3.43.39 \(\int \frac {e^{2 x} (4 x-x^3)+e^{2 x} (-8 x-8 x^2+4 x^3+2 x^4) \log (x)+(1+e^{2 x} (x+6 x^2+4 x^3)) \log ^2(x)+e^{2 x} (2 x+2 x^2) \log ^3(x)}{\log ^2(x)} \, dx\)

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

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Rubi [A]  time = 1.44, antiderivative size = 43, normalized size of antiderivative = 1.65, number of steps used = 3, number of rules used = 2, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6742, 2288} \begin {gather*} x-\frac {e^{2 x} x \left (x^3 (-\log (x))-2 x^2 \log ^2(x)-x \log ^3(x)+4 x \log (x)\right )}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(4*x - x^3) + E^(2*x)*(-8*x - 8*x^2 + 4*x^3 + 2*x^4)*Log[x] + (1 + E^(2*x)*(x + 6*x^2 + 4*x^3))*L
og[x]^2 + E^(2*x)*(2*x + 2*x^2)*Log[x]^3)/Log[x]^2,x]

[Out]

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

Rule 2288

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,
x], w*y]] /; FreeQ[F, x]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {e^{2 x} x \left (4-x^2-8 \log (x)-8 x \log (x)+4 x^2 \log (x)+2 x^3 \log (x)+\log ^2(x)+6 x \log ^2(x)+4 x^2 \log ^2(x)+2 \log ^3(x)+2 x \log ^3(x)\right )}{\log ^2(x)}\right ) \, dx\\ &=x+\int \frac {e^{2 x} x \left (4-x^2-8 \log (x)-8 x \log (x)+4 x^2 \log (x)+2 x^3 \log (x)+\log ^2(x)+6 x \log ^2(x)+4 x^2 \log ^2(x)+2 \log ^3(x)+2 x \log ^3(x)\right )}{\log ^2(x)} \, dx\\ &=x-\frac {e^{2 x} x \left (4 x \log (x)-x^3 \log (x)-2 x^2 \log ^2(x)-x \log ^3(x)\right )}{\log ^2(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 41, normalized size = 1.58 \begin {gather*} x+2 e^{2 x} x^3+\frac {e^{2 x} x^2 \left (-4+x^2\right )}{\log (x)}+e^{2 x} x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(4*x - x^3) + E^(2*x)*(-8*x - 8*x^2 + 4*x^3 + 2*x^4)*Log[x] + (1 + E^(2*x)*(x + 6*x^2 + 4*x
^3))*Log[x]^2 + E^(2*x)*(2*x + 2*x^2)*Log[x]^3)/Log[x]^2,x]

[Out]

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

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fricas [B]  time = 0.48, size = 46, normalized size = 1.77 \begin {gather*} \frac {x^{2} e^{\left (2 \, x\right )} \log \relax (x)^{2} + {\left (x^{4} - 4 \, x^{2}\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{3} e^{\left (2 \, x\right )} + x\right )} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(2*x)*log(x)^3+((4*x^3+6*x^2+x)*exp(2*x)+1)*log(x)^2+(2*x^4+4*x^3-8*x^2-8*x)*exp(2*x
)*log(x)+(-x^3+4*x)*exp(2*x))/log(x)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.20, size = 50, normalized size = 1.92 \begin {gather*} \frac {x^{4} e^{\left (2 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} \log \relax (x) + x^{2} e^{\left (2 \, x\right )} \log \relax (x)^{2} - 4 \, x^{2} e^{\left (2 \, x\right )} + x \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(2*x)*log(x)^3+((4*x^3+6*x^2+x)*exp(2*x)+1)*log(x)^2+(2*x^4+4*x^3-8*x^2-8*x)*exp(2*x
)*log(x)+(-x^3+4*x)*exp(2*x))/log(x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 39, normalized size = 1.50

method	result	size
risch	\(2 \,{\mathrm e}^{2 x} x^{3}+x^{2} {\mathrm e}^{2 x} \ln \relax (x )+x +\frac {x^{2} {\mathrm e}^{2 x} \left (x^{2}-4\right )}{\ln \relax (x )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2*x)*exp(2*x)*ln(x)^3+((4*x^3+6*x^2+x)*exp(2*x)+1)*ln(x)^2+(2*x^4+4*x^3-8*x^2-8*x)*exp(2*x)*ln(x)+
(-x^3+4*x)*exp(2*x))/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

2*exp(2*x)*x^3+x^2*exp(2*x)*ln(x)+x+x^2*exp(2*x)*(x^2-4)/ln(x)

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maxima [B]  time = 0.38, size = 89, normalized size = 3.42 \begin {gather*} \frac {1}{2} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {3}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{4} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + x + \frac {{\left (4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} - 16 \, x^{2} - {\left (2 \, x - 1\right )} \log \relax (x)\right )} e^{\left (2 \, x\right )}}{4 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(2*x)*log(x)^3+((4*x^3+6*x^2+x)*exp(2*x)+1)*log(x)^2+(2*x^4+4*x^3-8*x^2-8*x)*exp(2*x
)*log(x)+(-x^3+4*x)*exp(2*x))/log(x)^2,x, algorithm="maxima")

[Out]

1/2*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 3/2*(2*x^2 - 2*x + 1)*e^(2*x) + 1/4*(2*x - 1)*e^(2*x) + x + 1/4*(4*x^4
+ 4*x^2*log(x)^2 - 16*x^2 - (2*x - 1)*log(x))*e^(2*x)/log(x)

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mupad [B]  time = 2.90, size = 46, normalized size = 1.77 \begin {gather*} x+2\,x^3\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{2\,x}\,\ln \relax (x)-\frac {4\,x^2\,{\mathrm {e}}^{2\,x}}{\ln \relax (x)}+\frac {x^4\,{\mathrm {e}}^{2\,x}}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(4*x - x^3) + log(x)^2*(exp(2*x)*(x + 6*x^2 + 4*x^3) + 1) + exp(2*x)*log(x)^3*(2*x + 2*x^2) - ex
p(2*x)*log(x)*(8*x + 8*x^2 - 4*x^3 - 2*x^4))/log(x)^2,x)

[Out]

x + 2*x^3*exp(2*x) + x^2*exp(2*x)*log(x) - (4*x^2*exp(2*x))/log(x) + (x^4*exp(2*x))/log(x)

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sympy [A]  time = 0.37, size = 34, normalized size = 1.31 \begin {gather*} x + \frac {\left (x^{4} + 2 x^{3} \log {\relax (x )} + x^{2} \log {\relax (x )}^{2} - 4 x^{2}\right ) e^{2 x}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2*x)*exp(2*x)*ln(x)**3+((4*x**3+6*x**2+x)*exp(2*x)+1)*ln(x)**2+(2*x**4+4*x**3-8*x**2-8*x)*e
xp(2*x)*ln(x)+(-x**3+4*x)*exp(2*x))/ln(x)**2,x)

[Out]

x + (x**4 + 2*x**3*log(x) + x**2*log(x)**2 - 4*x**2)*exp(2*x)/log(x)

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