∫ ex(−8x+4x2+28x6−4x7−12x11+x12+log(4))______ 16x4−32x9+24x14−8x19+x24+(8x2−8x7+2x12) log(4)+log2(4) dx [6401]

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(20)=40\).
time = 0.64, antiderivative size = 68, normalized size of antiderivative = 3.40, number of steps used = 1, number of rules used = 1, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2326} \begin {gather*} \frac {e^x \left (x^{12}-4 x^7+4 x^2+\log (4)\right )}{x^{24}-8 x^{19}+24 x^{14}-32 x^9+16 x^4+2 \left (x^{12}-4 x^7+4 x^2\right ) \log (4)+\log ^2(4)} \end {gather*}

Int[(E^x*(-8*x + 4*x^2 + 28*x^6 - 4*x^7 - 12*x^11 + x^12 + Log[4]))/(16*x^4 - 32*x^9 + 24*x^14 - 8*x^19 + x^24

(E^x*(4*x^2 - 4*x^7 + x^12 + Log[4]))/(16*x^4 - 32*x^9 + 24*x^14 - 8*x^19 + x^24 + 2*(4*x^2 - 4*x^7 + x^12)*Lo

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^x \left (4 x^2-4 x^7+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+2 \left (4 x^2-4 x^7+x^{12}\right ) \log (4)+\log ^2(4)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 1.01, size = 22, normalized size = 1.10 \begin {gather*} \frac {e^x}{4 x^2-4 x^7+x^{12}+\log (4)} \end {gather*}

Integrate[(E^x*(-8*x + 4*x^2 + 28*x^6 - 4*x^7 - 12*x^11 + x^12 + Log[4]))/(16*x^4 - 32*x^9 + 24*x^14 - 8*x^19

+ x^24 + (8*x^2 - 8*x^7 + 2*x^12)*Log[4] + Log[4]^2),x]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.95, size = 3222, normalized size = 161.10


method	result	size

gosper	\(\frac {{\mathrm e}^{x}}{x^{12}-4 x^{7}+4 x^{2}+2 \ln \left (2\right )}\)	\(24\)
risch	\(\frac {{\mathrm e}^{x}}{x^{12}-4 x^{7}+4 x^{2}+2 \ln \left (2\right )}\)	\(24\)
default	\(\text {Expression too large to display}\)	\(3222\)

int((2*ln(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*ln(2)^2+2*(2*x^12-8*x^7+8*x^2)*ln(2)+x^24-8*x^19+2

-exp(x)*(787320*ln(2)^4*x^10+2834352*ln(2)^5*x^8-703125*ln(2)*x^11+506250*ln(2)^2*x^9-364500*x^7*ln(2)^3-28868

40*x^5*ln(2)^4-4723920*x^3*ln(2)^5+1953125*x^8+2578125*x^6*ln(2)-1856250*x^4*ln(2)^2+1336500*x^2*ln(2)^3+21870

00*ln(2)^4-3906250*x^3-1953125*x*ln(2))/ln(2)/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))-exp(x)*(-6

561000*ln(2)^3*x^11+4723920*x^9*ln(2)^4-4218750*ln(2)*x^10+3037500*ln(2)^2*x^8+24057000*x^6*ln(2)^3-17321040*l

n(2)^4*x^4+5668704*x^2*ln(2)^5+1953125*x^7+15468750*x^5*ln(2)-11137500*x^3*ln(2)^2-18225000*x*ln(2)^3-3906250*

x^2-11718750*ln(2))/ln(2)/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))+7*exp(x)*(-1093500*ln(2)^3*x^1

1+787320*x^9*ln(2)^4+2834352*ln(2)^5*x^7-703125*ln(2)*x^10+506250*ln(2)^2*x^8+4009500*x^6*ln(2)^3-2886840*ln(2

)^4*x^4-4723920*x^2*ln(2)^5+1953125*x^7+2578125*x^5*ln(2)-1856250*x^3*ln(2)^2-3037500*x*ln(2)^3-3906250*x^2-19

53125*ln(2))/ln(2)/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))-1/16/ln(2)*sum((5668704*ln(2)^6-72900

0*ln(2)^4+3906250*ln(2)+3906250*ln(2)*_R1-607500*ln(2)^3*_R1^6+16061328*ln(2)^5*_R1^2+468750*ln(2)*_R1^5+20250

00*_R1*ln(2)^3-6613488*ln(2)^5*_R1^8+7812500*_R1^2-3906250*_R1^8+15625000*_R1^7+7812500*_R1^3-168750*ln(2)^2*_

R1^9+121500*ln(2)^3*_R1^7+962280*ln(2)^4*_R1^5+12912048*ln(2)^5*_R1^3+5668704*ln(2)^6*_R1-445500*ln(2)^3*_R1^2

-262440*ln(2)^4*_R1^10+524880*ln(2)^4*_R1^9+26453952*ln(2)^5*_R1^7-234375*ln(2)*_R1^10+506250*ln(2)^2*_R1^8-14

87160*ln(2)^4*_R1^4+618750*_R1^4*ln(2)^2-1575000*_R1^3*ln(2)^2+234375*ln(2)*_R1^11-859375*_R1^6*ln(2))/(170061

12*ln(2)^5+9765625)/(3*_R1^10-7*_R1^5+2)/_R1*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^12-4*_Z^7+4*_Z^2+2*ln(2)))-16

40250*exp(x)*ln(2)^3/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))*x^10+1/8/ln(2)*sum((13122000*ln(2)^

4-11718750*ln(2)-7812500*_R2^7-3906250*_R2^2+1953125*_R2^8-3906250*_R2^3-9447840*ln(2)^4*_R2^9+4218750*ln(2)*_

R2^10-9112500*ln(2)^2*_R2^8+10935000*ln(2)^3*_R2^6+26768880*ln(2)^4*_R2^4-51018336*ln(2)^5*_R2^2-8437500*ln(2)

*_R2^5+28350000*_R2^3*ln(2)^2-36450000*_R2*ln(2)^3-11718750*_R2*ln(2)+3037500*ln(2)^2*_R2^9-2187000*ln(2)^3*_R

2^7-17321040*ln(2)^4*_R2^5+5668704*ln(2)^5*_R2^3+8019000*ln(2)^3*_R2^2+4723920*ln(2)^4*_R2^10-11137500*_R2^4*l

n(2)^2-4218750*ln(2)*_R2^11+15468750*_R2^6*ln(2))/(17006112*ln(2)^5+9765625)/(3*_R2^10-7*_R2^5+2)/_R2*exp(_R2)

*Ei(1,-x+_R2),_R2=RootOf(_Z^12-4*_Z^7+4*_Z^2+2*ln(2)))+7/4/ln(2)*sum((-1953125*ln(2)-3037500*ln(2)^3+4009500*l

n(2)^3*_R3^6-4723920*ln(2)^5*_R3^2+2578125*ln(2)*_R3^5+6581250*ln(2)^2*_R3^2-3037500*_R3*ln(2)^3+787320*ln(2)^

4*_R3^9+2834352*ln(2)^5*_R3^7-703125*ln(2)*_R3^10+506250*ln(2)^2*_R3^8-2886840*ln(2)^4*_R3^4+7348320*_R3^3*ln(

2)^4-1856250*_R3^3*ln(2)^2+1093500*ln(2)^3*_R3^10-2361960*ln(2)^4*_R3^8+1406250*ln(2)*_R3^9-2025000*ln(2)^2*_R

3^7-2187000*ln(2)^3*_R3^5-9447840*ln(2)^5*_R3-3984375*ln(2)*_R3^4-14171760*ln(2)^5*_R3^6-1093500*ln(2)^3*_R3^1

1-3906250*_R3^2-9765625*_R3^6+1953125*_R3^7)/(17006112*ln(2)^5+9765625)/(3*_R3^10-7*_R3^5+2)/_R3*exp(_R3)*Ei(1

,-x+_R3),_R3=RootOf(_Z^12-4*_Z^7+4*_Z^2+2*ln(2)))+3*exp(x)*(364500*ln(2)^3*x^11-262440*x^9*ln(2)^4-6613488*ln(

2)^5*x^7+234375*ln(2)*x^10-168750*ln(2)^2*x^8-1336500*x^6*ln(2)^3+962280*ln(2)^4*x^4+12912048*x^2*ln(2)^5-3906

250*x^7+5668704*ln(2)^6-859375*x^5*ln(2)+618750*x^3*ln(2)^2+1012500*x*ln(2)^3+7812500*x^2+3906250*ln(2))/ln(2)

/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))+5859375/4*exp(x)/(17006112*ln(2)^5+9765625)/(x^12-4*x^7

+4*x^2+2*ln(2))*x^11-21484375/4*exp(x)/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))*x^6-4556250*exp(x

)*ln(2)^3/(17006112*ln(2)^5+9765625)/(x^12-4*x^7+4*x^2+2*ln(2))+9765625/2*exp(x)/(17006112*ln(2)^5+9765625)/(x

^12-4*x^7+4*x^2+2*ln(2))*x-1/4/ln(2)*sum((-6561000*ln(2)^3*_R2^11+4723920*ln(2)^4*_R2^9+6561000*ln(2)^3*_R2^10

-14171760*ln(2)^4*_R2^8-4218750*ln(2)*_R2^10+3037500*ln(2)^2*_R2^8+8437500*ln(2)*_R2^9+24057000*ln(2)^3*_R2^6-

12150000*ln(2)^2*_R2^7-17321040*ln(2)^4*_R2^4-13122000*ln(2)^3*_R2^5+5668704*ln(2)^5*_R2^2+44089920*_R2^3*ln(2

)^4+1953125*_R2^7-56687040*ln(2)^5*_R2+15468750*ln(2)*_R2^5-9765625*_R2^6-11137500*_R2^3*ln(2)^2-23906250*ln(2

)*_R2^4-18225000*_R2*ln(2)^3+39487500*ln(2)^2*_R2^2-18225000*ln(2)^3-3906250*_R2^2-11718750*ln(2))/(17006112*l

n(2)^5+9765625)/(3*_R2^10-7*_R2^5+2)/_R2*exp(_R2)*Ei(1,-x+_R2),_R2=RootOf(_Z^12-4*_Z^7+4*_Z^2+2*ln(2)))+3/4/ln

(2)*sum((5668704*ln(2)^6+3906250*ln(2)+1012500*ln(2)^3+33067440*ln(2)^5*_R2^6-3906250*_R2^7+19531250*_R2^6+781

2500*_R2^2-6613488*ln(2)^5*_R2^7+364500*ln(2)^3*_R2^11-262440*ln(2)^4*_R2^9-364500*ln(2)^3*_R2^10+787320*ln(2)

^4*_R2^8+234375*ln(2)*_R2^10-168750*ln(2)^2*_R2^8-468750*ln(2)*_R2^9-1336500*ln(2)^3*_R2^6+675000*ln(2)^2*_R2^

7+962280*ln(2)^4*_R2^4+729000*ln(2)^3*_R2^5+12912048*ln(2)^5*_R2^2-2449440*_R2^3*ln(2)^4+3149280*ln(2)^5*_R2-8

59375*ln(2)*_R2^5+618750*_R2^3*ln(2)^2+1328125*ln(2)*_R2^4+1012500*_R2*ln(2)^3-2193750*ln(2)^2*_R2^2)/(1700611

2*ln(2)^5+9765625)/(3*_R2^10-7*_R2^5+2)/_R2*exp(_R2)*Ei(1,-x+_R2),_R2=RootOf(_Z^12-4*_Z^7+4*_Z^2+2*ln(2)))-1/4

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Maxima [A]
time = 0.54, size = 23, normalized size = 1.15 \begin {gather*} \frac {e^{x}}{x^{12} - 4 \, x^{7} + 4 \, x^{2} + 2 \, \log \left (2\right )} \end {gather*}

integrate((2*log(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*log(2)^2+2*(2*x^12-8*x^7+8*x^2)*log(2)+x^24

-8*x^19+24*x^14-32*x^9+16*x^4),x, algorithm="maxima")

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Fricas [A]
time = 0.42, size = 23, normalized size = 1.15 \begin {gather*} \frac {e^{x}}{x^{12} - 4 \, x^{7} + 4 \, x^{2} + 2 \, \log \left (2\right )} \end {gather*}

integrate((2*log(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*log(2)^2+2*(2*x^12-8*x^7+8*x^2)*log(2)+x^24

-8*x^19+24*x^14-32*x^9+16*x^4),x, algorithm="fricas")

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Sympy [A]
time = 0.10, size = 20, normalized size = 1.00 \begin {gather*} \frac {e^{x}}{x^{12} - 4 x^{7} + 4 x^{2} + 2 \log {\left (2 \right )}} \end {gather*}

integrate((2*ln(2)+x**12-12*x**11-4*x**7+28*x**6+4*x**2-8*x)*exp(x)/(4*ln(2)**2+2*(2*x**12-8*x**7+8*x**2)*ln(2

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Giac [A]
time = 0.54, size = 23, normalized size = 1.15 \begin {gather*} \frac {e^{x}}{x^{12} - 4 \, x^{7} + 4 \, x^{2} + 2 \, \log \left (2\right )} \end {gather*}

integrate((2*log(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*log(2)^2+2*(2*x^12-8*x^7+8*x^2)*log(2)+x^24

-8*x^19+24*x^14-32*x^9+16*x^4),x, algorithm="giac")

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Mupad [B]
time = 4.61, size = 21, normalized size = 1.05 \begin {gather*} \frac {{\mathrm {e}}^x}{x^{12}-4\,x^7+4\,x^2+\ln \left (4\right )} \end {gather*}

int((exp(x)*(2*log(2) - 8*x + 4*x^2 + 28*x^6 - 4*x^7 - 12*x^11 + x^12))/(4*log(2)^2 + 2*log(2)*(8*x^2 - 8*x^7

+ 2*x^12) + 16*x^4 - 32*x^9 + 24*x^14 - 8*x^19 + x^24),x)

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3.65.1 \(\int \frac {e^x (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4))}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+(8 x^2-8 x^7+2 x^{12}) \log (4)+\log ^2(4)} \, dx\) [6401]