3.98.77 \(\int \frac {e^{\frac {x^2+\log (\frac {1}{4} (-6+4 e^{4+x}+x-4 x^2))}{x}} (x-14 x^2+x^3-4 x^4+e^{4+x} (4 x+4 x^2)+(6-4 e^{4+x}-x+4 x^2) \log (\frac {1}{4} (-6+4 e^{4+x}+x-4 x^2)))}{-6 x^2+4 e^{4+x} x^2+x^3-4 x^4} \, dx\)

Optimal. Leaf size=33 \[ e^{x+\frac {\log \left (e^{4+x}+\frac {1}{4} \left (x-4 x \left (\frac {3}{2 x}+x\right )\right )\right )}{x}} \]

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Rubi [A]  time = 6.94, antiderivative size = 30, normalized size of antiderivative = 0.91, number of steps used = 1, number of rules used = 1, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6706} \begin {gather*} 4^{-1/x} e^x \left (-4 x^2+x+4 e^{x+4}-6\right )^{\frac {1}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((x^2 + Log[(-6 + 4*E^(4 + x) + x - 4*x^2)/4])/x)*(x - 14*x^2 + x^3 - 4*x^4 + E^(4 + x)*(4*x + 4*x^2) +
(6 - 4*E^(4 + x) - x + 4*x^2)*Log[(-6 + 4*E^(4 + x) + x - 4*x^2)/4]))/(-6*x^2 + 4*E^(4 + x)*x^2 + x^3 - 4*x^4
),x]

[Out]

(E^x*(-6 + 4*E^(4 + x) + x - 4*x^2)^x^(-1))/4^x^(-1)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4^{-1/x} e^x \left (-6+4 e^{4+x}+x-4 x^2\right )^{\frac {1}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 30, normalized size = 0.91 \begin {gather*} 4^{-1/x} e^x \left (-6+4 e^{4+x}+x-4 x^2\right )^{\frac {1}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x^2 + Log[(-6 + 4*E^(4 + x) + x - 4*x^2)/4])/x)*(x - 14*x^2 + x^3 - 4*x^4 + E^(4 + x)*(4*x + 4*
x^2) + (6 - 4*E^(4 + x) - x + 4*x^2)*Log[(-6 + 4*E^(4 + x) + x - 4*x^2)/4]))/(-6*x^2 + 4*E^(4 + x)*x^2 + x^3 -
4*x^4),x]

[Out]

(E^x*(-6 + 4*E^(4 + x) + x - 4*x^2)^x^(-1))/4^x^(-1)

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fricas [A]  time = 0.76, size = 24, normalized size = 0.73 \begin {gather*} e^{\left (\frac {x^{2} + \log \left (-x^{2} + \frac {1}{4} \, x + e^{\left (x + 4\right )} - \frac {3}{2}\right )}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4+x)+4*x^2-x+6)*log(exp(4+x)-x^2+1/4*x-3/2)+(4*x^2+4*x)*exp(4+x)-4*x^4+x^3-14*x^2+x)*exp((l
og(exp(4+x)-x^2+1/4*x-3/2)+x^2)/x)/(4*x^2*exp(4+x)-4*x^4+x^3-6*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.63, size = 22, normalized size = 0.67 \begin {gather*} e^{\left (x + \frac {\log \left (-x^{2} + \frac {1}{4} \, x + e^{\left (x + 4\right )} - \frac {3}{2}\right )}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4+x)+4*x^2-x+6)*log(exp(4+x)-x^2+1/4*x-3/2)+(4*x^2+4*x)*exp(4+x)-4*x^4+x^3-14*x^2+x)*exp((l
og(exp(4+x)-x^2+1/4*x-3/2)+x^2)/x)/(4*x^2*exp(4+x)-4*x^4+x^3-6*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 22, normalized size = 0.67

method	result	size
risch	\(\left ({\mathrm e}^{4+x}-x^{2}+\frac {x}{4}-\frac {3}{2}\right )^{\frac {1}{x}} {\mathrm e}^{x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(4+x)+4*x^2-x+6)*ln(exp(4+x)-x^2+1/4*x-3/2)+(4*x^2+4*x)*exp(4+x)-4*x^4+x^3-14*x^2+x)*exp((ln(exp(4
+x)-x^2+1/4*x-3/2)+x^2)/x)/(4*x^2*exp(4+x)-4*x^4+x^3-6*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.56, size = 29, normalized size = 0.88 \begin {gather*} e^{\left (x - \frac {2 \, \log \relax (2)}{x} + \frac {\log \left (-4 \, x^{2} + x + 4 \, e^{\left (x + 4\right )} - 6\right )}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4+x)+4*x^2-x+6)*log(exp(4+x)-x^2+1/4*x-3/2)+(4*x^2+4*x)*exp(4+x)-4*x^4+x^3-14*x^2+x)*exp((l
og(exp(4+x)-x^2+1/4*x-3/2)+x^2)/x)/(4*x^2*exp(4+x)-4*x^4+x^3-6*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.96, size = 21, normalized size = 0.64 \begin {gather*} {\mathrm {e}}^x\,{\left (\frac {x}{4}+{\mathrm {e}}^{x+4}-x^2-\frac {3}{2}\right )}^{1/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x/4 + exp(x + 4) - x^2 - 3/2) + x^2)/x)*(x + exp(x + 4)*(4*x + 4*x^2) - log(x/4 + exp(x + 4) - x
^2 - 3/2)*(x + 4*exp(x + 4) - 4*x^2 - 6) - 14*x^2 + x^3 - 4*x^4))/(4*x^2*exp(x + 4) - 6*x^2 + x^3 - 4*x^4),x)

[Out]

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

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sympy [F(-1)]  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(((-4*exp(4+x)+4*x**2-x+6)*ln(exp(4+x)-x**2+1/4*x-3/2)+(4*x**2+4*x)*exp(4+x)-4*x**4+x**3-14*x**2+x)*e
xp((ln(exp(4+x)-x**2+1/4*x-3/2)+x**2)/x)/(4*x**2*exp(4+x)-4*x**4+x**3-6*x**2),x)

[Out]

Timed out

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