Integrand size = 129, antiderivative size = 18 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {1}{3 \left (\frac {3}{4}+x+e^x x+\log (3)\right )^2} \] Output:
1/3/(ln(3)+3/4+x+exp(x)*x)^2
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {16}{3 \left (3+4 \left (1+e^x\right ) x+\log (81)\right )^2} \] Input:
Integrate[(-128 + E^x*(-128 - 128*x))/(81 + 324*x + 432*x^2 + 192*x^3 + 19 2*E^(3*x)*x^3 + (324 + 864*x + 576*x^2)*Log[3] + (432 + 576*x)*Log[3]^2 + 192*Log[3]^3 + E^(2*x)*(432*x^2 + 576*x^3 + 576*x^2*Log[3]) + E^x*(324*x + 864*x^2 + 576*x^3 + (864*x + 1152*x^2)*Log[3] + 576*x*Log[3]^2)),x]
Output:
16/(3*(3 + 4*(1 + E^x)*x + Log[81])^2)
Time = 0.50 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {7239, 27, 25, 7237}
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.
\(\displaystyle \int \frac {e^x (-128 x-128)-128}{192 e^{3 x} x^3+192 x^3+432 x^2+\left (576 x^2+864 x+324\right ) \log (3)+e^x \left (576 x^3+864 x^2+\left (1152 x^2+864 x\right ) \log (3)+324 x+576 x \log ^2(3)\right )+e^{2 x} \left (576 x^3+432 x^2+576 x^2 \log (3)\right )+324 x+(576 x+432) \log ^2(3)+81+192 \log ^3(3)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {128 \left (-e^x (x+1)-1\right )}{3 \left (4 \left (e^x+1\right ) x+3 \left (1+\frac {\log (81)}{3}\right )\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {128}{3} \int -\frac {e^x (x+1)+1}{\left (4 \left (1+e^x\right ) x+\log (81)+3\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {128}{3} \int \frac {e^x (x+1)+1}{\left (4 \left (1+e^x\right ) x+\log (81)+3\right )^3}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {16}{3 \left (4 \left (e^x+1\right ) x+3+\log (81)\right )^2}\) |
Input:
Int[(-128 + E^x*(-128 - 128*x))/(81 + 324*x + 432*x^2 + 192*x^3 + 192*E^(3 *x)*x^3 + (324 + 864*x + 576*x^2)*Log[3] + (432 + 576*x)*Log[3]^2 + 192*Lo g[3]^3 + E^(2*x)*(432*x^2 + 576*x^3 + 576*x^2*Log[3]) + E^x*(324*x + 864*x ^2 + 576*x^3 + (864*x + 1152*x^2)*Log[3] + 576*x*Log[3]^2)),x]
Output:
16/(3*(3 + 4*(1 + E^x)*x + Log[81])^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
norman | \(\frac {16}{3 \left (4 \,{\mathrm e}^{x} x +4 \ln \left (3\right )+4 x +3\right )^{2}}\) | \(19\) |
risch | \(\frac {16}{3 \left (4 \,{\mathrm e}^{x} x +4 \ln \left (3\right )+4 x +3\right )^{2}}\) | \(19\) |
parallelrisch | \(\frac {16}{3 \left (16 \,{\mathrm e}^{2 x} x^{2}+32 x \ln \left (3\right ) {\mathrm e}^{x}+32 \,{\mathrm e}^{x} x^{2}+16 \ln \left (3\right )^{2}+32 x \ln \left (3\right )+16 x^{2}+24 \,{\mathrm e}^{x} x +24 \ln \left (3\right )+24 x +9\right )}\) | \(58\) |
Input:
int(((-128*x-128)*exp(x)-128)/(192*x^3*exp(x)^3+(576*x^2*ln(3)+576*x^3+432 *x^2)*exp(x)^2+(576*x*ln(3)^2+(1152*x^2+864*x)*ln(3)+576*x^3+864*x^2+324*x )*exp(x)+192*ln(3)^3+(576*x+432)*ln(3)^2+(576*x^2+864*x+324)*ln(3)+192*x^3 +432*x^2+324*x+81),x,method=_RETURNVERBOSE)
Output:
16/3/(4*exp(x)*x+4*ln(3)+4*x+3)^2
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {16}{3 \, {\left (16 \, x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 8 \, {\left (4 \, x^{2} + 4 \, x \log \left (3\right ) + 3 \, x\right )} e^{x} + 8 \, {\left (4 \, x + 3\right )} \log \left (3\right ) + 16 \, \log \left (3\right )^{2} + 24 \, x + 9\right )}} \] Input:
integrate(((-128*x-128)*exp(x)-128)/(192*x^3*exp(x)^3+(576*x^2*log(3)+576* x^3+432*x^2)*exp(x)^2+(576*x*log(3)^2+(1152*x^2+864*x)*log(3)+576*x^3+864* x^2+324*x)*exp(x)+192*log(3)^3+(576*x+432)*log(3)^2+(576*x^2+864*x+324)*lo g(3)+192*x^3+432*x^2+324*x+81),x, algorithm="fricas")
Output:
16/3/(16*x^2*e^(2*x) + 16*x^2 + 8*(4*x^2 + 4*x*log(3) + 3*x)*e^x + 8*(4*x + 3)*log(3) + 16*log(3)^2 + 24*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {16}{48 x^{2} e^{2 x} + 48 x^{2} + 72 x + 96 x \log {\left (3 \right )} + \left (96 x^{2} + 72 x + 96 x \log {\left (3 \right )}\right ) e^{x} + 27 + 48 \log {\left (3 \right )}^{2} + 72 \log {\left (3 \right )}} \] Input:
integrate(((-128*x-128)*exp(x)-128)/(192*x**3*exp(x)**3+(576*x**2*ln(3)+57 6*x**3+432*x**2)*exp(x)**2+(576*x*ln(3)**2+(1152*x**2+864*x)*ln(3)+576*x** 3+864*x**2+324*x)*exp(x)+192*ln(3)**3+(576*x+432)*ln(3)**2+(576*x**2+864*x +324)*ln(3)+192*x**3+432*x**2+324*x+81),x)
Output:
16/(48*x**2*exp(2*x) + 48*x**2 + 72*x + 96*x*log(3) + (96*x**2 + 72*x + 96 *x*log(3))*exp(x) + 27 + 48*log(3)**2 + 72*log(3))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (18) = 36\).
Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.17 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {16}{3 \, {\left (16 \, x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 8 \, x {\left (4 \, \log \left (3\right ) + 3\right )} + 8 \, {\left (4 \, x^{2} + x {\left (4 \, \log \left (3\right ) + 3\right )}\right )} e^{x} + 16 \, \log \left (3\right )^{2} + 24 \, \log \left (3\right ) + 9\right )}} \] Input:
integrate(((-128*x-128)*exp(x)-128)/(192*x^3*exp(x)^3+(576*x^2*log(3)+576* x^3+432*x^2)*exp(x)^2+(576*x*log(3)^2+(1152*x^2+864*x)*log(3)+576*x^3+864* x^2+324*x)*exp(x)+192*log(3)^3+(576*x+432)*log(3)^2+(576*x^2+864*x+324)*lo g(3)+192*x^3+432*x^2+324*x+81),x, algorithm="maxima")
Output:
16/3/(16*x^2*e^(2*x) + 16*x^2 + 8*x*(4*log(3) + 3) + 8*(4*x^2 + x*(4*log(3 ) + 3))*e^x + 16*log(3)^2 + 24*log(3) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (18) = 36\).
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.17 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {16}{3 \, {\left (16 \, x^{2} e^{\left (2 \, x\right )} + 32 \, x^{2} e^{x} + 32 \, x e^{x} \log \left (3\right ) + 16 \, x^{2} + 24 \, x e^{x} + 32 \, x \log \left (3\right ) + 16 \, \log \left (3\right )^{2} + 24 \, x + 24 \, \log \left (3\right ) + 9\right )}} \] Input:
integrate(((-128*x-128)*exp(x)-128)/(192*x^3*exp(x)^3+(576*x^2*log(3)+576* x^3+432*x^2)*exp(x)^2+(576*x*log(3)^2+(1152*x^2+864*x)*log(3)+576*x^3+864* x^2+324*x)*exp(x)+192*log(3)^3+(576*x+432)*log(3)^2+(576*x^2+864*x+324)*lo g(3)+192*x^3+432*x^2+324*x+81),x, algorithm="giac")
Output:
16/3/(16*x^2*e^(2*x) + 32*x^2*e^x + 32*x*e^x*log(3) + 16*x^2 + 24*x*e^x + 32*x*log(3) + 16*log(3)^2 + 24*x + 24*log(3) + 9)
Time = 3.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 6.83 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=-\frac {\frac {128\,x}{4\,\ln \left (3\right )+3}+\frac {256\,x^2}{{\left (4\,\ln \left (3\right )+3\right )}^2}+\frac {512\,x^2\,{\mathrm {e}}^x}{{\left (4\,\ln \left (3\right )+3\right )}^2}+\frac {256\,x^2\,{\mathrm {e}}^{2\,x}}{{\left (\ln \left (81\right )+3\right )}^2}+\frac {128\,x\,{\mathrm {e}}^x}{\ln \left (81\right )+3}}{72\,x+72\,\ln \left (3\right )+96\,x^2\,{\mathrm {e}}^x+96\,x\,\ln \left (3\right )+48\,x^2\,{\mathrm {e}}^{2\,x}+72\,x\,{\mathrm {e}}^x+48\,{\ln \left (3\right )}^2+48\,x^2+96\,x\,{\mathrm {e}}^x\,\ln \left (3\right )+27} \] Input:
int(-(exp(x)*(128*x + 128) + 128)/(324*x + exp(2*x)*(576*x^2*log(3) + 432* x^2 + 576*x^3) + log(3)*(864*x + 576*x^2 + 324) + log(3)^2*(576*x + 432) + 192*x^3*exp(3*x) + exp(x)*(324*x + log(3)*(864*x + 1152*x^2) + 576*x*log( 3)^2 + 864*x^2 + 576*x^3) + 192*log(3)^3 + 432*x^2 + 192*x^3 + 81),x)
Output:
-((128*x)/(4*log(3) + 3) + (256*x^2)/(4*log(3) + 3)^2 + (512*x^2*exp(x))/( 4*log(3) + 3)^2 + (256*x^2*exp(2*x))/(log(81) + 3)^2 + (128*x*exp(x))/(log (81) + 3))/(72*x + 72*log(3) + 96*x^2*exp(x) + 96*x*log(3) + 48*x^2*exp(2* x) + 72*x*exp(x) + 48*log(3)^2 + 48*x^2 + 96*x*exp(x)*log(3) + 27)
Time = 0.20 (sec) , antiderivative size = 207, normalized size of antiderivative = 11.50 \[ \int \frac {-128+e^x (-128-128 x)}{81+324 x+432 x^2+192 x^3+192 e^{3 x} x^3+\left (324+864 x+576 x^2\right ) \log (3)+(432+576 x) \log ^2(3)+192 \log ^3(3)+e^{2 x} \left (432 x^2+576 x^3+576 x^2 \log (3)\right )+e^x \left (324 x+864 x^2+576 x^3+\left (864 x+1152 x^2\right ) \log (3)+576 x \log ^2(3)\right )} \, dx=\frac {128 x \left (-2 e^{2 x} x -4 e^{x} \mathrm {log}\left (3\right )-4 e^{x} x -3 e^{x}-4 \,\mathrm {log}\left (3\right )-2 x -3\right )}{768 e^{2 x} \mathrm {log}\left (3\right )^{2} x^{2}+1152 e^{2 x} \mathrm {log}\left (3\right ) x^{2}+432 e^{2 x} x^{2}+1536 e^{x} \mathrm {log}\left (3\right )^{3} x +1536 e^{x} \mathrm {log}\left (3\right )^{2} x^{2}+3456 e^{x} \mathrm {log}\left (3\right )^{2} x +2304 e^{x} \mathrm {log}\left (3\right ) x^{2}+2592 e^{x} \mathrm {log}\left (3\right ) x +864 e^{x} x^{2}+648 e^{x} x +768 \mathrm {log}\left (3\right )^{4}+1536 \mathrm {log}\left (3\right )^{3} x +2304 \mathrm {log}\left (3\right )^{3}+768 \mathrm {log}\left (3\right )^{2} x^{2}+3456 \mathrm {log}\left (3\right )^{2} x +2592 \mathrm {log}\left (3\right )^{2}+1152 \,\mathrm {log}\left (3\right ) x^{2}+2592 \,\mathrm {log}\left (3\right ) x +1296 \,\mathrm {log}\left (3\right )+432 x^{2}+648 x +243} \] Input:
int(((-128*x-128)*exp(x)-128)/(192*x^3*exp(x)^3+(576*x^2*log(3)+576*x^3+43 2*x^2)*exp(x)^2+(576*x*log(3)^2+(1152*x^2+864*x)*log(3)+576*x^3+864*x^2+32 4*x)*exp(x)+192*log(3)^3+(576*x+432)*log(3)^2+(576*x^2+864*x+324)*log(3)+1 92*x^3+432*x^2+324*x+81),x)
Output:
(128*x*( - 2*e**(2*x)*x - 4*e**x*log(3) - 4*e**x*x - 3*e**x - 4*log(3) - 2 *x - 3))/(3*(256*e**(2*x)*log(3)**2*x**2 + 384*e**(2*x)*log(3)*x**2 + 144* e**(2*x)*x**2 + 512*e**x*log(3)**3*x + 512*e**x*log(3)**2*x**2 + 1152*e**x *log(3)**2*x + 768*e**x*log(3)*x**2 + 864*e**x*log(3)*x + 288*e**x*x**2 + 216*e**x*x + 256*log(3)**4 + 512*log(3)**3*x + 768*log(3)**3 + 256*log(3)* *2*x**2 + 1152*log(3)**2*x + 864*log(3)**2 + 384*log(3)*x**2 + 864*log(3)* x + 432*log(3) + 144*x**2 + 216*x + 81))