∫ e2e6x−4e3xx2+2x4 ((24−8x2) log(3+x2−x log(2) x )+(−96x4−32x6+32x5 log(2)+e6x(−144x−48x3+48x2 log(2))+e3x(96x2+144x3+32x4+48x5+(−32x3−48x4) log(2))) log2(3+x2−x log(2) x )) _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3.46.84 \(\int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} ((24-8 x^2) \log (\frac {3+x^2-x \log (2)}{x})+(-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} (-144 x-48 x^3+48 x^2 \log (2))+e^{3 x} (96 x^2+144 x^3+32 x^4+48 x^5+(-32 x^3-48 x^4) \log (2))) \log ^2(\frac {3+x^2-x \log (2)}{x}))}{-3 x-x^3+x^2 \log (2)} \, dx\) [4584]

Optimal. Leaf size=36 \[ 4 e^{2 \left (e^{3 x}-x^2\right )^2} \log ^2\left (\frac {3+x^2}{x}-\log (2)\right ) \]

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(36)=72\).
time = 1.09, antiderivative size = 174, normalized size of antiderivative = 4.83, number of steps used = 2, number of rules used = 2, integrand size = 166, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {1608, 2326} \begin {gather*} \frac {4 e^{2 x^4-4 e^{3 x} x^2+2 e^{6 x}} \left (2 x^6-2 x^5 \log (2)+6 x^4+3 e^{6 x} \left (x^3-x^2 \log (2)+3 x\right )-e^{3 x} \left (3 x^5+2 x^4+9 x^3+6 x^2-\left (3 x^4+2 x^3\right ) \log (2)\right )\right ) \log ^2\left (\frac {x^2-x \log (2)+3}{x}\right )}{x \left (2 x^3-3 e^{3 x} x^2-2 e^{3 x} x+3 e^{6 x}\right ) \left (x^2-x \log (2)+3\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(2*E^(6*x) - 4*E^(3*x)*x^2 + 2*x^4)*((24 - 8*x^2)*Log[(3 + x^2 - x*Log[2])/x] + (-96*x^4 - 32*x^6 + 32*

x^5*Log[2] + E^(6*x)*(-144*x - 48*x^3 + 48*x^2*Log[2]) + E^(3*x)*(96*x^2 + 144*x^3 + 32*x^4 + 48*x^5 + (-32*x^

3 - 48*x^4)*Log[2]))*Log[(3 + x^2 - x*Log[2])/x]^2))/(-3*x - x^3 + x^2*Log[2]),x]

[Out]

(4*E^(2*E^(6*x) - 4*E^(3*x)*x^2 + 2*x^4)*(6*x^4 + 2*x^6 - 2*x^5*Log[2] + 3*E^(6*x)*(3*x + x^3 - x^2*Log[2]) -

E^(3*x)*(6*x^2 + 9*x^3 + 2*x^4 + 3*x^5 - (2*x^3 + 3*x^4)*Log[2]))*Log[(3 + x^2 - x*Log[2])/x]^2)/(x*(3*E^(6*x)

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

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^

(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2326

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{x \left (-3-x^2+x \log (2)\right )} \, dx\\ &=\frac {4 e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (6 x^4+2 x^6-2 x^5 \log (2)+3 e^{6 x} \left (3 x+x^3-x^2 \log (2)\right )-e^{3 x} \left (6 x^2+9 x^3+2 x^4+3 x^5-\left (2 x^3+3 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )}{x \left (3 e^{6 x}-2 e^{3 x} x-3 e^{3 x} x^2+2 x^3\right ) \left (3+x^2-x \log (2)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.12, size = 33, normalized size = 0.92 \begin {gather*} 4 e^{2 \left (e^{3 x}-x^2\right )^2} \log ^2\left (\frac {3}{x}+x-\log (2)\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^(2*E^(6*x) - 4*E^(3*x)*x^2 + 2*x^4)*((24 - 8*x^2)*Log[(3 + x^2 - x*Log[2])/x] + (-96*x^4 - 32*x^6

+ 32*x^5*Log[2] + E^(6*x)*(-144*x - 48*x^3 + 48*x^2*Log[2]) + E^(3*x)*(96*x^2 + 144*x^3 + 32*x^4 + 48*x^5 + (

-32*x^3 - 48*x^4)*Log[2]))*Log[(3 + x^2 - x*Log[2])/x]^2))/(-3*x - x^3 + x^2*Log[2]),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.87, size = 1142, normalized size = 31.72


method	result	size

risch	\(\text {Expression too large to display}\)	\(1142\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((48*x^2*ln(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*ln(2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp(3*x)+32

*x^5*ln(2)-32*x^6-96*x^4)*ln((-x*ln(2)+x^2+3)/x)^2+(-8*x^2+24)*ln((-x*ln(2)+x^2+3)/x))*exp(exp(3*x)^2-2*x^2*ex

p(3*x)+x^4)^2/(x^2*ln(2)-x^3-3*x),x,method=_RETURNVERBOSE)

[Out]

(-4*Pi^2+4*ln(x)^2+4*ln(x*ln(2)-x^2-3)^2-8*I*Pi*ln(x)-4*I*ln(x)*Pi*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x

^2-3))^2-4*I*ln(x)*Pi*csgn(I/x*(x*ln(2)-x^2-3))^2*csgn(I/x)+4*I*ln(x*ln(2)-x^2-3)*Pi*csgn(I*(x*ln(2)-x^2-3))*c

sgn(I/x*(x*ln(2)-x^2-3))^2+4*I*ln(x*ln(2)-x^2-3)*Pi*csgn(I/x*(x*ln(2)-x^2-3))^3+8*Pi^2*csgn(I/x*(x*ln(2)-x^2-3

))^2+4*I*ln(x*ln(2)-x^2-3)*Pi*csgn(I/x*(x*ln(2)-x^2-3))^2*csgn(I/x)-8*ln(x)*ln(x*ln(2)-x^2-3)-4*Pi^2*csgn(I/x*

(x*ln(2)-x^2-3))^3-Pi^2*csgn(I/x*(x*ln(2)-x^2-3))^6+4*Pi^2*csgn(I/x*(x*ln(2)-x^2-3))^5-4*Pi^2*csgn(I/x*(x*ln(2

)-x^2-3))^4+8*I*ln(x)*Pi*csgn(I/x*(x*ln(2)-x^2-3))^2-8*I*ln(x*ln(2)-x^2-3)*Pi*csgn(I/x*(x*ln(2)-x^2-3))^2-4*Pi

^2*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x^2-3))^3*csgn(I/x)+2*Pi^2*csgn(I*(x*ln(2)-x^2-3))^2*csgn(I/x*(x*

ln(2)-x^2-3))^3*csgn(I/x)+2*Pi^2*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x^2-3))^3*csgn(I/x)^2-Pi^2*csgn(I*(

x*ln(2)-x^2-3))^2*csgn(I/x*(x*ln(2)-x^2-3))^2*csgn(I/x)^2+4*Pi^2*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x^2

-3))*csgn(I/x)+8*I*Pi*ln(x*ln(2)-x^2-3)+4*Pi^2*csgn(I/x*(x*ln(2)-x^2-3))^4*csgn(I/x)+4*Pi^2*csgn(I*(x*ln(2)-x^

2-3))*csgn(I/x*(x*ln(2)-x^2-3))^4-2*Pi^2*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x^2-3))^5-2*Pi^2*csgn(I/x*(

x*ln(2)-x^2-3))^5*csgn(I/x)-Pi^2*csgn(I*(x*ln(2)-x^2-3))^2*csgn(I/x*(x*ln(2)-x^2-3))^4-Pi^2*csgn(I/x*(x*ln(2)-

x^2-3))^4*csgn(I/x)^2-4*Pi^2*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x^2-3))^2-4*Pi^2*csgn(I/x*(x*ln(2)-x^2-

3))^2*csgn(I/x)-4*I*ln(x)*Pi*csgn(I/x*(x*ln(2)-x^2-3))^3+4*I*ln(x)*Pi*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2

)-x^2-3))*csgn(I/x)-4*I*ln(x*ln(2)-x^2-3)*Pi*csgn(I*(x*ln(2)-x^2-3))*csgn(I/x*(x*ln(2)-x^2-3))*csgn(I/x))*exp(

2*exp(6*x)-4*x^2*exp(3*x)+2*x^4)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (34) = 68\).
time = 0.69, size = 103, normalized size = 2.86 \begin {gather*} 4 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (x^{2} - x \log \left (2\right ) + 3\right )^{2} - 8 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (x^{2} - x \log \left (2\right ) + 3\right ) \log \left (x\right ) + 4 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (x\right )^{2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((48*x^2*log(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*log(2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp

(3*x)+32*x^5*log(2)-32*x^6-96*x^4)*log((-x*log(2)+x^2+3)/x)^2+(-8*x^2+24)*log((-x*log(2)+x^2+3)/x))*exp(exp(3*

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

[Out]

4*e^(2*x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))*log(x^2 - x*log(2) + 3)^2 - 8*e^(2*x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))*lo

g(x^2 - x*log(2) + 3)*log(x) + 4*e^(2*x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))*log(x)^2

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Fricas [A]
time = 0.35, size = 41, normalized size = 1.14 \begin {gather*} 4 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (\frac {x^{2} - x \log \left (2\right ) + 3}{x}\right )^{2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((48*x^2*log(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*log(2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp

(3*x)+32*x^5*log(2)-32*x^6-96*x^4)*log((-x*log(2)+x^2+3)/x)^2+(-8*x^2+24)*log((-x*log(2)+x^2+3)/x))*exp(exp(3*

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((48*x**2*ln(2)-48*x**3-144*x)*exp(3*x)**2+((-48*x**4-32*x**3)*ln(2)+48*x**5+32*x**4+144*x**3+96*x*

*2)*exp(3*x)+32*x**5*ln(2)-32*x**6-96*x**4)*ln((-x*ln(2)+x**2+3)/x)**2+(-8*x**2+24)*ln((-x*ln(2)+x**2+3)/x))*e

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((48*x^2*log(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*log(2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp

(3*x)+32*x^5*log(2)-32*x^6-96*x^4)*log((-x*log(2)+x^2+3)/x)^2+(-8*x^2+24)*log((-x*log(2)+x^2+3)/x))*exp(exp(3*

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

[Out]

integrate(8*(2*(2*x^6 - 2*x^5*log(2) + 6*x^4 + 3*(x^3 - x^2*log(2) + 3*x)*e^(6*x) - (3*x^5 + 2*x^4 + 9*x^3 + 6

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{6\,x}-4\,x^2\,{\mathrm {e}}^{3\,x}+2\,x^4}\,\left (\left ({\mathrm {e}}^{6\,x}\,\left (48\,x^3-48\,\ln \left (2\right )\,x^2+144\,x\right )-32\,x^5\,\ln \left (2\right )-{\mathrm {e}}^{3\,x}\,\left (96\,x^2-\ln \left (2\right )\,\left (48\,x^4+32\,x^3\right )+144\,x^3+32\,x^4+48\,x^5\right )+96\,x^4+32\,x^6\right )\,{\ln \left (\frac {x^2-\ln \left (2\right )\,x+3}{x}\right )}^2+\left (8\,x^2-24\right )\,\ln \left (\frac {x^2-\ln \left (2\right )\,x+3}{x}\right )\right )}{x^3-\ln \left (2\right )\,x^2+3\,x} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(6*x) - 4*x^2*exp(3*x) + 2*x^4)*(log((x^2 - x*log(2) + 3)/x)*(8*x^2 - 24) + log((x^2 - x*log(2)

+ 3)/x)^2*(exp(6*x)*(144*x - 48*x^2*log(2) + 48*x^3) - 32*x^5*log(2) - exp(3*x)*(96*x^2 - log(2)*(32*x^3 + 48*

x^4) + 144*x^3 + 32*x^4 + 48*x^5) + 96*x^4 + 32*x^6)))/(3*x - x^2*log(2) + x^3),x)

[Out]

int((exp(2*exp(6*x) - 4*x^2*exp(3*x) + 2*x^4)*(log((x^2 - x*log(2) + 3)/x)*(8*x^2 - 24) + log((x^2 - x*log(2)

+ 3)/x)^2*(exp(6*x)*(144*x - 48*x^2*log(2) + 48*x^3) - 32*x^5*log(2) - exp(3*x)*(96*x^2 - log(2)*(32*x^3 + 48*

x^4) + 144*x^3 + 32*x^4 + 48*x^5) + 96*x^4 + 32*x^6)))/(3*x - x^2*log(2) + x^3), x)

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