Integrand size = 83, antiderivative size = 29 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=e^{\frac {4 (4-x)^2 \left (1+x^2 (2+x)\right )^2 \log ^2(3)}{x^2}} \] Output:
exp(4*(4-x)^2/x^2*ln(3)^2*(x^2*(2+x)+1)^2)
Time = 2.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=e^{\frac {4 \left (-4+x-8 x^2-2 x^3+x^4\right )^2 \log ^2(3)}{x^2}} \] Input:
Integrate[(E^(((64 - 32*x + 260*x^2 + 208*x^4 + 136*x^5 - 48*x^6 - 16*x^7 + 4*x^8)*Log[3]^2)/x^2)*(-128 + 32*x + 416*x^4 + 408*x^5 - 192*x^6 - 80*x^ 7 + 24*x^8)*Log[3]^2)/x^3,x]
Output:
E^((4*(-4 + x - 8*x^2 - 2*x^3 + x^4)^2*Log[3]^2)/x^2)
Time = 2.89 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {27, 27, 7257}
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.
\(\displaystyle \int \frac {\left (24 x^8-80 x^7-192 x^6+408 x^5+416 x^4+32 x-128\right ) \log ^2(3) \exp \left (\frac {\left (4 x^8-16 x^7-48 x^6+136 x^5+208 x^4+260 x^2-32 x+64\right ) \log ^2(3)}{x^2}\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \log ^2(3) \int -\frac {8 \exp \left (\frac {4 \left (x^8-4 x^7-12 x^6+34 x^5+52 x^4+65 x^2-8 x+16\right ) \log ^2(3)}{x^2}\right ) \left (-3 x^8+10 x^7+24 x^6-51 x^5-52 x^4-4 x+16\right )}{x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -8 \log ^2(3) \int \frac {\exp \left (\frac {4 \left (x^8-4 x^7-12 x^6+34 x^5+52 x^4+65 x^2-8 x+16\right ) \log ^2(3)}{x^2}\right ) \left (-3 x^8+10 x^7+24 x^6-51 x^5-52 x^4-4 x+16\right )}{x^3}dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle \exp \left (\frac {4 \left (x^8-4 x^7-12 x^6+34 x^5+52 x^4+65 x^2-8 x+16\right ) \log ^2(3)}{x^2}\right )\) |
Input:
Int[(E^(((64 - 32*x + 260*x^2 + 208*x^4 + 136*x^5 - 48*x^6 - 16*x^7 + 4*x^ 8)*Log[3]^2)/x^2)*(-128 + 32*x + 416*x^4 + 408*x^5 - 192*x^6 - 80*x^7 + 24 *x^8)*Log[3]^2)/x^3,x]
Output:
E^((4*(16 - 8*x + 65*x^2 + 52*x^4 + 34*x^5 - 12*x^6 - 4*x^7 + x^8)*Log[3]^ 2)/x^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 0.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
risch | \({\mathrm e}^{\frac {4 \left (x -4\right )^{2} \left (x^{3}+2 x^{2}+1\right )^{2} \ln \left (3\right )^{2}}{x^{2}}}\) | \(28\) |
gosper | \({\mathrm e}^{\frac {4 \left (x^{8}-4 x^{7}-12 x^{6}+34 x^{5}+52 x^{4}+65 x^{2}-8 x +16\right ) \ln \left (3\right )^{2}}{x^{2}}}\) | \(44\) |
parallelrisch | \({\mathrm e}^{\frac {4 \left (x^{8}-4 x^{7}-12 x^{6}+34 x^{5}+52 x^{4}+65 x^{2}-8 x +16\right ) \ln \left (3\right )^{2}}{x^{2}}}\) | \(44\) |
default | \({\mathrm e}^{\frac {\left (4 x^{8}-16 x^{7}-48 x^{6}+136 x^{5}+208 x^{4}+260 x^{2}-32 x +64\right ) \ln \left (3\right )^{2}}{x^{2}}}\) | \(45\) |
norman | \({\mathrm e}^{\frac {\left (4 x^{8}-16 x^{7}-48 x^{6}+136 x^{5}+208 x^{4}+260 x^{2}-32 x +64\right ) \ln \left (3\right )^{2}}{x^{2}}}\) | \(45\) |
Input:
int((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*ln(3)^2*exp((4*x^8-16 *x^7-48*x^6+136*x^5+208*x^4+260*x^2-32*x+64)*ln(3)^2/x^2)/x^3,x,method=_RE TURNVERBOSE)
Output:
exp(4*(x-4)^2*(x^3+2*x^2+1)^2*ln(3)^2/x^2)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=e^{\left (\frac {4 \, {\left (x^{8} - 4 \, x^{7} - 12 \, x^{6} + 34 \, x^{5} + 52 \, x^{4} + 65 \, x^{2} - 8 \, x + 16\right )} \log \left (3\right )^{2}}{x^{2}}\right )} \] Input:
integrate((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4 *x^8-16*x^7-48*x^6+136*x^5+208*x^4+260*x^2-32*x+64)*log(3)^2/x^2)/x^3,x, a lgorithm="fricas")
Output:
e^(4*(x^8 - 4*x^7 - 12*x^6 + 34*x^5 + 52*x^4 + 65*x^2 - 8*x + 16)*log(3)^2 /x^2)
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=e^{\frac {\left (4 x^{8} - 16 x^{7} - 48 x^{6} + 136 x^{5} + 208 x^{4} + 260 x^{2} - 32 x + 64\right ) \log {\left (3 \right )}^{2}}{x^{2}}} \] Input:
integrate((24*x**8-80*x**7-192*x**6+408*x**5+416*x**4+32*x-128)*ln(3)**2*e xp((4*x**8-16*x**7-48*x**6+136*x**5+208*x**4+260*x**2-32*x+64)*ln(3)**2/x* *2)/x**3,x)
Output:
exp((4*x**8 - 16*x**7 - 48*x**6 + 136*x**5 + 208*x**4 + 260*x**2 - 32*x + 64)*log(3)**2/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=e^{\left (4 \, x^{6} \log \left (3\right )^{2} - 16 \, x^{5} \log \left (3\right )^{2} - 48 \, x^{4} \log \left (3\right )^{2} + 136 \, x^{3} \log \left (3\right )^{2} + 208 \, x^{2} \log \left (3\right )^{2} + 260 \, \log \left (3\right )^{2} - \frac {32 \, \log \left (3\right )^{2}}{x} + \frac {64 \, \log \left (3\right )^{2}}{x^{2}}\right )} \] Input:
integrate((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4 *x^8-16*x^7-48*x^6+136*x^5+208*x^4+260*x^2-32*x+64)*log(3)^2/x^2)/x^3,x, a lgorithm="maxima")
Output:
e^(4*x^6*log(3)^2 - 16*x^5*log(3)^2 - 48*x^4*log(3)^2 + 136*x^3*log(3)^2 + 208*x^2*log(3)^2 + 260*log(3)^2 - 32*log(3)^2/x + 64*log(3)^2/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=e^{\left (4 \, x^{6} \log \left (3\right )^{2} - 16 \, x^{5} \log \left (3\right )^{2} - 48 \, x^{4} \log \left (3\right )^{2} + 136 \, x^{3} \log \left (3\right )^{2} + 208 \, x^{2} \log \left (3\right )^{2} + 260 \, \log \left (3\right )^{2} - \frac {32 \, \log \left (3\right )^{2}}{x} + \frac {64 \, \log \left (3\right )^{2}}{x^{2}}\right )} \] Input:
integrate((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4 *x^8-16*x^7-48*x^6+136*x^5+208*x^4+260*x^2-32*x+64)*log(3)^2/x^2)/x^3,x, a lgorithm="giac")
Output:
e^(4*x^6*log(3)^2 - 16*x^5*log(3)^2 - 48*x^4*log(3)^2 + 136*x^3*log(3)^2 + 208*x^2*log(3)^2 + 260*log(3)^2 - 32*log(3)^2/x + 64*log(3)^2/x^2)
Time = 4.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx={\mathrm {e}}^{4\,x^6\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{-16\,x^5\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{-\frac {32\,{\ln \left (3\right )}^2}{x}}\,{\mathrm {e}}^{-48\,x^4\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{\frac {64\,{\ln \left (3\right )}^2}{x^2}}\,{\mathrm {e}}^{136\,x^3\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{208\,x^2\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{260\,{\ln \left (3\right )}^2} \] Input:
int((exp((log(3)^2*(260*x^2 - 32*x + 208*x^4 + 136*x^5 - 48*x^6 - 16*x^7 + 4*x^8 + 64))/x^2)*log(3)^2*(32*x + 416*x^4 + 408*x^5 - 192*x^6 - 80*x^7 + 24*x^8 - 128))/x^3,x)
Output:
exp(4*x^6*log(3)^2)*exp(-16*x^5*log(3)^2)*exp(-(32*log(3)^2)/x)*exp(-48*x^ 4*log(3)^2)*exp((64*log(3)^2)/x^2)*exp(136*x^3*log(3)^2)*exp(208*x^2*log(3 )^2)*exp(260*log(3)^2)
\[ \int \frac {e^{\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}} \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right ) \log ^2(3)}{x^3} \, dx=\int \frac {\left (24 x^{8}-80 x^{7}-192 x^{6}+408 x^{5}+416 x^{4}+32 x -128\right ) \mathrm {log}\left (3\right )^{2} {\mathrm e}^{\frac {\left (4 x^{8}-16 x^{7}-48 x^{6}+136 x^{5}+208 x^{4}+260 x^{2}-32 x +64\right ) \mathrm {log}\left (3\right )^{2}}{x^{2}}}}{x^{3}}d x \] Input:
int((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4*x^8-1 6*x^7-48*x^6+136*x^5+208*x^4+260*x^2-32*x+64)*log(3)^2/x^2)/x^3,x)
Output:
int((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4*x^8-1 6*x^7-48*x^6+136*x^5+208*x^4+260*x^2-32*x+64)*log(3)^2/x^2)/x^3,x)