3.2.14 \(\int \frac {(-5632-12288 x-6144 x^2) \log ^2(4)+e^{2 x} (-384 x^2-256 x^3) \log ^2(4)+e^x (3072 x+4608 x^2+1024 x^3) \log ^2(4)}{64000+211200 x+462720 x^2+668864 x^3+724224 x^4+581376 x^5+344064 x^6+144384 x^7+36864 x^8+4096 x^9+e^{6 x} x^9+e^x (-57600 x^2-165120 x^3-292416 x^4-336768 x^5-265728 x^6-144384 x^7-46080 x^8-6144 x^9)+e^{3 x} (-2880 x^5-6816 x^6-9024 x^7-5760 x^8-1280 x^9)+e^{5 x} (-36 x^8-24 x^9)+e^{4 x} (120 x^6+564 x^7+720 x^8+240 x^9)+e^{2 x} (4800 x^3+27840 x^4+59376 x^5+70272 x^6+54144 x^7+23040 x^8+3840 x^9)} \, dx\)
Optimal. Leaf size=30 \[ \frac {\log ^2(4)}{\left (5+x+\frac {1}{2} x \left (3+2 x-\frac {e^x x}{2}\right )^2\right )^2} \]
________________________________________________________________________________________
Rubi [A] time = 1.22, antiderivative size = 38, normalized size of antiderivative = 1.27,
number of steps used = 3, number of rules used = 3, integrand size = 279, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used
= {6688, 12, 6686} \begin {gather*} \frac {64 \log ^2(4)}{\left (\left (4-e^x\right )^2 x^3+12 \left (4-e^x\right ) x^2+44 x+40\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((-5632 - 12288*x - 6144*x^2)*Log[4]^2 + E^(2*x)*(-384*x^2 - 256*x^3)*Log[4]^2 + E^x*(3072*x + 4608*x^2 +
1024*x^3)*Log[4]^2)/(64000 + 211200*x + 462720*x^2 + 668864*x^3 + 724224*x^4 + 581376*x^5 + 344064*x^6 + 14438
4*x^7 + 36864*x^8 + 4096*x^9 + E^(6*x)*x^9 + E^x*(-57600*x^2 - 165120*x^3 - 292416*x^4 - 336768*x^5 - 265728*x
^6 - 144384*x^7 - 46080*x^8 - 6144*x^9) + E^(3*x)*(-2880*x^5 - 6816*x^6 - 9024*x^7 - 5760*x^8 - 1280*x^9) + E^
(5*x)*(-36*x^8 - 24*x^9) + E^(4*x)*(120*x^6 + 564*x^7 + 720*x^8 + 240*x^9) + E^(2*x)*(4800*x^3 + 27840*x^4 + 5
9376*x^5 + 70272*x^6 + 54144*x^7 + 23040*x^8 + 3840*x^9)),x]
[Out]
(64*Log[4]^2)/(40 + 44*x + 12*(4 - E^x)*x^2 + (4 - E^x)^2*x^3)^2
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 6686
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /; !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {128 \left (-44+24 \left (-4+e^x\right ) x-3 \left (16-12 e^x+e^{2 x}\right ) x^2-2 e^x \left (-4+e^x\right ) x^3\right ) \log ^2(4)}{\left (40+44 x-12 \left (-4+e^x\right ) x^2+\left (-4+e^x\right )^2 x^3\right )^3} \, dx\\ &=\left (128 \log ^2(4)\right ) \int \frac {-44+24 \left (-4+e^x\right ) x-3 \left (16-12 e^x+e^{2 x}\right ) x^2-2 e^x \left (-4+e^x\right ) x^3}{\left (40+44 x-12 \left (-4+e^x\right ) x^2+\left (-4+e^x\right )^2 x^3\right )^3} \, dx\\ &=\frac {64 \log ^2(4)}{\left (40+44 x+12 \left (4-e^x\right ) x^2+\left (4-e^x\right )^2 x^3\right )^2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 34, normalized size = 1.13 \begin {gather*} \frac {64 \log ^2(4)}{\left (40+44 x-12 \left (-4+e^x\right ) x^2+\left (-4+e^x\right )^2 x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((-5632 - 12288*x - 6144*x^2)*Log[4]^2 + E^(2*x)*(-384*x^2 - 256*x^3)*Log[4]^2 + E^x*(3072*x + 4608*
x^2 + 1024*x^3)*Log[4]^2)/(64000 + 211200*x + 462720*x^2 + 668864*x^3 + 724224*x^4 + 581376*x^5 + 344064*x^6 +
144384*x^7 + 36864*x^8 + 4096*x^9 + E^(6*x)*x^9 + E^x*(-57600*x^2 - 165120*x^3 - 292416*x^4 - 336768*x^5 - 26
5728*x^6 - 144384*x^7 - 46080*x^8 - 6144*x^9) + E^(3*x)*(-2880*x^5 - 6816*x^6 - 9024*x^7 - 5760*x^8 - 1280*x^9
) + E^(5*x)*(-36*x^8 - 24*x^9) + E^(4*x)*(120*x^6 + 564*x^7 + 720*x^8 + 240*x^9) + E^(2*x)*(4800*x^3 + 27840*x
^4 + 59376*x^5 + 70272*x^6 + 54144*x^7 + 23040*x^8 + 3840*x^9)),x]
[Out]
(64*Log[4]^2)/(40 + 44*x - 12*(-4 + E^x)*x^2 + (-4 + E^x)^2*x^3)^2
________________________________________________________________________________________
fricas [B] time = 0.82, size = 120, normalized size = 4.00 \begin {gather*} \frac {256 \, \log \relax (2)^{2}}{x^{6} e^{\left (4 \, x\right )} + 256 \, x^{6} + 1536 \, x^{5} + 3712 \, x^{4} + 5504 \, x^{3} + 5776 \, x^{2} - 8 \, {\left (2 \, x^{6} + 3 \, x^{5}\right )} e^{\left (3 \, x\right )} + 8 \, {\left (12 \, x^{6} + 36 \, x^{5} + 29 \, x^{4} + 10 \, x^{3}\right )} e^{\left (2 \, x\right )} - 32 \, {\left (8 \, x^{6} + 36 \, x^{5} + 58 \, x^{4} + 53 \, x^{3} + 30 \, x^{2}\right )} e^{x} + 3520 \, x + 1600} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*(-256*x^3-384*x^2)*log(2)^2*exp(x)^2+4*(1024*x^3+4608*x^2+3072*x)*log(2)^2*exp(x)+4*(-6144*x^2-12
288*x-5632)*log(2)^2)/(x^9*exp(x)^6+(-24*x^9-36*x^8)*exp(x)^5+(240*x^9+720*x^8+564*x^7+120*x^6)*exp(x)^4+(-128
0*x^9-5760*x^8-9024*x^7-6816*x^6-2880*x^5)*exp(x)^3+(3840*x^9+23040*x^8+54144*x^7+70272*x^6+59376*x^5+27840*x^
4+4800*x^3)*exp(x)^2+(-6144*x^9-46080*x^8-144384*x^7-265728*x^6-336768*x^5-292416*x^4-165120*x^3-57600*x^2)*ex
p(x)+4096*x^9+36864*x^8+144384*x^7+344064*x^6+581376*x^5+724224*x^4+668864*x^3+462720*x^2+211200*x+64000),x, a
lgorithm="fricas")
[Out]
256*log(2)^2/(x^6*e^(4*x) + 256*x^6 + 1536*x^5 + 3712*x^4 + 5504*x^3 + 5776*x^2 - 8*(2*x^6 + 3*x^5)*e^(3*x) +
8*(12*x^6 + 36*x^5 + 29*x^4 + 10*x^3)*e^(2*x) - 32*(8*x^6 + 36*x^5 + 58*x^4 + 53*x^3 + 30*x^2)*e^x + 3520*x +
1600)
________________________________________________________________________________________
giac [B] time = 1.82, size = 135, normalized size = 4.50 \begin {gather*} \frac {512 \, \log \relax (2)^{2}}{x^{6} e^{\left (4 \, x\right )} - 16 \, x^{6} e^{\left (3 \, x\right )} + 96 \, x^{6} e^{\left (2 \, x\right )} - 256 \, x^{6} e^{x} + 256 \, x^{6} - 24 \, x^{5} e^{\left (3 \, x\right )} + 288 \, x^{5} e^{\left (2 \, x\right )} - 1152 \, x^{5} e^{x} + 1536 \, x^{5} + 232 \, x^{4} e^{\left (2 \, x\right )} - 1856 \, x^{4} e^{x} + 3712 \, x^{4} + 80 \, x^{3} e^{\left (2 \, x\right )} - 1696 \, x^{3} e^{x} + 5504 \, x^{3} - 960 \, x^{2} e^{x} + 5776 \, x^{2} + 3520 \, x + 1600} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*(-256*x^3-384*x^2)*log(2)^2*exp(x)^2+4*(1024*x^3+4608*x^2+3072*x)*log(2)^2*exp(x)+4*(-6144*x^2-12
288*x-5632)*log(2)^2)/(x^9*exp(x)^6+(-24*x^9-36*x^8)*exp(x)^5+(240*x^9+720*x^8+564*x^7+120*x^6)*exp(x)^4+(-128
0*x^9-5760*x^8-9024*x^7-6816*x^6-2880*x^5)*exp(x)^3+(3840*x^9+23040*x^8+54144*x^7+70272*x^6+59376*x^5+27840*x^
4+4800*x^3)*exp(x)^2+(-6144*x^9-46080*x^8-144384*x^7-265728*x^6-336768*x^5-292416*x^4-165120*x^3-57600*x^2)*ex
p(x)+4096*x^9+36864*x^8+144384*x^7+344064*x^6+581376*x^5+724224*x^4+668864*x^3+462720*x^2+211200*x+64000),x, a
lgorithm="giac")
[Out]
512*log(2)^2/(x^6*e^(4*x) - 16*x^6*e^(3*x) + 96*x^6*e^(2*x) - 256*x^6*e^x + 256*x^6 - 24*x^5*e^(3*x) + 288*x^5
*e^(2*x) - 1152*x^5*e^x + 1536*x^5 + 232*x^4*e^(2*x) - 1856*x^4*e^x + 3712*x^4 + 80*x^3*e^(2*x) - 1696*x^3*e^x
+ 5504*x^3 - 960*x^2*e^x + 5776*x^2 + 3520*x + 1600)
________________________________________________________________________________________
maple [A] time = 0.07, size = 46, normalized size = 1.53
|
|
|
| method |
result |
size |
|
|
|
| risch |
\(\frac {256 \ln \relax (2)^{2}}{\left ({\mathrm e}^{2 x} x^{3}-8 \,{\mathrm e}^{x} x^{3}-12 \,{\mathrm e}^{x} x^{2}+16 x^{3}+48 x^{2}+44 x +40\right )^{2}}\) |
\(46\) |
|
|
|
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((4*(-256*x^3-384*x^2)*ln(2)^2*exp(x)^2+4*(1024*x^3+4608*x^2+3072*x)*ln(2)^2*exp(x)+4*(-6144*x^2-12288*x-56
32)*ln(2)^2)/(x^9*exp(x)^6+(-24*x^9-36*x^8)*exp(x)^5+(240*x^9+720*x^8+564*x^7+120*x^6)*exp(x)^4+(-1280*x^9-576
0*x^8-9024*x^7-6816*x^6-2880*x^5)*exp(x)^3+(3840*x^9+23040*x^8+54144*x^7+70272*x^6+59376*x^5+27840*x^4+4800*x^
3)*exp(x)^2+(-6144*x^9-46080*x^8-144384*x^7-265728*x^6-336768*x^5-292416*x^4-165120*x^3-57600*x^2)*exp(x)+4096
*x^9+36864*x^8+144384*x^7+344064*x^6+581376*x^5+724224*x^4+668864*x^3+462720*x^2+211200*x+64000),x,method=_RET
URNVERBOSE)
[Out]
256*ln(2)^2/(exp(2*x)*x^3-8*exp(x)*x^3-12*exp(x)*x^2+16*x^3+48*x^2+44*x+40)^2
________________________________________________________________________________________
maxima [B] time = 1.40, size = 120, normalized size = 4.00 \begin {gather*} \frac {256 \, \log \relax (2)^{2}}{x^{6} e^{\left (4 \, x\right )} + 256 \, x^{6} + 1536 \, x^{5} + 3712 \, x^{4} + 5504 \, x^{3} + 5776 \, x^{2} - 8 \, {\left (2 \, x^{6} + 3 \, x^{5}\right )} e^{\left (3 \, x\right )} + 8 \, {\left (12 \, x^{6} + 36 \, x^{5} + 29 \, x^{4} + 10 \, x^{3}\right )} e^{\left (2 \, x\right )} - 32 \, {\left (8 \, x^{6} + 36 \, x^{5} + 58 \, x^{4} + 53 \, x^{3} + 30 \, x^{2}\right )} e^{x} + 3520 \, x + 1600} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*(-256*x^3-384*x^2)*log(2)^2*exp(x)^2+4*(1024*x^3+4608*x^2+3072*x)*log(2)^2*exp(x)+4*(-6144*x^2-12
288*x-5632)*log(2)^2)/(x^9*exp(x)^6+(-24*x^9-36*x^8)*exp(x)^5+(240*x^9+720*x^8+564*x^7+120*x^6)*exp(x)^4+(-128
0*x^9-5760*x^8-9024*x^7-6816*x^6-2880*x^5)*exp(x)^3+(3840*x^9+23040*x^8+54144*x^7+70272*x^6+59376*x^5+27840*x^
4+4800*x^3)*exp(x)^2+(-6144*x^9-46080*x^8-144384*x^7-265728*x^6-336768*x^5-292416*x^4-165120*x^3-57600*x^2)*ex
p(x)+4096*x^9+36864*x^8+144384*x^7+344064*x^6+581376*x^5+724224*x^4+668864*x^3+462720*x^2+211200*x+64000),x, a
lgorithm="maxima")
[Out]
256*log(2)^2/(x^6*e^(4*x) + 256*x^6 + 1536*x^5 + 3712*x^4 + 5504*x^3 + 5776*x^2 - 8*(2*x^6 + 3*x^5)*e^(3*x) +
8*(12*x^6 + 36*x^5 + 29*x^4 + 10*x^3)*e^(2*x) - 32*(8*x^6 + 36*x^5 + 58*x^4 + 53*x^3 + 30*x^2)*e^x + 3520*x +
1600)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {4\,{\ln \relax (2)}^2\,\left (6144\,x^2+12288\,x+5632\right )+4\,{\mathrm {e}}^{2\,x}\,{\ln \relax (2)}^2\,\left (256\,x^3+384\,x^2\right )-4\,{\mathrm {e}}^x\,{\ln \relax (2)}^2\,\left (1024\,x^3+4608\,x^2+3072\,x\right )}{211200\,x-{\mathrm {e}}^{3\,x}\,\left (1280\,x^9+5760\,x^8+9024\,x^7+6816\,x^6+2880\,x^5\right )-{\mathrm {e}}^x\,\left (6144\,x^9+46080\,x^8+144384\,x^7+265728\,x^6+336768\,x^5+292416\,x^4+165120\,x^3+57600\,x^2\right )-{\mathrm {e}}^{5\,x}\,\left (24\,x^9+36\,x^8\right )+x^9\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{2\,x}\,\left (3840\,x^9+23040\,x^8+54144\,x^7+70272\,x^6+59376\,x^5+27840\,x^4+4800\,x^3\right )+{\mathrm {e}}^{4\,x}\,\left (240\,x^9+720\,x^8+564\,x^7+120\,x^6\right )+462720\,x^2+668864\,x^3+724224\,x^4+581376\,x^5+344064\,x^6+144384\,x^7+36864\,x^8+4096\,x^9+64000} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(4*log(2)^2*(12288*x + 6144*x^2 + 5632) + 4*exp(2*x)*log(2)^2*(384*x^2 + 256*x^3) - 4*exp(x)*log(2)^2*(30
72*x + 4608*x^2 + 1024*x^3))/(211200*x - exp(3*x)*(2880*x^5 + 6816*x^6 + 9024*x^7 + 5760*x^8 + 1280*x^9) - exp
(x)*(57600*x^2 + 165120*x^3 + 292416*x^4 + 336768*x^5 + 265728*x^6 + 144384*x^7 + 46080*x^8 + 6144*x^9) - exp(
5*x)*(36*x^8 + 24*x^9) + x^9*exp(6*x) + exp(2*x)*(4800*x^3 + 27840*x^4 + 59376*x^5 + 70272*x^6 + 54144*x^7 + 2
3040*x^8 + 3840*x^9) + exp(4*x)*(120*x^6 + 564*x^7 + 720*x^8 + 240*x^9) + 462720*x^2 + 668864*x^3 + 724224*x^4
+ 581376*x^5 + 344064*x^6 + 144384*x^7 + 36864*x^8 + 4096*x^9 + 64000),x)
[Out]
int(-(4*log(2)^2*(12288*x + 6144*x^2 + 5632) + 4*exp(2*x)*log(2)^2*(384*x^2 + 256*x^3) - 4*exp(x)*log(2)^2*(30
72*x + 4608*x^2 + 1024*x^3))/(211200*x - exp(3*x)*(2880*x^5 + 6816*x^6 + 9024*x^7 + 5760*x^8 + 1280*x^9) - exp
(x)*(57600*x^2 + 165120*x^3 + 292416*x^4 + 336768*x^5 + 265728*x^6 + 144384*x^7 + 46080*x^8 + 6144*x^9) - exp(
5*x)*(36*x^8 + 24*x^9) + x^9*exp(6*x) + exp(2*x)*(4800*x^3 + 27840*x^4 + 59376*x^5 + 70272*x^6 + 54144*x^7 + 2
3040*x^8 + 3840*x^9) + exp(4*x)*(120*x^6 + 564*x^7 + 720*x^8 + 240*x^9) + 462720*x^2 + 668864*x^3 + 724224*x^4
+ 581376*x^5 + 344064*x^6 + 144384*x^7 + 36864*x^8 + 4096*x^9 + 64000), x)
________________________________________________________________________________________
sympy [B] time = 0.62, size = 117, normalized size = 3.90 \begin {gather*} \frac {256 \log {\relax (2 )}^{2}}{x^{6} e^{4 x} + 256 x^{6} + 1536 x^{5} + 3712 x^{4} + 5504 x^{3} + 5776 x^{2} + 3520 x + \left (- 16 x^{6} - 24 x^{5}\right ) e^{3 x} + \left (96 x^{6} + 288 x^{5} + 232 x^{4} + 80 x^{3}\right ) e^{2 x} + \left (- 256 x^{6} - 1152 x^{5} - 1856 x^{4} - 1696 x^{3} - 960 x^{2}\right ) e^{x} + 1600} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*(-256*x**3-384*x**2)*ln(2)**2*exp(x)**2+4*(1024*x**3+4608*x**2+3072*x)*ln(2)**2*exp(x)+4*(-6144*x
**2-12288*x-5632)*ln(2)**2)/(x**9*exp(x)**6+(-24*x**9-36*x**8)*exp(x)**5+(240*x**9+720*x**8+564*x**7+120*x**6)
*exp(x)**4+(-1280*x**9-5760*x**8-9024*x**7-6816*x**6-2880*x**5)*exp(x)**3+(3840*x**9+23040*x**8+54144*x**7+702
72*x**6+59376*x**5+27840*x**4+4800*x**3)*exp(x)**2+(-6144*x**9-46080*x**8-144384*x**7-265728*x**6-336768*x**5-
292416*x**4-165120*x**3-57600*x**2)*exp(x)+4096*x**9+36864*x**8+144384*x**7+344064*x**6+581376*x**5+724224*x**
4+668864*x**3+462720*x**2+211200*x+64000),x)
[Out]
256*log(2)**2/(x**6*exp(4*x) + 256*x**6 + 1536*x**5 + 3712*x**4 + 5504*x**3 + 5776*x**2 + 3520*x + (-16*x**6 -
24*x**5)*exp(3*x) + (96*x**6 + 288*x**5 + 232*x**4 + 80*x**3)*exp(2*x) + (-256*x**6 - 1152*x**5 - 1856*x**4 -
1696*x**3 - 960*x**2)*exp(x) + 1600)
________________________________________________________________________________________