3.27.25 \(\int \frac {2-2 x+e^{16 x^2+8 x^3+x^4} (-2 x+193 x^2-16 x^3-64 x^4+4 x^5+4 x^6)}{e^{32 x^2+16 x^3+2 x^4} (-18 x+21 x^2-8 x^3+x^4)+e^{16 x^2+8 x^3+x^4} (-12 x+10 x^2-2 x^3) \log (-2 x+x^2)+(-2 x+x^2) \log ^2(-2 x+x^2)} \, dx\)

Optimal. Leaf size=30 \[ -1+\frac {1}{e^{x^2 (4+x)^2} (3-x)+\log \left (-2 x+x^2\right )} \]

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Rubi [A]  time = 1.36, antiderivative size = 39, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, integrand size = 148, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6741, 6686} \begin {gather*} \frac {1}{-e^{x^2 (x+4)^2} x+3 e^{x^2 (x+4)^2}+\log (-((2-x) x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 2*x + E^(16*x^2 + 8*x^3 + x^4)*(-2*x + 193*x^2 - 16*x^3 - 64*x^4 + 4*x^5 + 4*x^6))/(E^(32*x^2 + 16*x^
3 + 2*x^4)*(-18*x + 21*x^2 - 8*x^3 + x^4) + E^(16*x^2 + 8*x^3 + x^4)*(-12*x + 10*x^2 - 2*x^3)*Log[-2*x + x^2]
+ (-2*x + x^2)*Log[-2*x + x^2]^2),x]

[Out]

(3*E^(x^2*(4 + x)^2) - E^(x^2*(4 + x)^2)*x + Log[-((2 - x)*x)])^(-1)

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+2 x-e^{16 x^2+8 x^3+x^4} \left (-2 x+193 x^2-16 x^3-64 x^4+4 x^5+4 x^6\right )}{(2-x) x \left (3 e^{x^2 (4+x)^2}-e^{x^2 (4+x)^2} x+\log ((-2+x) x)\right )^2} \, dx\\ &=\frac {1}{3 e^{x^2 (4+x)^2}-e^{x^2 (4+x)^2} x+\log (-((2-x) x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 28, normalized size = 0.93 \begin {gather*} -\frac {1}{e^{x^2 (4+x)^2} (-3+x)-\log ((-2+x) x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 2*x + E^(16*x^2 + 8*x^3 + x^4)*(-2*x + 193*x^2 - 16*x^3 - 64*x^4 + 4*x^5 + 4*x^6))/(E^(32*x^2 +
16*x^3 + 2*x^4)*(-18*x + 21*x^2 - 8*x^3 + x^4) + E^(16*x^2 + 8*x^3 + x^4)*(-12*x + 10*x^2 - 2*x^3)*Log[-2*x +
x^2] + (-2*x + x^2)*Log[-2*x + x^2]^2),x]

[Out]

-(E^(x^2*(4 + x)^2)*(-3 + x) - Log[(-2 + x)*x])^(-1)

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fricas [A]  time = 0.53, size = 34, normalized size = 1.13 \begin {gather*} -\frac {1}{{\left (x - 3\right )} e^{\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} - \log \left (x^{2} - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^6+4*x^5-64*x^4-16*x^3+193*x^2-2*x)*exp(x^4+8*x^3+16*x^2)-2*x+2)/((x^2-2*x)*log(x^2-2*x)^2+(-2*
x^3+10*x^2-12*x)*exp(x^4+8*x^3+16*x^2)*log(x^2-2*x)+(x^4-8*x^3+21*x^2-18*x)*exp(x^4+8*x^3+16*x^2)^2),x, algori
thm="fricas")

[Out]

-1/((x - 3)*e^(x^4 + 8*x^3 + 16*x^2) - log(x^2 - 2*x))

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giac [A]  time = 4.15, size = 49, normalized size = 1.63 \begin {gather*} -\frac {1}{x e^{\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} - 3 \, e^{\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} - \log \left (x^{2} - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^6+4*x^5-64*x^4-16*x^3+193*x^2-2*x)*exp(x^4+8*x^3+16*x^2)-2*x+2)/((x^2-2*x)*log(x^2-2*x)^2+(-2*
x^3+10*x^2-12*x)*exp(x^4+8*x^3+16*x^2)*log(x^2-2*x)+(x^4-8*x^3+21*x^2-18*x)*exp(x^4+8*x^3+16*x^2)^2),x, algori
thm="giac")

[Out]

-1/(x*e^(x^4 + 8*x^3 + 16*x^2) - 3*e^(x^4 + 8*x^3 + 16*x^2) - log(x^2 - 2*x))

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maple [C]  time = 0.08, size = 118, normalized size = 3.93

method	result	size
risch	\(\frac {2 i}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (i \left (x -2\right ) x \right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -2\right ) x \right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (i \left (x -2\right ) x \right )^{2}+\pi \mathrm {csgn}\left (i \left (x -2\right ) x \right )^{3}-2 i x \,{\mathrm e}^{\left (4+x \right )^{2} x^{2}}+6 i {\mathrm e}^{\left (4+x \right )^{2} x^{2}}+2 i \ln \relax (x )+2 i \ln \left (x -2\right )}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^6+4*x^5-64*x^4-16*x^3+193*x^2-2*x)*exp(x^4+8*x^3+16*x^2)-2*x+2)/((x^2-2*x)*ln(x^2-2*x)^2+(-2*x^3+10*
x^2-12*x)*exp(x^4+8*x^3+16*x^2)*ln(x^2-2*x)+(x^4-8*x^3+21*x^2-18*x)*exp(x^4+8*x^3+16*x^2)^2),x,method=_RETURNV
ERBOSE)

[Out]

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

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maxima [A]  time = 0.73, size = 34, normalized size = 1.13 \begin {gather*} -\frac {1}{{\left (x - 3\right )} e^{\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} - \log \left (x - 2\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^6+4*x^5-64*x^4-16*x^3+193*x^2-2*x)*exp(x^4+8*x^3+16*x^2)-2*x+2)/((x^2-2*x)*log(x^2-2*x)^2+(-2*
x^3+10*x^2-12*x)*exp(x^4+8*x^3+16*x^2)*log(x^2-2*x)+(x^4-8*x^3+21*x^2-18*x)*exp(x^4+8*x^3+16*x^2)^2),x, algori
thm="maxima")

[Out]

-1/((x - 3)*e^(x^4 + 8*x^3 + 16*x^2) - log(x - 2) - log(x))

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mupad [B]  time = 1.85, size = 31, normalized size = 1.03 \begin {gather*} \frac {1}{\ln \left (x^2-2\,x\right )-{\mathrm {e}}^{x^4+8\,x^3+16\,x^2}\,\left (x-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(16*x^2 + 8*x^3 + x^4)*(2*x - 193*x^2 + 16*x^3 + 64*x^4 - 4*x^5 - 4*x^6) - 2)/(exp(32*x^2 + 16*x
^3 + 2*x^4)*(18*x - 21*x^2 + 8*x^3 - x^4) + log(x^2 - 2*x)^2*(2*x - x^2) + exp(16*x^2 + 8*x^3 + x^4)*log(x^2 -
2*x)*(12*x - 10*x^2 + 2*x^3)),x)

[Out]

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

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sympy [A]  time = 0.47, size = 29, normalized size = 0.97 \begin {gather*} - \frac {1}{\left (x - 3\right ) e^{x^{4} + 8 x^{3} + 16 x^{2}} - \log {\left (x^{2} - 2 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**6+4*x**5-64*x**4-16*x**3+193*x**2-2*x)*exp(x**4+8*x**3+16*x**2)-2*x+2)/((x**2-2*x)*ln(x**2-2*
x)**2+(-2*x**3+10*x**2-12*x)*exp(x**4+8*x**3+16*x**2)*ln(x**2-2*x)+(x**4-8*x**3+21*x**2-18*x)*exp(x**4+8*x**3+
16*x**2)**2),x)

[Out]

-1/((x - 3)*exp(x**4 + 8*x**3 + 16*x**2) - log(x**2 - 2*x))

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