Integrand size = 35, antiderivative size = 28 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {1}{4} e^{2 x \left (\frac {2}{x}+x+3 \left (5+\frac {\log ^2(9)}{144}\right )\right )} \] Output:
1/4*exp(x*(2*x+4/x+30+1/6*ln(3)^2))
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {1}{4} e^{4+2 x^2+\frac {1}{24} x \left (720+\log ^2(9)\right )} \] Input:
Integrate[(E^((96 + 720*x + 48*x^2 + x*Log[9]^2)/24)*(720 + 96*x + Log[9]^ 2))/96,x]
Output:
E^(4 + 2*x^2 + (x*(720 + Log[9]^2))/24)/4
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {27, 2674, 2666}
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.
\(\displaystyle \int \frac {1}{96} e^{\frac {1}{24} \left (48 x^2+720 x+x \log ^2(9)+96\right )} \left (96 x+720+\log ^2(9)\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{96} \int e^{\frac {1}{24} \left (48 x^2+\log ^2(9) x+720 x+96\right )} \left (96 x+\log ^2(9)+720\right )dx\) |
\(\Big \downarrow \) 2674 |
\(\displaystyle \frac {1}{96} \int e^{2 x^2+\frac {1}{24} \left (720+\log ^2(9)\right ) x+4} \left (96 x+\log ^2(9)+720\right )dx\) |
\(\Big \downarrow \) 2666 |
\(\displaystyle \frac {1}{4} e^{2 x^2+\frac {1}{24} x \left (720+\log ^2(9)\right )+4}\) |
Input:
Int[(E^((96 + 720*x + 48*x^2 + x*Log[9]^2)/24)*(720 + 96*x + Log[9]^2))/96 ,x]
Output:
E^(4 + 2*x^2 + (x*(720 + Log[9]^2))/24)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSu m[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !( LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
gosper | \(\frac {{\mathrm e}^{\frac {x \ln \left (3\right )^{2}}{6}+2 x^{2}+30 x +4}}{4}\) | \(21\) |
norman | \(\frac {{\mathrm e}^{\frac {x \ln \left (3\right )^{2}}{6}+2 x^{2}+30 x +4}}{4}\) | \(21\) |
risch | \(\frac {{\mathrm e}^{\frac {x \ln \left (3\right )^{2}}{6}+2 x^{2}+30 x +4}}{4}\) | \(21\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {x \ln \left (3\right )^{2}}{6}+2 x^{2}+30 x +4}}{4}\) | \(21\) |
orering | \(\frac {\left (4 \ln \left (3\right )^{2}+96 x +720\right ) {\mathrm e}^{\frac {x \ln \left (3\right )^{2}}{6}+2 x^{2}+30 x +4}}{16 \ln \left (3\right )^{2}+384 x +2880}\) | \(43\) |
default | \(-\frac {i \ln \left (3\right )^{2} {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right )}{96}-\frac {15 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right )}{8}+{\mathrm e}^{4} \left (\frac {{\mathrm e}^{2 x^{2}+\left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) x}}{4}+\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right )}{16}\right )\) | \(176\) |
parts | \(-\frac {i \sqrt {\pi }\, {\mathrm e}^{4-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right ) \ln \left (3\right )^{2}}{96}-\frac {i \sqrt {\pi }\, {\mathrm e}^{4-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right ) x}{4}-\frac {15 i \sqrt {\pi }\, {\mathrm e}^{4-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right )}{8}+\frac {\sqrt {\pi }\, {\mathrm e}^{4-\frac {\left (\frac {\ln \left (3\right )^{2}}{6}+30\right )^{2}}{8}} \left (\operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right ) \left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right )+\frac {{\mathrm e}^{-{\left (i \sqrt {2}\, x +\frac {i \left (\frac {\ln \left (3\right )^{2}}{6}+30\right ) \sqrt {2}}{4}\right )}^{2}}}{\sqrt {\pi }}\right )}{4}\) | \(246\) |
Input:
int(1/96*(4*ln(3)^2+96*x+720)*exp(1/6*x*ln(3)^2+2*x^2+30*x+4),x,method=_RE TURNVERBOSE)
Output:
1/4*exp(1/6*x*ln(3)^2+2*x^2+30*x+4)
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {1}{4} \, e^{\left (\frac {1}{6} \, x \log \left (3\right )^{2} + 2 \, x^{2} + 30 \, x + 4\right )} \] Input:
integrate(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x, a lgorithm="fricas")
Output:
1/4*e^(1/6*x*log(3)^2 + 2*x^2 + 30*x + 4)
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {e^{2 x^{2} + \frac {x \log {\left (3 \right )}^{2}}{6} + 30 x + 4}}{4} \] Input:
integrate(1/96*(4*ln(3)**2+96*x+720)*exp(1/6*x*ln(3)**2+2*x**2+30*x+4),x)
Output:
exp(2*x**2 + x*log(3)**2/6 + 30*x + 4)/4
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {1}{4} \, e^{\left (\frac {1}{6} \, x \log \left (3\right )^{2} + 2 \, x^{2} + 30 \, x + 4\right )} \] Input:
integrate(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x, a lgorithm="maxima")
Output:
1/4*e^(1/6*x*log(3)^2 + 2*x^2 + 30*x + 4)
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {1}{4} \, e^{\left (\frac {1}{6} \, x \log \left (3\right )^{2} + 2 \, x^{2} + 30 \, x + 4\right )} \] Input:
integrate(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x, a lgorithm="giac")
Output:
1/4*e^(1/6*x*log(3)^2 + 2*x^2 + 30*x + 4)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {{\mathrm {e}}^{\frac {x\,{\ln \left (3\right )}^2}{6}}\,{\mathrm {e}}^{30\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{2\,x^2}}{4} \] Input:
int((exp(30*x + (x*log(3)^2)/6 + 2*x^2 + 4)*(96*x + 4*log(3)^2 + 720))/96, x)
Output:
(exp((x*log(3)^2)/6)*exp(30*x)*exp(4)*exp(2*x^2))/4
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{96} e^{\frac {1}{24} \left (96+720 x+48 x^2+x \log ^2(9)\right )} \left (720+96 x+\log ^2(9)\right ) \, dx=\frac {e^{\frac {\mathrm {log}\left (3\right )^{2} x}{6}+2 x^{2}+30 x} e^{4}}{4} \] Input:
int(1/96*(4*log(3)^2+96*x+720)*exp(1/6*x*log(3)^2+2*x^2+30*x+4),x)
Output:
(e**((log(3)**2*x + 12*x**2 + 180*x)/6)*e**4)/4