3.62.56 \(\int \frac {e^{\frac {1}{4} (4-x^2)} ((-4+4 e^5-8 x) \log (20+x-e^5 x+x^2)+(20 x+x^2-e^5 x^2+x^3) \log ^2(20+x-e^5 x+x^2))}{-40-2 x+2 e^5 x-2 x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{1-\frac {x^2}{4}} \log ^2\left (20+x-e^5 x+x^2\right ) \]

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Rubi [B]  time = 0.12, antiderivative size = 66, normalized size of antiderivative = 2.44, number of steps used = 2, number of rules used = 2, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 2288} \begin {gather*} \frac {e^{\frac {1}{4} \left (4-x^2\right )} \left (x^3-e^5 x^2+x^2+20 x\right ) \log ^2\left (x^2-e^5 x+x+20\right )}{x \left (x^2+\left (1-e^5\right ) x+20\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((4 - x^2)/4)*((-4 + 4*E^5 - 8*x)*Log[20 + x - E^5*x + x^2] + (20*x + x^2 - E^5*x^2 + x^3)*Log[20 + x -
E^5*x + x^2]^2))/(-40 - 2*x + 2*E^5*x - 2*x^2),x]

[Out]

(E^((4 - x^2)/4)*(20*x + x^2 - E^5*x^2 + x^3)*Log[20 + x - E^5*x + x^2]^2)/(x*(20 + (1 - E^5)*x + x^2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{4} \left (4-x^2\right )} \left (\left (-4+4 e^5-8 x\right ) \log \left (20+x-e^5 x+x^2\right )+\left (20 x+x^2-e^5 x^2+x^3\right ) \log ^2\left (20+x-e^5 x+x^2\right )\right )}{-40+\left (-2+2 e^5\right ) x-2 x^2} \, dx\\ &=\frac {e^{\frac {1}{4} \left (4-x^2\right )} \left (20 x+x^2-e^5 x^2+x^3\right ) \log ^2\left (20+x-e^5 x+x^2\right )}{x \left (20+\left (1-e^5\right ) x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.00 \begin {gather*} e^{1-\frac {x^2}{4}} \log ^2\left (20+x-e^5 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 - x^2)/4)*((-4 + 4*E^5 - 8*x)*Log[20 + x - E^5*x + x^2] + (20*x + x^2 - E^5*x^2 + x^3)*Log[20
+ x - E^5*x + x^2]^2))/(-40 - 2*x + 2*E^5*x - 2*x^2),x]

[Out]

E^(1 - x^2/4)*Log[20 + x - E^5*x + x^2]^2

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fricas [A]  time = 0.75, size = 23, normalized size = 0.85 \begin {gather*} e^{\left (-\frac {1}{4} \, x^{2} + 1\right )} \log \left (x^{2} - x e^{5} + x + 20\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(5)+x^3+x^2+20*x)*log(-x*exp(5)+x^2+x+20)^2+(4*exp(5)-8*x-4)*log(-x*exp(5)+x^2+x+20))/(2*x
*exp(5)-2*x^2-2*x-40)/exp(1/4*x^2-1),x, algorithm="fricas")

[Out]

e^(-1/4*x^2 + 1)*log(x^2 - x*e^5 + x + 20)^2

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giac [A]  time = 0.39, size = 23, normalized size = 0.85 \begin {gather*} e^{\left (-\frac {1}{4} \, x^{2} + 1\right )} \log \left (x^{2} - x e^{5} + x + 20\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(5)+x^3+x^2+20*x)*log(-x*exp(5)+x^2+x+20)^2+(4*exp(5)-8*x-4)*log(-x*exp(5)+x^2+x+20))/(2*x
*exp(5)-2*x^2-2*x-40)/exp(1/4*x^2-1),x, algorithm="giac")

[Out]

e^(-1/4*x^2 + 1)*log(x^2 - x*e^5 + x + 20)^2

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maple [A]  time = 0.62, size = 25, normalized size = 0.93

method	result	size
risch	\(\ln \left (-x \,{\mathrm e}^{5}+x^{2}+x +20\right )^{2} {\mathrm e}^{-\frac {\left (x -2\right ) \left (2+x \right )}{4}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*exp(5)+x^3+x^2+20*x)*ln(-x*exp(5)+x^2+x+20)^2+(4*exp(5)-8*x-4)*ln(-x*exp(5)+x^2+x+20))/(2*x*exp(5)-
2*x^2-2*x-40)/exp(1/4*x^2-1),x,method=_RETURNVERBOSE)

[Out]

ln(-x*exp(5)+x^2+x+20)^2*exp(-1/4*(x-2)*(2+x))

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maxima [A]  time = 0.44, size = 24, normalized size = 0.89 \begin {gather*} e^{\left (-\frac {1}{4} \, x^{2} + 1\right )} \log \left (x^{2} - x {\left (e^{5} - 1\right )} + 20\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(5)+x^3+x^2+20*x)*log(-x*exp(5)+x^2+x+20)^2+(4*exp(5)-8*x-4)*log(-x*exp(5)+x^2+x+20))/(2*x
*exp(5)-2*x^2-2*x-40)/exp(1/4*x^2-1),x, algorithm="maxima")

[Out]

e^(-1/4*x^2 + 1)*log(x^2 - x*(e^5 - 1) + 20)^2

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mupad [B]  time = 4.28, size = 23, normalized size = 0.85 \begin {gather*} {\ln \left (x-x\,{\mathrm {e}}^5+x^2+20\right )}^2\,\mathrm {e}\,{\mathrm {e}}^{-\frac {x^2}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1 - x^2/4)*(log(x - x*exp(5) + x^2 + 20)^2*(20*x - x^2*exp(5) + x^2 + x^3) - log(x - x*exp(5) + x^2
+ 20)*(8*x - 4*exp(5) + 4)))/(2*x - 2*x*exp(5) + 2*x^2 + 40),x)

[Out]

log(x - x*exp(5) + x^2 + 20)^2*exp(1)*exp(-x^2/4)

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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(((-x**2*exp(5)+x**3+x**2+20*x)*ln(-x*exp(5)+x**2+x+20)**2+(4*exp(5)-8*x-4)*ln(-x*exp(5)+x**2+x+20))/
(2*x*exp(5)-2*x**2-2*x-40)/exp(1/4*x**2-1),x)

[Out]

Timed out

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