3.88.48 \(\int \frac {x+e^{6-2 x} (-773768 x+2488 x^2-2 x^3)+2 \log (x)+e^{6-2 x} (2488 x-4 x^2) \log ^2(x)-2 e^{6-2 x} x \log ^4(x)}{386884 x-1244 x^2+x^3+(-1244 x+2 x^2) \log ^2(x)+x \log ^4(x)} \, dx\)

Optimal. Leaf size=29 \[ e^{6-2 x}+\frac {x}{-x+x \left (623-x-\log ^2(x)\right )} \]

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Rubi [A]  time = 1.26, antiderivative size = 21, normalized size of antiderivative = 0.72, number of steps used = 5, number of rules used = 4, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6688, 6742, 2194, 6686} \begin {gather*} e^{6-2 x}+\frac {1}{-x-\log ^2(x)+622} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + E^(6 - 2*x)*(-773768*x + 2488*x^2 - 2*x^3) + 2*Log[x] + E^(6 - 2*x)*(2488*x - 4*x^2)*Log[x]^2 - 2*E^(
6 - 2*x)*x*Log[x]^4)/(386884*x - 1244*x^2 + x^3 + (-1244*x + 2*x^2)*Log[x]^2 + x*Log[x]^4),x]

[Out]

E^(6 - 2*x) + (622 - x - Log[x]^2)^(-1)

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 \frac {e^{-2 x} \left (\left (e^{2 x}-2 e^6 (-622+x)^2\right ) x+2 e^{2 x} \log (x)-4 e^6 (-622+x) x \log ^2(x)-2 e^6 x \log ^4(x)\right )}{x \left (622-x-\log ^2(x)\right )^2} \, dx\\ &=\int \left (-2 e^{6-2 x}+\frac {x+2 \log (x)}{x \left (-622+x+\log ^2(x)\right )^2}\right ) \, dx\\ &=-\left (2 \int e^{6-2 x} \, dx\right )+\int \frac {x+2 \log (x)}{x \left (-622+x+\log ^2(x)\right )^2} \, dx\\ &=e^{6-2 x}+\frac {1}{622-x-\log ^2(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 19, normalized size = 0.66 \begin {gather*} e^{6-2 x}-\frac {1}{-622+x+\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + E^(6 - 2*x)*(-773768*x + 2488*x^2 - 2*x^3) + 2*Log[x] + E^(6 - 2*x)*(2488*x - 4*x^2)*Log[x]^2 -
2*E^(6 - 2*x)*x*Log[x]^4)/(386884*x - 1244*x^2 + x^3 + (-1244*x + 2*x^2)*Log[x]^2 + x*Log[x]^4),x]

[Out]

E^(6 - 2*x) - (-622 + x + Log[x]^2)^(-1)

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fricas [A]  time = 0.43, size = 33, normalized size = 1.14 \begin {gather*} \frac {e^{\left (-2 \, x + 6\right )} \log \relax (x)^{2} + {\left (x - 622\right )} e^{\left (-2 \, x + 6\right )} - 1}{\log \relax (x)^{2} + x - 622} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(6-2*x)*log(x)^4+(-4*x^2+2488*x)*exp(6-2*x)*log(x)^2+2*log(x)+(-2*x^3+2488*x^2-773768*x)*ex
p(6-2*x)+x)/(x*log(x)^4+(2*x^2-1244*x)*log(x)^2+x^3-1244*x^2+386884*x),x, algorithm="fricas")

[Out]

(e^(-2*x + 6)*log(x)^2 + (x - 622)*e^(-2*x + 6) - 1)/(log(x)^2 + x - 622)

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giac [A]  time = 0.54, size = 39, normalized size = 1.34 \begin {gather*} \frac {e^{\left (-2 \, x + 6\right )} \log \relax (x)^{2} + x e^{\left (-2 \, x + 6\right )} - 622 \, e^{\left (-2 \, x + 6\right )} - 1}{\log \relax (x)^{2} + x - 622} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(6-2*x)*log(x)^4+(-4*x^2+2488*x)*exp(6-2*x)*log(x)^2+2*log(x)+(-2*x^3+2488*x^2-773768*x)*ex
p(6-2*x)+x)/(x*log(x)^4+(2*x^2-1244*x)*log(x)^2+x^3-1244*x^2+386884*x),x, algorithm="giac")

[Out]

(e^(-2*x + 6)*log(x)^2 + x*e^(-2*x + 6) - 622*e^(-2*x + 6) - 1)/(log(x)^2 + x - 622)

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maple [A]  time = 0.06, size = 19, normalized size = 0.66

method	result	size
risch	\({\mathrm e}^{6-2 x}-\frac {1}{\ln \relax (x )^{2}+x -622}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(6-2*x)*ln(x)^4+(-4*x^2+2488*x)*exp(6-2*x)*ln(x)^2+2*ln(x)+(-2*x^3+2488*x^2-773768*x)*exp(6-2*x)+
x)/(x*ln(x)^4+(2*x^2-1244*x)*ln(x)^2+x^3-1244*x^2+386884*x),x,method=_RETURNVERBOSE)

[Out]

exp(6-2*x)-1/(ln(x)^2+x-622)

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maxima [A]  time = 0.38, size = 36, normalized size = 1.24 \begin {gather*} \frac {{\left (e^{6} \log \relax (x)^{2} + x e^{6} - 622 \, e^{6} - e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{\log \relax (x)^{2} + x - 622} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(6-2*x)*log(x)^4+(-4*x^2+2488*x)*exp(6-2*x)*log(x)^2+2*log(x)+(-2*x^3+2488*x^2-773768*x)*ex
p(6-2*x)+x)/(x*log(x)^4+(2*x^2-1244*x)*log(x)^2+x^3-1244*x^2+386884*x),x, algorithm="maxima")

[Out]

(e^6*log(x)^2 + x*e^6 - 622*e^6 - e^(2*x))*e^(-2*x)/(log(x)^2 + x - 622)

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mupad [B]  time = 5.51, size = 19, normalized size = 0.66 \begin {gather*} {\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^6-\frac {1}{{\ln \relax (x)}^2+x-622} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*log(x) - exp(6 - 2*x)*(773768*x - 2488*x^2 + 2*x^3) + exp(6 - 2*x)*log(x)^2*(2488*x - 4*x^2) - 2*x*
exp(6 - 2*x)*log(x)^4)/(386884*x - log(x)^2*(1244*x - 2*x^2) + x*log(x)^4 - 1244*x^2 + x^3),x)

[Out]

exp(-2*x)*exp(6) - 1/(x + log(x)^2 - 622)

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sympy [A]  time = 0.35, size = 15, normalized size = 0.52 \begin {gather*} e^{6 - 2 x} - \frac {1}{x + \log {\relax (x )}^{2} - 622} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(6-2*x)*ln(x)**4+(-4*x**2+2488*x)*exp(6-2*x)*ln(x)**2+2*ln(x)+(-2*x**3+2488*x**2-773768*x)*
exp(6-2*x)+x)/(x*ln(x)**4+(2*x**2-1244*x)*ln(x)**2+x**3-1244*x**2+386884*x),x)

[Out]

exp(6 - 2*x) - 1/(x + log(x)**2 - 622)

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