**3-6*x)/x/exp(x)**6,x)
Output:
9*x**2 - 6*x*log(x**2/4) + 42*x + (6*x - 2*log(x**2/4) + 14)*exp(-3*x) - 2 8*log(x) + log(x**2/4)**2 + exp(-6*x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (19) = 38\).
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=9 \, x^{2} + 2 \, {\left (3 \, x + 1\right )} e^{\left (-3 \, x\right )} - 6 \, x \log \left (\frac {1}{4} \, x^{2}\right ) - 2 \, e^{\left (-3 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + \log \left (\frac {1}{4} \, x^{2}\right )^{2} + 42 \, x + 12 \, e^{\left (-3 \, x\right )} + e^{\left (-6 \, x\right )} - 28 \, \log \left (x\right ) \] Input:
integrate((((-6*x+4)*exp(x)^6+6*x*exp(x)^3)*log(1/4*x^2)+(18*x^2+30*x-28)* exp(x)^6+(-18*x^2-36*x-4)*exp(x)^3-6*x)/x/exp(x)^6,x, algorithm="maxima")
Output:
9*x^2 + 2*(3*x + 1)*e^(-3*x) - 6*x*log(1/4*x^2) - 2*e^(-3*x)*log(1/4*x^2) + log(1/4*x^2)^2 + 42*x + 12*e^(-3*x) + e^(-6*x) - 28*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (19) = 38\).
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx={\left (9 \, x^{2} e^{\left (6 \, x\right )} - 6 \, x e^{\left (6 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + e^{\left (6 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right )^{2} + 42 \, x e^{\left (6 \, x\right )} + 6 \, x e^{\left (3 \, x\right )} - 2 \, e^{\left (3 \, x\right )} \log \left (\frac {1}{4} \, x^{2}\right ) - 28 \, e^{\left (6 \, x\right )} \log \left (x\right ) + 14 \, e^{\left (3 \, x\right )} + 1\right )} e^{\left (-6 \, x\right )} \] Input:
integrate((((-6*x+4)*exp(x)^6+6*x*exp(x)^3)*log(1/4*x^2)+(18*x^2+30*x-28)* exp(x)^6+(-18*x^2-36*x-4)*exp(x)^3-6*x)/x/exp(x)^6,x, algorithm="giac")
Output:
(9*x^2*e^(6*x) - 6*x*e^(6*x)*log(1/4*x^2) + e^(6*x)*log(1/4*x^2)^2 + 42*x* e^(6*x) + 6*x*e^(3*x) - 2*e^(3*x)*log(1/4*x^2) - 28*e^(6*x)*log(x) + 14*e^ (3*x) + 1)*e^(-6*x)
Time = 3.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=30\,x+14\,{\mathrm {e}}^{-3\,x}+{\mathrm {e}}^{-6\,x}-28\,\ln \left (x\right )+{\ln \left (\frac {x^2}{4}\right )}^2+6\,x\,{\mathrm {e}}^{-3\,x}-x\,\left (6\,\ln \left (\frac {x^2}{4}\right )-12\right )-2\,{\mathrm {e}}^{-3\,x}\,\ln \left (\frac {x^2}{4}\right )+9\,x^2 \] Input:
int(-(exp(-6*x)*(6*x + exp(3*x)*(36*x + 18*x^2 + 4) - exp(6*x)*(30*x + 18* x^2 - 28) - log(x^2/4)*(6*x*exp(3*x) - exp(6*x)*(6*x - 4))))/x,x)
Output:
30*x + 14*exp(-3*x) + exp(-6*x) - 28*log(x) + log(x^2/4)^2 + 6*x*exp(-3*x) - x*(6*log(x^2/4) - 12) - 2*exp(-3*x)*log(x^2/4) + 9*x^2
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.41 \[ \int \frac {e^{-6 x} \left (-6 x+e^{3 x} \left (-4-36 x-18 x^2\right )+e^{6 x} \left (-28+30 x+18 x^2\right )+\left (e^{6 x} (4-6 x)+6 e^{3 x} x\right ) \log \left (\frac {x^2}{4}\right )\right )}{x} \, dx=\frac {e^{6 x} \mathrm {log}\left (\frac {x^{2}}{4}\right )^{2}-6 e^{6 x} \mathrm {log}\left (\frac {x^{2}}{4}\right ) x -14 e^{6 x} \mathrm {log}\left (\frac {x^{2}}{4}\right )+9 e^{6 x} x^{2}+42 e^{6 x} x -2 e^{3 x} \mathrm {log}\left (\frac {x^{2}}{4}\right )+6 e^{3 x} x +14 e^{3 x}+1}{e^{6 x}} \] Input:
int((((-6*x+4)*exp(x)^6+6*x*exp(x)^3)*log(1/4*x^2)+(18*x^2+30*x-28)*exp(x) ^6+(-18*x^2-36*x-4)*exp(x)^3-6*x)/x/exp(x)^6,x)
Output:
(e**(6*x)*log(x**2/4)**2 - 6*e**(6*x)*log(x**2/4)*x - 14*e**(6*x)*log(x**2 /4) + 9*e**(6*x)*x**2 + 42*e**(6*x)*x - 2*e**(3*x)*log(x**2/4) + 6*e**(3*x )*x + 14*e**(3*x) + 1)/e**(6*x)