∫ 32x3−16x5+2x7+(16x2−4x4) log(4)+2x log2(4)+ex(4x2+2x3−x4+(1+x) log(4))_________________________________________________________ 16x2−8x4+x6+(8x−2x3) log(4)+log2(4) dx [4663]

Optimal. Leaf size=26 \[ -80+x^2+\frac {e^x x}{-x-x \left (-5+x^2\right )+\log (4)} \]

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Rubi [A]
time = 0.40, antiderivative size = 36, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 2, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6820, 2326} \begin {gather*} x^2+\frac {e^x \left (-x^4+4 x^2+x \log (4)\right )}{\left (-x^3+4 x+\log (4)\right )^2} \end {gather*}

Int[(32*x^3 - 16*x^5 + 2*x^7 + (16*x^2 - 4*x^4)*Log[4] + 2*x*Log[4]^2 + E^x*(4*x^2 + 2*x^3 - x^4 + (1 + x)*Log

[4]))/(16*x^2 - 8*x^4 + x^6 + (8*x - 2*x^3)*Log[4] + Log[4]^2),x]

x^2 + (E^x*(4*x^2 - x^4 + x*Log[4]))/(4*x - x^3 + Log[4])^2

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,

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x+\frac {e^x \left (4 x^2+2 x^3-x^4+\log (4)+x \log (4)\right )}{\left (4 x-x^3+\log (4)\right )^2}\right ) \, dx\\ &=x^2+\int \frac {e^x \left (4 x^2+2 x^3-x^4+\log (4)+x \log (4)\right )}{\left (4 x-x^3+\log (4)\right )^2} \, dx\\ &=x^2+\frac {e^x \left (4 x^2-x^4+x \log (4)\right )}{\left (4 x-x^3+\log (4)\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, size = 21, normalized size = 0.81 \begin {gather*} x \left (x+\frac {e^x}{4 x-x^3+\log (4)}\right ) \end {gather*}

Integrate[(32*x^3 - 16*x^5 + 2*x^7 + (16*x^2 - 4*x^4)*Log[4] + 2*x*Log[4]^2 + E^x*(4*x^2 + 2*x^3 - x^4 + (1 +

x)*Log[4]))/(16*x^2 - 8*x^4 + x^6 + (8*x - 2*x^3)*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 = 0.44, size = 1409, normalized size = 54.19


method	result	size

norman	\(\frac {16 x +{\mathrm e}^{x} x -x^{5}+2 x^{2} \ln \left (2\right )+8 \ln \left (2\right )}{-x^{3}+2 \ln \left (2\right )+4 x}\)	\(41\)
default	\(\text {Expression too large to display}\)	\(1409\)

int(((2*ln(2)*(x+1)-x^4+2*x^3+4*x^2)*exp(x)+8*x*ln(2)^2+2*(-4*x^4+16*x^2)*ln(2)+2*x^7-16*x^5+32*x^3)/(4*ln(2)^

2+2*(-2*x^3+8*x)*ln(2)+x^6-8*x^4+16*x^2),x,method=_RETURNVERBOSE)

-24*ln(2)^2*exp(x)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)+32*ln(2)*exp(x)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)-exp(x

)*(9*x^2*ln(2)^2+12*ln(2)^2+8*x*ln(2)-32*x^2)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)+2*exp(x)*(9*x*ln(2)^2+6*x^2*l

n(2)-16*ln(2)-32*x)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)+4*exp(x)*(9*ln(2)^2+6*x*ln(2)-8*x^2)/(27*ln(2)^2-64)/(-

x^3+2*ln(2)+4*x)-18*ln(2)^3/(-1/2*x^3+ln(2)+2*x)/(27*ln(2)^2-64)*x^2+32*ln(2)^2/(-1/2*x^3+ln(2)+2*x)/(27*ln(2)

^2-64)*x+x^2+8/(54*ln(2)^2-128)*sum((45*_R*ln(2)^3+108*ln(2)^2*_R^2+8*ln(2)^2-112*_R*ln(2)-256*_R^2)/(3*_R^2-4

)*ln(x-_R),_R=RootOf(_Z^3-2*ln(2)-4*_Z))+32*ln(2)/(54*ln(2)^2-128)*sum((-9*ln(2)^2*_R+8*ln(2)+16*_R)/(3*_R^2-4

)*ln(x-_R),_R=RootOf(_Z^3-2*ln(2)-4*_Z))+9*ln(2)^2*exp(x)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)*x^2-16*exp(x)/(27

*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)*x*ln(2)-192*ln(2)/(54*ln(2)^2-128)*sum((-3*ln(2)+2*_R)/(3*_R^2-4)*ln(x-_R),_R=

RootOf(_Z^3-2*ln(2)-4*_Z))-8*(-1/4*ln(2)*(9*ln(2)^2-16)/(27*ln(2)^2-64)*x^2-2*(13*ln(2)^2-32)/(27*ln(2)^2-64)*

x-4*ln(2)*(3*ln(2)^2-8)/(27*ln(2)^2-64))/(-1/2*x^3+ln(2)+2*x)+9*exp(x)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)*x*ln

(2)^2-ln(2)*sum((9*ln(2)*_R1-12*_R1^2-18*ln(2)+12*_R1+32)/(27*ln(2)^2-64)/(3*_R1^2-4)*exp(_R1)*Ei(1,-x+_R1),_R

1=RootOf(_Z^3-2*ln(2)-4*_Z))+sum((9*ln(2)^2*_R1^2+18*ln(2)^2*_R1+12*ln(2)^2+8*ln(2)*_R1-32*_R1^2-16*ln(2)-32*_

R1)/(27*ln(2)^2-64)/(3*_R1^2-4)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-2*ln(2)-4*_Z))+48*ln(2)^3/(-1/2*x^3+ln(2

)+2*x)/(27*ln(2)^2-64)-8*ln(2)^2/(54*ln(2)^2-128)*sum((9*_R*ln(2)-32)/(3*_R^2-4)*ln(x-_R),_R=RootOf(_Z^3-2*ln(

2)-4*_Z))-16*(4/(27*ln(2)^2-64)*ln(2)*x^2+8*(3*ln(2)^2-8)/(27*ln(2)^2-64)*x+ln(2)*(9*ln(2)^2-32)/(27*ln(2)^2-6

4))/(-1/2*x^3+ln(2)+2*x)+32/(54*ln(2)^2-128)*sum((-27*ln(2)^2*_R^2-12*ln(2)^2+8*_R*ln(2)+64*_R^2)/(3*_R^2-4)*l

n(x-_R),_R=RootOf(_Z^3-2*ln(2)-4*_Z))-ln(2)*sum((9*ln(2)*_R1^2-9*ln(2)*_R1-24*ln(2)-16*_R1+32)/(27*ln(2)^2-64)

/(3*_R1^2-4)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-2*ln(2)-4*_Z))-256*ln(2)/(54*ln(2)^2-128)*sum((3*ln(2)-2*_R

)/(3*_R^2-4)*ln(x-_R),_R=RootOf(_Z^3-2*ln(2)-4*_Z))-4*sum((9*ln(2)^2+6*ln(2)*_R1-8*_R1^2-12*ln(2)+8*_R1)/(27*l

n(2)^2-64)/(3*_R1^2-4)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-2*ln(2)-4*_Z))+32*(3/(27*ln(2)^2-64)*ln(2)*x^2+1/

2*(9*ln(2)^2-32)/(27*ln(2)^2-64)*x-8/(27*ln(2)^2-64)*ln(2))/(-1/2*x^3+ln(2)+2*x)-2*sum((9*ln(2)^2*_R1+6*ln(2)*

_R1^2+9*ln(2)^2-6*ln(2)*_R1-16*ln(2)-32*_R1)/(27*ln(2)^2-64)/(3*_R1^2-4)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3

-2*ln(2)-4*_Z))-12*ln(2)*exp(x)/(27*ln(2)^2-64)/(-x^3+2*ln(2)+4*x)*x^2

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Maxima [A]
time = 0.51, size = 35, normalized size = 1.35 \begin {gather*} \frac {x^{5} - 4 \, x^{3} - 2 \, x^{2} \log \left (2\right ) - x e^{x}}{x^{3} - 4 \, x - 2 \, \log \left (2\right )} \end {gather*}

integrate(((2*log(2)*(1+x)-x^4+2*x^3+4*x^2)*exp(x)+8*x*log(2)^2+2*(-4*x^4+16*x^2)*log(2)+2*x^7-16*x^5+32*x^3)/

(4*log(2)^2+2*(-2*x^3+8*x)*log(2)+x^6-8*x^4+16*x^2),x, algorithm="maxima")

(x^5 - 4*x^3 - 2*x^2*log(2) - x*e^x)/(x^3 - 4*x - 2*log(2))

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Fricas [A]
time = 0.35, size = 35, normalized size = 1.35 \begin {gather*} \frac {x^{5} - 4 \, x^{3} - 2 \, x^{2} \log \left (2\right ) - x e^{x}}{x^{3} - 4 \, x - 2 \, \log \left (2\right )} \end {gather*}

integrate(((2*log(2)*(1+x)-x^4+2*x^3+4*x^2)*exp(x)+8*x*log(2)^2+2*(-4*x^4+16*x^2)*log(2)+2*x^7-16*x^5+32*x^3)/

(4*log(2)^2+2*(-2*x^3+8*x)*log(2)+x^6-8*x^4+16*x^2),x, algorithm="fricas")

(x^5 - 4*x^3 - 2*x^2*log(2) - x*e^x)/(x^3 - 4*x - 2*log(2))

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Sympy [A]
time = 0.08, size = 19, normalized size = 0.73 \begin {gather*} x^{2} - \frac {x e^{x}}{x^{3} - 4 x - 2 \log {\left (2 \right )}} \end {gather*}

integrate(((2*ln(2)*(1+x)-x**4+2*x**3+4*x**2)*exp(x)+8*x*ln(2)**2+2*(-4*x**4+16*x**2)*ln(2)+2*x**7-16*x**5+32*

x**3)/(4*ln(2)**2+2*(-2*x**3+8*x)*ln(2)+x**6-8*x**4+16*x**2),x)

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Giac [A]
time = 0.42, size = 35, normalized size = 1.35 \begin {gather*} \frac {x^{5} - 4 \, x^{3} - 2 \, x^{2} \log \left (2\right ) - x e^{x}}{x^{3} - 4 \, x - 2 \, \log \left (2\right )} \end {gather*}

integrate(((2*log(2)*(1+x)-x^4+2*x^3+4*x^2)*exp(x)+8*x*log(2)^2+2*(-4*x^4+16*x^2)*log(2)+2*x^7-16*x^5+32*x^3)/

(4*log(2)^2+2*(-2*x^3+8*x)*log(2)+x^6-8*x^4+16*x^2),x, algorithm="giac")

(x^5 - 4*x^3 - 2*x^2*log(2) - x*e^x)/(x^3 - 4*x - 2*log(2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,\ln \left (2\right )\,\left (16\,x^2-4\,x^4\right )+8\,x\,{\ln \left (2\right )}^2+32\,x^3-16\,x^5+2\,x^7+{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (x+1\right )+4\,x^2+2\,x^3-x^4\right )}{2\,\ln \left (2\right )\,\left (8\,x-2\,x^3\right )+4\,{\ln \left (2\right )}^2+16\,x^2-8\,x^4+x^6} \,d x \end {gather*}

int((2*log(2)*(16*x^2 - 4*x^4) + 8*x*log(2)^2 + 32*x^3 - 16*x^5 + 2*x^7 + exp(x)*(2*log(2)*(x + 1) + 4*x^2 + 2

*x^3 - x^4))/(2*log(2)*(8*x - 2*x^3) + 4*log(2)^2 + 16*x^2 - 8*x^4 + x^6),x)

int((2*log(2)*(16*x^2 - 4*x^4) + 8*x*log(2)^2 + 32*x^3 - 16*x^5 + 2*x^7 + exp(x)*(2*log(2)*(x + 1) + 4*x^2 + 2

*x^3 - x^4))/(2*log(2)*(8*x - 2*x^3) + 4*log(2)^2 + 16*x^2 - 8*x^4 + x^6), x)

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3.47.63 \(\int \frac {32 x^3-16 x^5+2 x^7+(16 x^2-4 x^4) \log (4)+2 x \log ^2(4)+e^x (4 x^2+2 x^3-x^4+(1+x) \log (4))}{16 x^2-8 x^4+x^6+(8 x-2 x^3) \log (4)+\log ^2(4)} \, dx\) [4663]