Integrand size = 126, antiderivative size = 23 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=\left (\frac {1}{2}-e^x-x\right ) \left (-x^2+\log (3)\right )^4 \] Output:
(1/2-x-exp(x))*(ln(3)-x^2)^4
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(23)=46\).
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.70 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=\frac {x^8}{2}-x^9-e^x \left (x^2-\log (3)\right )^4-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3) \] Input:
Integrate[4*x^7 - 9*x^8 + (-12*x^5 + 28*x^6)*Log[3] + (12*x^3 - 30*x^4)*Lo g[3]^2 + (-4*x + 12*x^2)*Log[3]^3 - Log[3]^4 + E^x*(-8*x^7 - x^8 + (24*x^5 + 4*x^6)*Log[3] + (-24*x^3 - 6*x^4)*Log[3]^2 + (8*x + 4*x^2)*Log[3]^3 - L og[3]^4),x]
Output:
x^8/2 - x^9 - E^x*(x^2 - Log[3])^4 - 2*x^6*Log[3] + 4*x^7*Log[3] + 3*x^4*L og[3]^2 - 6*x^5*Log[3]^2 - 2*x^2*Log[3]^3 + 4*x^3*Log[3]^3 - x*Log[3]^4
Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(23)=46\).
Time = 0.73 (sec) , antiderivative size = 121, normalized size of antiderivative = 5.26, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {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.
\(\displaystyle \int \left (-9 x^8+4 x^7+\left (12 x^2-4 x\right ) \log ^3(3)+\left (28 x^6-12 x^5\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+e^x \left (-x^8-8 x^7+\left (4 x^2+8 x\right ) \log ^3(3)+\left (4 x^6+24 x^5\right ) \log (3)+\left (-6 x^4-24 x^3\right ) \log ^2(3)-\log ^4(3)\right )-\log ^4(3)\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^9-e^x x^8+\frac {x^8}{2}+4 x^7 \log (3)+4 e^x x^6 \log (3)-2 x^6 \log (3)-6 x^5 \log ^2(3)-6 e^x x^4 \log ^2(3)+3 x^4 \log ^2(3)+4 x^3 \log ^3(3)+4 e^x x^2 \log ^3(3)-2 x^2 \log ^3(3)-x \log ^4(3)-e^x \log ^4(3)\) |
Input:
Int[4*x^7 - 9*x^8 + (-12*x^5 + 28*x^6)*Log[3] + (12*x^3 - 30*x^4)*Log[3]^2 + (-4*x + 12*x^2)*Log[3]^3 - Log[3]^4 + E^x*(-8*x^7 - x^8 + (24*x^5 + 4*x ^6)*Log[3] + (-24*x^3 - 6*x^4)*Log[3]^2 + (8*x + 4*x^2)*Log[3]^3 - Log[3]^ 4),x]
Output:
x^8/2 - E^x*x^8 - x^9 - 2*x^6*Log[3] + 4*E^x*x^6*Log[3] + 4*x^7*Log[3] + 3 *x^4*Log[3]^2 - 6*E^x*x^4*Log[3]^2 - 6*x^5*Log[3]^2 - 2*x^2*Log[3]^3 + 4*E ^x*x^2*Log[3]^3 + 4*x^3*Log[3]^3 - E^x*Log[3]^4 - x*Log[3]^4
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(20)=40\).
Time = 1.71 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.74
method | result | size |
risch | \(\left (-x^{8}+4 \ln \left (3\right ) x^{6}-6 x^{4} \ln \left (3\right )^{2}+4 x^{2} \ln \left (3\right )^{3}-\ln \left (3\right )^{4}\right ) {\mathrm e}^{x}-\ln \left (3\right )^{4} x +4 x^{3} \ln \left (3\right )^{3}-2 x^{2} \ln \left (3\right )^{3}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 x^{7} \ln \left (3\right )-2 \ln \left (3\right ) x^{6}-x^{9}+\frac {x^{8}}{2}\) | \(109\) |
default | \(-{\mathrm e}^{x} \ln \left (3\right )^{4}-{\mathrm e}^{x} x^{8}+4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}+4 x^{3} \ln \left (3\right )^{3}-2 x^{2} \ln \left (3\right )^{3}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 x^{7} \ln \left (3\right )-2 \ln \left (3\right ) x^{6}+\frac {x^{8}}{2}-x^{9}-\ln \left (3\right )^{4} x\) | \(115\) |
norman | \(-{\mathrm e}^{x} \ln \left (3\right )^{4}-{\mathrm e}^{x} x^{8}+4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}+4 x^{3} \ln \left (3\right )^{3}-2 x^{2} \ln \left (3\right )^{3}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 x^{7} \ln \left (3\right )-2 \ln \left (3\right ) x^{6}+\frac {x^{8}}{2}-x^{9}-\ln \left (3\right )^{4} x\) | \(115\) |
parallelrisch | \(-{\mathrm e}^{x} \ln \left (3\right )^{4}-{\mathrm e}^{x} x^{8}+4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}+4 x^{3} \ln \left (3\right )^{3}-2 x^{2} \ln \left (3\right )^{3}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 x^{7} \ln \left (3\right )-2 \ln \left (3\right ) x^{6}+\frac {x^{8}}{2}-x^{9}-\ln \left (3\right )^{4} x\) | \(115\) |
parts | \(-{\mathrm e}^{x} \ln \left (3\right )^{4}-{\mathrm e}^{x} x^{8}+4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}+4 x^{3} \ln \left (3\right )^{3}-2 x^{2} \ln \left (3\right )^{3}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 x^{7} \ln \left (3\right )-2 \ln \left (3\right ) x^{6}+\frac {x^{8}}{2}-x^{9}-\ln \left (3\right )^{4} x\) | \(115\) |
Input:
int((-ln(3)^4+(4*x^2+8*x)*ln(3)^3+(-6*x^4-24*x^3)*ln(3)^2+(4*x^6+24*x^5)*l n(3)-x^8-8*x^7)*exp(x)-ln(3)^4+(12*x^2-4*x)*ln(3)^3+(-30*x^4+12*x^3)*ln(3) ^2+(28*x^6-12*x^5)*ln(3)-9*x^8+4*x^7,x,method=_RETURNVERBOSE)
Output:
(-x^8+4*ln(3)*x^6-6*x^4*ln(3)^2+4*x^2*ln(3)^3-ln(3)^4)*exp(x)-ln(3)^4*x+4* x^3*ln(3)^3-2*x^2*ln(3)^3-6*x^5*ln(3)^2+3*x^4*ln(3)^2+4*x^7*ln(3)-2*ln(3)* x^6-x^9+1/2*x^8
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^{9} + \frac {1}{2} \, x^{8} - x \log \left (3\right )^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \left (3\right )^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \left (3\right )^{2} - {\left (x^{8} - 4 \, x^{6} \log \left (3\right ) + 6 \, x^{4} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right )^{3} + \log \left (3\right )^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \left (3\right ) \] Input:
integrate((-log(3)^4+(4*x^2+8*x)*log(3)^3+(-6*x^4-24*x^3)*log(3)^2+(4*x^6+ 24*x^5)*log(3)-x^8-8*x^7)*exp(x)-log(3)^4+(12*x^2-4*x)*log(3)^3+(-30*x^4+1 2*x^3)*log(3)^2+(28*x^6-12*x^5)*log(3)-9*x^8+4*x^7,x, algorithm="fricas")
Output:
-x^9 + 1/2*x^8 - x*log(3)^4 + 2*(2*x^3 - x^2)*log(3)^3 - 3*(2*x^5 - x^4)*l og(3)^2 - (x^8 - 4*x^6*log(3) + 6*x^4*log(3)^2 - 4*x^2*log(3)^3 + log(3)^4 )*e^x + 2*(2*x^7 - x^6)*log(3)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (15) = 30\).
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.87 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=- x^{9} + \frac {x^{8}}{2} + 4 x^{7} \log {\left (3 \right )} - 2 x^{6} \log {\left (3 \right )} - 6 x^{5} \log {\left (3 \right )}^{2} + 3 x^{4} \log {\left (3 \right )}^{2} + 4 x^{3} \log {\left (3 \right )}^{3} - 2 x^{2} \log {\left (3 \right )}^{3} - x \log {\left (3 \right )}^{4} + \left (- x^{8} + 4 x^{6} \log {\left (3 \right )} - 6 x^{4} \log {\left (3 \right )}^{2} + 4 x^{2} \log {\left (3 \right )}^{3} - \log {\left (3 \right )}^{4}\right ) e^{x} \] Input:
integrate((-ln(3)**4+(4*x**2+8*x)*ln(3)**3+(-6*x**4-24*x**3)*ln(3)**2+(4*x **6+24*x**5)*ln(3)-x**8-8*x**7)*exp(x)-ln(3)**4+(12*x**2-4*x)*ln(3)**3+(-3 0*x**4+12*x**3)*ln(3)**2+(28*x**6-12*x**5)*ln(3)-9*x**8+4*x**7,x)
Output:
-x**9 + x**8/2 + 4*x**7*log(3) - 2*x**6*log(3) - 6*x**5*log(3)**2 + 3*x**4 *log(3)**2 + 4*x**3*log(3)**3 - 2*x**2*log(3)**3 - x*log(3)**4 + (-x**8 + 4*x**6*log(3) - 6*x**4*log(3)**2 + 4*x**2*log(3)**3 - log(3)**4)*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^{9} + \frac {1}{2} \, x^{8} - x \log \left (3\right )^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \left (3\right )^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \left (3\right )^{2} - {\left (x^{8} - 4 \, x^{6} \log \left (3\right ) + 6 \, x^{4} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right )^{3} + \log \left (3\right )^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \left (3\right ) \] Input:
integrate((-log(3)^4+(4*x^2+8*x)*log(3)^3+(-6*x^4-24*x^3)*log(3)^2+(4*x^6+ 24*x^5)*log(3)-x^8-8*x^7)*exp(x)-log(3)^4+(12*x^2-4*x)*log(3)^3+(-30*x^4+1 2*x^3)*log(3)^2+(28*x^6-12*x^5)*log(3)-9*x^8+4*x^7,x, algorithm="maxima")
Output:
-x^9 + 1/2*x^8 - x*log(3)^4 + 2*(2*x^3 - x^2)*log(3)^3 - 3*(2*x^5 - x^4)*l og(3)^2 - (x^8 - 4*x^6*log(3) + 6*x^4*log(3)^2 - 4*x^2*log(3)^3 + log(3)^4 )*e^x + 2*(2*x^7 - x^6)*log(3)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^{9} + \frac {1}{2} \, x^{8} - x \log \left (3\right )^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \left (3\right )^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \left (3\right )^{2} - {\left (x^{8} - 4 \, x^{6} \log \left (3\right ) + 6 \, x^{4} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right )^{3} + \log \left (3\right )^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \left (3\right ) \] Input:
integrate((-log(3)^4+(4*x^2+8*x)*log(3)^3+(-6*x^4-24*x^3)*log(3)^2+(4*x^6+ 24*x^5)*log(3)-x^8-8*x^7)*exp(x)-log(3)^4+(12*x^2-4*x)*log(3)^3+(-30*x^4+1 2*x^3)*log(3)^2+(28*x^6-12*x^5)*log(3)-9*x^8+4*x^7,x, algorithm="giac")
Output:
-x^9 + 1/2*x^8 - x*log(3)^4 + 2*(2*x^3 - x^2)*log(3)^3 - 3*(2*x^5 - x^4)*l og(3)^2 - (x^8 - 4*x^6*log(3) + 6*x^4*log(3)^2 - 4*x^2*log(3)^3 + log(3)^4 )*e^x + 2*(2*x^7 - x^6)*log(3)
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.96 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=4\,x^3\,{\ln \left (3\right )}^3-2\,x^2\,{\ln \left (3\right )}^3+3\,x^4\,{\ln \left (3\right )}^2-6\,x^5\,{\ln \left (3\right )}^2-{\mathrm {e}}^x\,{\ln \left (3\right )}^4-x^8\,{\mathrm {e}}^x-x\,{\ln \left (3\right )}^4-2\,x^6\,\ln \left (3\right )+4\,x^7\,\ln \left (3\right )+\frac {x^8}{2}-x^9+4\,x^6\,{\mathrm {e}}^x\,\ln \left (3\right )+4\,x^2\,{\mathrm {e}}^x\,{\ln \left (3\right )}^3-6\,x^4\,{\mathrm {e}}^x\,{\ln \left (3\right )}^2 \] Input:
int(4*x^7 - log(3)*(12*x^5 - 28*x^6) - exp(x)*(log(3)^4 - log(3)*(24*x^5 + 4*x^6) - log(3)^3*(8*x + 4*x^2) + 8*x^7 + x^8 + log(3)^2*(24*x^3 + 6*x^4) ) - log(3)^4 - log(3)^3*(4*x - 12*x^2) - 9*x^8 + log(3)^2*(12*x^3 - 30*x^4 ),x)
Output:
4*x^3*log(3)^3 - 2*x^2*log(3)^3 + 3*x^4*log(3)^2 - 6*x^5*log(3)^2 - exp(x) *log(3)^4 - x^8*exp(x) - x*log(3)^4 - 2*x^6*log(3) + 4*x^7*log(3) + x^8/2 - x^9 + 4*x^6*exp(x)*log(3) + 4*x^2*exp(x)*log(3)^3 - 6*x^4*exp(x)*log(3)^ 2
Time = 0.16 (sec) , antiderivative size = 119, normalized size of antiderivative = 5.17 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-e^{x} \mathrm {log}\left (3\right )^{4}+4 e^{x} \mathrm {log}\left (3\right )^{3} x^{2}-6 e^{x} \mathrm {log}\left (3\right )^{2} x^{4}+4 e^{x} \mathrm {log}\left (3\right ) x^{6}-e^{x} x^{8}-\mathrm {log}\left (3\right )^{4} x +4 \mathrm {log}\left (3\right )^{3} x^{3}-2 \mathrm {log}\left (3\right )^{3} x^{2}-6 \mathrm {log}\left (3\right )^{2} x^{5}+3 x^{4} \mathrm {log}\left (3\right )^{2}+4 \,\mathrm {log}\left (3\right ) x^{7}-2 \,\mathrm {log}\left (3\right ) x^{6}-x^{9}+\frac {x^{8}}{2} \] Input:
int((-log(3)^4+(4*x^2+8*x)*log(3)^3+(-6*x^4-24*x^3)*log(3)^2+(4*x^6+24*x^5 )*log(3)-x^8-8*x^7)*exp(x)-log(3)^4+(12*x^2-4*x)*log(3)^3+(-30*x^4+12*x^3) *log(3)^2+(28*x^6-12*x^5)*log(3)-9*x^8+4*x^7,x)
Output:
( - 2*e**x*log(3)**4 + 8*e**x*log(3)**3*x**2 - 12*e**x*log(3)**2*x**4 + 8* e**x*log(3)*x**6 - 2*e**x*x**8 - 2*log(3)**4*x + 8*log(3)**3*x**3 - 4*log( 3)**3*x**2 - 12*log(3)**2*x**5 + 6*log(3)**2*x**4 + 8*log(3)*x**7 - 4*log( 3)*x**6 - 2*x**9 + x**8)/2