Integrand size = 96, antiderivative size = 26 \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\frac {\left (3+e^{10}+\frac {e^{6+2 x}}{x^2}-\log (x)\right )^2}{e^8} \] Output:
(exp(5)^2-ln(x)+3+exp(3+x)^2/x^2)^2/exp(4)^2
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\frac {\left (e^{6+2 x}+3 x^2+e^{10} x^2-x^2 \log (x)\right )^2}{e^8 x^4} \] Input:
Integrate[(-6*x^4 - 2*E^10*x^4 + E^(12 + 4*x)*(-4 + 4*x) + E^(6 + 2*x)*(-1 4*x^2 + 12*x^3 + E^10*(-4*x^2 + 4*x^3)) + (2*x^4 + E^(6 + 2*x)*(4*x^2 - 4* x^3))*Log[x])/(E^8*x^5),x]
Output:
(E^(6 + 2*x) + 3*x^2 + E^10*x^2 - x^2*Log[x])^2/(E^8*x^4)
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(26)=52\).
Time = 0.39 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {6, 27, 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.
\(\displaystyle \int \frac {-2 e^{10} x^4-6 x^4+e^{2 x+6} \left (12 x^3-14 x^2+e^{10} \left (4 x^3-4 x^2\right )\right )+\left (2 x^4+e^{2 x+6} \left (4 x^2-4 x^3\right )\right ) \log (x)+e^{4 x+12} (4 x-4)}{e^8 x^5} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (-6-2 e^{10}\right ) x^4+e^{2 x+6} \left (12 x^3-14 x^2+e^{10} \left (4 x^3-4 x^2\right )\right )+\left (2 x^4+e^{2 x+6} \left (4 x^2-4 x^3\right )\right ) \log (x)+e^{4 x+12} (4 x-4)}{e^8 x^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {2 \left (\left (3+e^{10}\right ) x^4+2 e^{4 x+12} (1-x)+e^{2 x+6} \left (-6 x^3+7 x^2+2 e^{10} \left (x^2-x^3\right )\right )-\left (x^4+2 e^{2 x+6} \left (x^2-x^3\right )\right ) \log (x)\right )}{x^5}dx}{e^8}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \int \frac {\left (3+e^{10}\right ) x^4+2 e^{4 x+12} (1-x)+e^{2 x+6} \left (-6 x^3+7 x^2+2 e^{10} \left (x^2-x^3\right )\right )-\left (x^4+2 e^{2 x+6} \left (x^2-x^3\right )\right ) \log (x)}{x^5}dx}{e^8}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {2 \int \left (-\frac {2 e^{4 x+12} (x-1)}{x^5}+\frac {3 \left (1+\frac {e^{10}}{3}\right )-\log (x)}{x}+\frac {e^{2 x+6} \left (2 \log (x) x-6 \left (1+\frac {e^{10}}{3}\right ) x-2 \log (x)+7 \left (1+\frac {2 e^{10}}{7}\right )\right )}{x^3}\right )dx}{e^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \left (-\frac {e^{4 x+12}}{2 x^4}-\frac {e^{2 x+6} \left (\left (3+e^{10}\right ) x-x \log (x)\right )}{x^3}-\frac {1}{2} \left (-\log (x)+e^{10}+3\right )^2\right )}{e^8}\) |
Input:
Int[(-6*x^4 - 2*E^10*x^4 + E^(12 + 4*x)*(-4 + 4*x) + E^(6 + 2*x)*(-14*x^2 + 12*x^3 + E^10*(-4*x^2 + 4*x^3)) + (2*x^4 + E^(6 + 2*x)*(4*x^2 - 4*x^3))* Log[x])/(E^8*x^5),x]
Output:
(-2*(-1/2*E^(12 + 4*x)/x^4 - (3 + E^10 - Log[x])^2/2 - (E^(6 + 2*x)*((3 + E^10)*x - x*Log[x]))/x^3))/E^8
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(27)=54\).
Time = 3.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23
method | result | size |
default | \({\mathrm e}^{-8} \left (\frac {\left (2 \,{\mathrm e}^{10}+6\right ) {\mathrm e}^{2 x +6}-2 \ln \left (x \right ) {\mathrm e}^{2 x +6}}{x^{2}}+\left (-2 \,{\mathrm e}^{10}-6\right ) \ln \left (x \right )+\ln \left (x \right )^{2}+\frac {{\mathrm e}^{4 x +12}}{x^{4}}\right )\) | \(58\) |
risch | \(\frac {-2 x^{4} \ln \left (x \right ) {\mathrm e}^{2}+{\mathrm e}^{-8} \ln \left (x \right )^{2} x^{4}-6 \,{\mathrm e}^{-8} \ln \left (x \right ) x^{4}+2 x^{2} {\mathrm e}^{2 x +8}+6 x^{2} {\mathrm e}^{-2+2 x}-2 \ln \left (x \right ) {\mathrm e}^{-2+2 x} x^{2}+{\mathrm e}^{4+4 x}}{x^{4}}\) | \(75\) |
parallelrisch | \(-\frac {{\mathrm e}^{-8} \left (2 \,{\mathrm e}^{10} \ln \left (x \right ) x^{4}-2 \,{\mathrm e}^{2 x +6} {\mathrm e}^{10} x^{2}-x^{4} \ln \left (x \right )^{2}+6 x^{4} \ln \left (x \right )+2 \,{\mathrm e}^{2 x +6} \ln \left (x \right ) x^{2}-6 \,{\mathrm e}^{2 x +6} x^{2}-{\mathrm e}^{4 x +12}\right )}{x^{4}}\) | \(85\) |
Input:
int((((-4*x^3+4*x^2)*exp(3+x)^2+2*x^4)*ln(x)+(-4+4*x)*exp(3+x)^4+((4*x^3-4 *x^2)*exp(5)^2+12*x^3-14*x^2)*exp(3+x)^2-2*x^4*exp(5)^2-6*x^4)/x^5/exp(4)^ 2,x,method=_RETURNVERBOSE)
Output:
1/exp(4)^2*(((2*exp(10)+6)*exp(2*x+6)-2*ln(x)*exp(2*x+6))/x^2+(-2*exp(10)- 6)*ln(x)+ln(x)^2+exp(4*x+12)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58 \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\frac {{\left (x^{4} \log \left (x\right )^{2} + 2 \, {\left (x^{2} e^{10} + 3 \, x^{2}\right )} e^{\left (2 \, x + 6\right )} - 2 \, {\left (x^{4} e^{10} + 3 \, x^{4} + x^{2} e^{\left (2 \, x + 6\right )}\right )} \log \left (x\right ) + e^{\left (4 \, x + 12\right )}\right )} e^{\left (-8\right )}}{x^{4}} \] Input:
integrate((((-4*x^3+4*x^2)*exp(3+x)^2+2*x^4)*log(x)+(-4+4*x)*exp(3+x)^4+(( 4*x^3-4*x^2)*exp(5)^2+12*x^3-14*x^2)*exp(3+x)^2-2*x^4*exp(5)^2-6*x^4)/x^5/ exp(4)^2,x, algorithm="fricas")
Output:
(x^4*log(x)^2 + 2*(x^2*e^10 + 3*x^2)*e^(2*x + 6) - 2*(x^4*e^10 + 3*x^4 + x ^2*e^(2*x + 6))*log(x) + e^(4*x + 12))*e^(-8)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (22) = 44\).
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.00 \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\frac {\log {\left (x \right )}^{2}}{e^{8}} + \frac {\left (- 2 e^{10} - 6\right ) \log {\left (x \right )}}{e^{8}} + \frac {x^{2} e^{8} e^{4 x + 12} + \left (- 2 x^{4} e^{8} \log {\left (x \right )} + 6 x^{4} e^{8} + 2 x^{4} e^{18}\right ) e^{2 x + 6}}{x^{6} e^{16}} \] Input:
integrate((((-4*x**3+4*x**2)*exp(3+x)**2+2*x**4)*ln(x)+(-4+4*x)*exp(3+x)** 4+((4*x**3-4*x**2)*exp(5)**2+12*x**3-14*x**2)*exp(3+x)**2-2*x**4*exp(5)**2 -6*x**4)/x**5/exp(4)**2,x)
Output:
exp(-8)*log(x)**2 + (-2*exp(10) - 6)*exp(-8)*log(x) + (x**2*exp(8)*exp(4*x + 12) + (-2*x**4*exp(8)*log(x) + 6*x**4*exp(8) + 2*x**4*exp(18))*exp(2*x + 6))*exp(-16)/x**6
\[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\int { -\frac {2 \, {\left (x^{4} e^{10} + 3 \, x^{4} - 2 \, {\left (x - 1\right )} e^{\left (4 \, x + 12\right )} - {\left (6 \, x^{3} - 7 \, x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{10}\right )} e^{\left (2 \, x + 6\right )} - {\left (x^{4} - 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x + 6\right )}\right )} \log \left (x\right )\right )} e^{\left (-8\right )}}{x^{5}} \,d x } \] Input:
integrate((((-4*x^3+4*x^2)*exp(3+x)^2+2*x^4)*log(x)+(-4+4*x)*exp(3+x)^4+(( 4*x^3-4*x^2)*exp(5)^2+12*x^3-14*x^2)*exp(3+x)^2-2*x^4*exp(5)^2-6*x^4)/x^5/ exp(4)^2,x, algorithm="maxima")
Output:
(8*e^16*gamma(-1, -2*x) + 24*e^6*gamma(-1, -2*x) + 16*e^16*gamma(-2, -2*x) + 56*e^6*gamma(-2, -2*x) + 256*e^12*gamma(-3, -4*x) + 1024*e^12*gamma(-4, -4*x) - 2*e^10*log(x) + log(x)^2 - 2*e^(2*x + 6)*log(x)/x^2 + 2*integrate (e^(2*x + 6)/x^3, x) - 6*log(x))*e^(-8)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=-\frac {{\left (2 \, x^{4} e^{10} \log \left (x\right ) - x^{4} \log \left (x\right )^{2} + 6 \, x^{4} \log \left (x\right ) + 2 \, x^{2} e^{\left (2 \, x + 6\right )} \log \left (x\right ) - 2 \, x^{2} e^{\left (2 \, x + 16\right )} - 6 \, x^{2} e^{\left (2 \, x + 6\right )} - e^{\left (4 \, x + 12\right )}\right )} e^{\left (-8\right )}}{x^{4}} \] Input:
integrate((((-4*x^3+4*x^2)*exp(3+x)^2+2*x^4)*log(x)+(-4+4*x)*exp(3+x)^4+(( 4*x^3-4*x^2)*exp(5)^2+12*x^3-14*x^2)*exp(3+x)^2-2*x^4*exp(5)^2-6*x^4)/x^5/ exp(4)^2,x, algorithm="giac")
Output:
-(2*x^4*e^10*log(x) - x^4*log(x)^2 + 6*x^4*log(x) + 2*x^2*e^(2*x + 6)*log( x) - 2*x^2*e^(2*x + 16) - 6*x^2*e^(2*x + 6) - e^(4*x + 12))*e^(-8)/x^4
Timed out. \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\int -\frac {{\mathrm {e}}^{-8}\,\left ({\mathrm {e}}^{2\,x+6}\,\left ({\mathrm {e}}^{10}\,\left (4\,x^2-4\,x^3\right )+14\,x^2-12\,x^3\right )-{\mathrm {e}}^{4\,x+12}\,\left (4\,x-4\right )+2\,x^4\,{\mathrm {e}}^{10}-\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x+6}\,\left (4\,x^2-4\,x^3\right )+2\,x^4\right )+6\,x^4\right )}{x^5} \,d x \] Input:
int(-(exp(-8)*(exp(2*x + 6)*(exp(10)*(4*x^2 - 4*x^3) + 14*x^2 - 12*x^3) - exp(4*x + 12)*(4*x - 4) + 2*x^4*exp(10) - log(x)*(exp(2*x + 6)*(4*x^2 - 4* x^3) + 2*x^4) + 6*x^4))/x^5,x)
Output:
int(-(exp(-8)*(exp(2*x + 6)*(exp(10)*(4*x^2 - 4*x^3) + 14*x^2 - 12*x^3) - exp(4*x + 12)*(4*x - 4) + 2*x^4*exp(10) - log(x)*(exp(2*x + 6)*(4*x^2 - 4* x^3) + 2*x^4) + 6*x^4))/x^5, x)
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {-6 x^4-2 e^{10} x^4+e^{12+4 x} (-4+4 x)+e^{6+2 x} \left (-14 x^2+12 x^3+e^{10} \left (-4 x^2+4 x^3\right )\right )+\left (2 x^4+e^{6+2 x} \left (4 x^2-4 x^3\right )\right ) \log (x)}{e^8 x^5} \, dx=\frac {e^{4 x} e^{12}-2 e^{2 x} \mathrm {log}\left (x \right ) e^{6} x^{2}+2 e^{2 x} e^{16} x^{2}+6 e^{2 x} e^{6} x^{2}+\mathrm {log}\left (x \right )^{2} x^{4}-2 \,\mathrm {log}\left (x \right ) e^{10} x^{4}-6 \,\mathrm {log}\left (x \right ) x^{4}}{e^{8} x^{4}} \] Input:
int((((-4*x^3+4*x^2)*exp(3+x)^2+2*x^4)*log(x)+(-4+4*x)*exp(3+x)^4+((4*x^3- 4*x^2)*exp(5)^2+12*x^3-14*x^2)*exp(3+x)^2-2*x^4*exp(5)^2-6*x^4)/x^5/exp(4) ^2,x)
Output:
(e**(4*x)*e**12 - 2*e**(2*x)*log(x)*e**6*x**2 + 2*e**(2*x)*e**16*x**2 + 6* e**(2*x)*e**6*x**2 + log(x)**2*x**4 - 2*log(x)*e**10*x**4 - 6*log(x)*x**4) /(e**8*x**4)