Integrand size = 85, antiderivative size = 25 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=\left (x-e^{-3+x} x\right ) \left (6 (4+x)+\frac {8}{\log (2)}\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(25)=50\).
Time = 5.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.56 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=-\frac {4 \left (-\frac {1}{2} e^3 x \left (2 x^2 \log ^2(8)+x \log (4096) (4+\log (4096))+2 (4+\log (4096))^2\right )+e^x x \left (16+6 \log ^2(8)+x^2 \log ^2(8)+21 \log (2) (4+\log (64))+\log (4096)-\log (8) \log (4096)+x \log (8) (8+\log (16777216))\right )\right )}{e^3 \log ^2(2)} \]
Integrate[(64*E^3 + E^3*(384 + 192*x)*Log[2] + E^3*(576 + 576*x + 108*x^2) *Log[2]^2 + E^x*(-64 - 64*x + (-384 - 576*x - 96*x^2)*Log[2] + (-576 - 115 2*x - 396*x^2 - 36*x^3)*Log[2]^2))/(E^3*Log[2]^2),x]
(-4*(-1/2*(E^3*x*(2*x^2*Log[8]^2 + x*Log[4096]*(4 + Log[4096]) + 2*(4 + Lo g[4096])^2)) + E^x*x*(16 + 6*Log[8]^2 + x^2*Log[8]^2 + 21*Log[2]*(4 + Log[ 64]) + Log[4096] - Log[8]*Log[4096] + x*Log[8]*(8 + Log[16777216]))))/(E^3 *Log[2]^2)
Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(25)=50\).
Time = 0.36 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {27, 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.
\(\displaystyle \int \frac {e^3 \left (108 x^2+576 x+576\right ) \log ^2(2)+e^x \left (\left (-96 x^2-576 x-384\right ) \log (2)+\left (-36 x^3-396 x^2-1152 x-576\right ) \log ^2(2)-64 x-64\right )+e^3 (192 x+384) \log (2)+64 e^3}{e^3 \log ^2(2)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (192 e^3 \log (2) (x+2)-4 e^x \left (16 x+9 \left (x^3+11 x^2+32 x+16\right ) \log ^2(2)+24 \left (x^2+6 x+4\right ) \log (2)+16\right )+36 e^3 \left (3 x^2+16 x+16\right ) \log ^2(2)+64 e^3\right )dx}{e^3 \log ^2(2)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-36 e^x x^3 \log ^2(2)+36 e^3 x^3 \log ^2(2)-288 e^x x^2 \log ^2(2)+288 e^3 x^2 \log ^2(2)-96 e^x x^2 \log (2)-64 e^x x+64 e^3 x-576 e^x x \log ^2(2)+576 e^3 x \log ^2(2)-384 e^x x \log (2)+96 e^3 (x+2)^2 \log (2)}{e^3 \log ^2(2)}\) |
Int[(64*E^3 + E^3*(384 + 192*x)*Log[2] + E^3*(576 + 576*x + 108*x^2)*Log[2 ]^2 + E^x*(-64 - 64*x + (-384 - 576*x - 96*x^2)*Log[2] + (-576 - 1152*x - 396*x^2 - 36*x^3)*Log[2]^2))/(E^3*Log[2]^2),x]
(64*E^3*x - 64*E^x*x - 384*E^x*x*Log[2] - 96*E^x*x^2*Log[2] + 96*E^3*(2 + x)^2*Log[2] + 576*E^3*x*Log[2]^2 - 576*E^x*x*Log[2]^2 + 288*E^3*x^2*Log[2] ^2 - 288*E^x*x^2*Log[2]^2 + 36*E^3*x^3*Log[2]^2 - 36*E^x*x^3*Log[2]^2)/(E^ 3*Log[2]^2)
3.20.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(23)=46\).
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.52
method | result | size |
risch | \(36 x^{3}+288 x^{2}+576 x +\frac {96 x^{2}}{\ln \left (2\right )}+\frac {384 x}{\ln \left (2\right )}+\frac {64 x}{\ln \left (2\right )^{2}}+\frac {\left (-36 x^{3} \ln \left (2\right )^{2}-288 x^{2} \ln \left (2\right )^{2}-576 x \ln \left (2\right )^{2}-96 x^{2} \ln \left (2\right )-384 x \ln \left (2\right )-64 x \right ) {\mathrm e}^{-3+x}}{\ln \left (2\right )^{2}}\) | \(88\) |
norman | \(\frac {\left (288 \ln \left (2\right )+96\right ) x^{2}+36 x^{3} \ln \left (2\right )+\frac {64 \left (9 \ln \left (2\right )^{2}+6 \ln \left (2\right )+1\right ) x}{\ln \left (2\right )}-96 \left (3 \ln \left (2\right )+1\right ) {\mathrm e}^{-3} x^{2} {\mathrm e}^{x}-36 \ln \left (2\right ) {\mathrm e}^{-3} x^{3} {\mathrm e}^{x}-\frac {64 \left (9 \ln \left (2\right )^{2}+6 \ln \left (2\right )+1\right ) {\mathrm e}^{-3} x \,{\mathrm e}^{x}}{\ln \left (2\right )}}{\ln \left (2\right )}\) | \(98\) |
default | \(\frac {{\mathrm e}^{-3} \left (-64 \,{\mathrm e}^{x} x -36 \,{\mathrm e}^{x} \ln \left (2\right )^{2} x^{3}-288 \,{\mathrm e}^{x} \ln \left (2\right )^{2} x^{2}-576 x \ln \left (2\right )^{2} {\mathrm e}^{x}-384 x \ln \left (2\right ) {\mathrm e}^{x}-96 x^{2} \ln \left (2\right ) {\mathrm e}^{x}+192 \,{\mathrm e}^{3} \ln \left (2\right ) \left (2 x +\frac {1}{2} x^{2}\right )+36 \,{\mathrm e}^{3} \ln \left (2\right )^{2} \left (x^{3}+8 x^{2}+16 x \right )+64 x \,{\mathrm e}^{3}\right )}{\ln \left (2\right )^{2}}\) | \(103\) |
parts | \(36 x^{3}+288 x^{2}+576 x +\frac {96 x^{2}}{\ln \left (2\right )}+\frac {384 x}{\ln \left (2\right )}+\frac {64 x}{\ln \left (2\right )^{2}}-36 \,{\mathrm e}^{-3} {\mathrm e}^{x} x^{3}-288 \,{\mathrm e}^{-3} x^{2} {\mathrm e}^{x}-576 \,{\mathrm e}^{-3} {\mathrm e}^{x} x -\frac {96 \,{\mathrm e}^{-3} x^{2} {\mathrm e}^{x}}{\ln \left (2\right )}-\frac {384 \,{\mathrm e}^{-3} x \,{\mathrm e}^{x}}{\ln \left (2\right )}-\frac {64 \,{\mathrm e}^{-3} x \,{\mathrm e}^{x}}{\ln \left (2\right )^{2}}\) | \(110\) |
parallelrisch | \(\frac {{\mathrm e}^{-3} \left (36 \,{\mathrm e}^{3} \ln \left (2\right )^{2} x^{3}-36 \,{\mathrm e}^{x} \ln \left (2\right )^{2} x^{3}+288 \,{\mathrm e}^{3} \ln \left (2\right )^{2} x^{2}-288 \,{\mathrm e}^{x} \ln \left (2\right )^{2} x^{2}+576 \,{\mathrm e}^{3} \ln \left (2\right )^{2} x +96 \,{\mathrm e}^{3} \ln \left (2\right ) x^{2}-576 x \ln \left (2\right )^{2} {\mathrm e}^{x}-96 x^{2} \ln \left (2\right ) {\mathrm e}^{x}+384 \,{\mathrm e}^{3} \ln \left (2\right ) x -384 x \ln \left (2\right ) {\mathrm e}^{x}-64 \,{\mathrm e}^{x} x +64 x \,{\mathrm e}^{3}\right )}{\ln \left (2\right )^{2}}\) | \(115\) |
int((((-36*x^3-396*x^2-1152*x-576)*ln(2)^2+(-96*x^2-576*x-384)*ln(2)-64*x- 64)*exp(x)+(108*x^2+576*x+576)*exp(3)*ln(2)^2+(192*x+384)*exp(3)*ln(2)+64* exp(3))/exp(3)/ln(2)^2,x,method=_RETURNVERBOSE)
36*x^3+288*x^2+576*x+96*x^2/ln(2)+384*x/ln(2)+64*x/ln(2)^2+(-36*x^3*ln(2)^ 2-288*x^2*ln(2)^2-576*x*ln(2)^2-96*x^2*ln(2)-384*x*ln(2)-64*x)/ln(2)^2*exp (-3+x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.36 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=\frac {4 \, {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{3} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} e^{3} \log \left (2\right ) + 16 \, x e^{3} - {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} \log \left (2\right ) + 16 \, x\right )} e^{x}\right )} e^{\left (-3\right )}}{\log \left (2\right )^{2}} \]
integrate((((-36*x^3-396*x^2-1152*x-576)*log(2)^2+(-96*x^2-576*x-384)*log( 2)-64*x-64)*exp(x)+(108*x^2+576*x+576)*exp(3)*log(2)^2+(192*x+384)*exp(3)* log(2)+64*exp(3))/exp(3)/log(2)^2,x, algorithm=\
4*(9*(x^3 + 8*x^2 + 16*x)*e^3*log(2)^2 + 24*(x^2 + 4*x)*e^3*log(2) + 16*x* e^3 - (9*(x^3 + 8*x^2 + 16*x)*log(2)^2 + 24*(x^2 + 4*x)*log(2) + 16*x)*e^x )*e^(-3)/log(2)^2
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=36 x^{3} + \frac {x^{2} \cdot \left (96 + 288 \log {\left (2 \right )}\right )}{\log {\left (2 \right )}} + \frac {x \left (64 + 384 \log {\left (2 \right )} + 576 \log {\left (2 \right )}^{2}\right )}{\log {\left (2 \right )}^{2}} + \frac {\left (- 36 x^{3} \log {\left (2 \right )}^{2} - 288 x^{2} \log {\left (2 \right )}^{2} - 96 x^{2} \log {\left (2 \right )} - 576 x \log {\left (2 \right )}^{2} - 384 x \log {\left (2 \right )} - 64 x\right ) e^{x}}{e^{3} \log {\left (2 \right )}^{2}} \]
integrate((((-36*x**3-396*x**2-1152*x-576)*ln(2)**2+(-96*x**2-576*x-384)*l n(2)-64*x-64)*exp(x)+(108*x**2+576*x+576)*exp(3)*ln(2)**2+(192*x+384)*exp( 3)*ln(2)+64*exp(3))/exp(3)/ln(2)**2,x)
36*x**3 + x**2*(96 + 288*log(2))/log(2) + x*(64 + 384*log(2) + 576*log(2)* *2)/log(2)**2 + (-36*x**3*log(2)**2 - 288*x**2*log(2)**2 - 96*x**2*log(2) - 576*x*log(2)**2 - 384*x*log(2) - 64*x)*exp(-3)*exp(x)/log(2)**2
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=\frac {4 \, {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{3} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} e^{3} \log \left (2\right ) + 16 \, x e^{3} - {\left (9 \, x^{3} \log \left (2\right )^{2} + 24 \, {\left (3 \, \log \left (2\right )^{2} + \log \left (2\right )\right )} x^{2} + 16 \, {\left (9 \, \log \left (2\right )^{2} + 6 \, \log \left (2\right ) + 1\right )} x\right )} e^{x}\right )} e^{\left (-3\right )}}{\log \left (2\right )^{2}} \]
integrate((((-36*x^3-396*x^2-1152*x-576)*log(2)^2+(-96*x^2-576*x-384)*log( 2)-64*x-64)*exp(x)+(108*x^2+576*x+576)*exp(3)*log(2)^2+(192*x+384)*exp(3)* log(2)+64*exp(3))/exp(3)/log(2)^2,x, algorithm=\
4*(9*(x^3 + 8*x^2 + 16*x)*e^3*log(2)^2 + 24*(x^2 + 4*x)*e^3*log(2) + 16*x* e^3 - (9*x^3*log(2)^2 + 24*(3*log(2)^2 + log(2))*x^2 + 16*(9*log(2)^2 + 6* log(2) + 1)*x)*e^x)*e^(-3)/log(2)^2
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.68 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=\frac {4 \, {\left (9 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{3} \log \left (2\right )^{2} + 24 \, {\left (x^{2} + 4 \, x\right )} e^{3} \log \left (2\right ) + 16 \, x e^{3} - {\left (9 \, x^{3} \log \left (2\right )^{2} + 72 \, x^{2} \log \left (2\right )^{2} + 24 \, x^{2} \log \left (2\right ) + 144 \, x \log \left (2\right )^{2} + 96 \, x \log \left (2\right ) + 16 \, x\right )} e^{x}\right )} e^{\left (-3\right )}}{\log \left (2\right )^{2}} \]
integrate((((-36*x^3-396*x^2-1152*x-576)*log(2)^2+(-96*x^2-576*x-384)*log( 2)-64*x-64)*exp(x)+(108*x^2+576*x+576)*exp(3)*log(2)^2+(192*x+384)*exp(3)* log(2)+64*exp(3))/exp(3)/log(2)^2,x, algorithm=\
4*(9*(x^3 + 8*x^2 + 16*x)*e^3*log(2)^2 + 24*(x^2 + 4*x)*e^3*log(2) + 16*x* e^3 - (9*x^3*log(2)^2 + 72*x^2*log(2)^2 + 24*x^2*log(2) + 144*x*log(2)^2 + 96*x*log(2) + 16*x)*e^x)*e^(-3)/log(2)^2
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {64 e^3+e^3 (384+192 x) \log (2)+e^3 \left (576+576 x+108 x^2\right ) \log ^2(2)+e^x \left (-64-64 x+\left (-384-576 x-96 x^2\right ) \log (2)+\left (-576-1152 x-396 x^2-36 x^3\right ) \log ^2(2)\right )}{e^3 \log ^2(2)} \, dx=\frac {36\,x^3\,{\ln \left (2\right )}^2+x\,\left (384\,\ln \left (2\right )+576\,{\ln \left (2\right )}^2+64\right )-64\,x\,{\mathrm {e}}^{x-3}\,\left (\ln \left (64\right )+9\,{\ln \left (2\right )}^2+1\right )-36\,x^3\,{\mathrm {e}}^{x-3}\,{\ln \left (2\right )}^2-96\,x^2\,{\mathrm {e}}^{x-3}\,\left (\ln \left (2\right )+3\,{\ln \left (2\right )}^2\right )+96\,x^2\,\ln \left (2\right )\,\left (3\,\ln \left (2\right )+1\right )}{{\ln \left (2\right )}^2} \]
int((exp(-3)*(64*exp(3) - exp(x)*(64*x + log(2)*(576*x + 96*x^2 + 384) + l og(2)^2*(1152*x + 396*x^2 + 36*x^3 + 576) + 64) + exp(3)*log(2)*(192*x + 3 84) + exp(3)*log(2)^2*(576*x + 108*x^2 + 576)))/log(2)^2,x)