3.33.5 $\int \frac{e^{-4+e^x}(16-16x)+e^{-4+e^x}(e^x(16x-16x^2)+4e^{x}x\log (-x^4))\log (4-4x+\log (-x^4))}{4x-4x^2+x\log (-x^4)}dx$

Optimal. Leaf size=21 $$4e^{-4+e^x}\log (4-4x+\log (-x^4))$$

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Rubi [A]  time = 0.22, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 80, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.038, Rules used = {6688, 12, 2288} $$\begin{array}{c}4e^{e^x-4}\log (\log (-x^4)-4x+4)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(-4 + E^x)*(16 - 16*x) + E^(-4 + E^x)*(E^x*(16*x - 16*x^2) + 4*E^x*x*Log[-x^4])*Log[4 - 4*x + Log[-x^4]
])/(4*x - 4*x^2 + x*Log[-x^4]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rule 6688

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int 4e^{-4+e^x}(\frac{4-4x}{4x-4x^2+x\log (-x^4)}+e^x\log (4-4x+\log (-x^4)))dx\\ & =4\int e^{-4+e^x}(\frac{4-4x}{4x-4x^2+x\log (-x^4)}+e^x\log (4-4x+\log (-x^4)))dx\\ & =4e^{-4+e^x}\log (4-4x+\log (-x^4))\end{array}\end{array}$$

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Mathematica [A]  time = 0.06, size = 21, normalized size = 1.00 $$\begin{array}{c}4e^{-4+e^x}\log (4-4x+\log (-x^4))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 + E^x)*(16 - 16*x) + E^(-4 + E^x)*(E^x*(16*x - 16*x^2) + 4*E^x*x*Log[-x^4])*Log[4 - 4*x + Log
[-x^4]])/(4*x - 4*x^2 + x*Log[-x^4]),x]

[Out]

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

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fricas [A]  time = 0.53, size = 19, normalized size = 0.90 $$\begin{array}{c}4e^{(e^x-4)}\log (-4x+\log (-x^4)+4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)*log(-x^4)+(-16*x^2+16*x)*exp(x))*exp(exp(x)-4)*log(log(-x^4)-4*x+4)+(-16*x+16)*exp(exp(
x)-4))/(x*log(-x^4)-4*x^2+4*x),x, algorithm="fricas")

[Out]

4*e^(e^x - 4)*log(-4*x + log(-x^4) + 4)

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giac [A]  time = 0.36, size = 19, normalized size = 0.90 $$\begin{array}{c}4e^{(e^x-4)}\log (-4x+\log (-x^4)+4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)*log(-x^4)+(-16*x^2+16*x)*exp(x))*exp(exp(x)-4)*log(log(-x^4)-4*x+4)+(-16*x+16)*exp(exp(
x)-4))/(x*log(-x^4)-4*x^2+4*x),x, algorithm="giac")

[Out]

4*e^(e^x - 4)*log(-4*x + log(-x^4) + 4)

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maple [C]  time = 0.19, size = 158, normalized size = 7.52

method	result	size
risch	$4{\mathrm{e}}^{{\mathrm{e}}^x-4}\ln (i\pi +4\ln (x)-\frac{i\pi \mathrm{csgn}\left(i{x}^2\right){(-\mathrm{csgn}\left(i{x}^2\right)+\mathrm{csgn}\left(ix\right))}^2}{2}-\frac{i\pi \mathrm{csgn}\left(i{x}^3\right)(-\mathrm{csgn}\left(i{x}^3\right)+\mathrm{csgn}\left(i{x}^2\right))(-\mathrm{csgn}\left(i{x}^3\right)+\mathrm{csgn}\left(ix\right))}{2}-\frac{i\pi \mathrm{csgn}\left(i{x}^4\right)(-\mathrm{csgn}\left(i{x}^4\right)+\mathrm{csgn}\left(i{x}^3\right))(-\mathrm{csgn}\left(i{x}^4\right)+\mathrm{csgn}\left(ix\right))}{2}+i\pi \mathrm{csgn}{\left(i{x}^4\right)}^2(\mathrm{csgn}\left(i{x}^4\right)-1)-4x+4)$	$158$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x*exp(x)*ln(-x^4)+(-16*x^2+16*x)*exp(x))*exp(exp(x)-4)*ln(ln(-x^4)-4*x+4)+(-16*x+16)*exp(exp(x)-4))/(x
*ln(-x^4)-4*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

4*exp(exp(x)-4)*ln(I*Pi+4*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*
x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(
I*x))+I*Pi*csgn(I*x^4)^2*(csgn(I*x^4)-1)-4*x+4)

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maxima [C]  time = 0.47, size = 20, normalized size = 0.95 $$\begin{array}{c}4e^{(e^x-4)}\log (i\pi -4x+4\log (x)+4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)*log(-x^4)+(-16*x^2+16*x)*exp(x))*exp(exp(x)-4)*log(log(-x^4)-4*x+4)+(-16*x+16)*exp(exp(
x)-4))/(x*log(-x^4)-4*x^2+4*x),x, algorithm="maxima")

[Out]

4*e^(e^x - 4)*log(I*pi - 4*x + 4*log(x) + 4)

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mupad [B]  time = 2.16, size = 19, normalized size = 0.90 $$\begin{array}{c}4\ln (\ln (-x^4)-4x+4){\mathrm{e}}^{{\mathrm{e}}^x-4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x) - 4)*(16*x - 16) - log(log(-x^4) - 4*x + 4)*exp(exp(x) - 4)*(exp(x)*(16*x - 16*x^2) + 4*x*exp
(x)*log(-x^4)))/(4*x + x*log(-x^4) - 4*x^2),x)

[Out]

4*log(log(-x^4) - 4*x + 4)*exp(exp(x) - 4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(x)*ln(-x**4)+(-16*x**2+16*x)*exp(x))*exp(exp(x)-4)*ln(ln(-x**4)-4*x+4)+(-16*x+16)*exp(exp(
x)-4))/(x*ln(-x**4)-4*x**2+4*x),x)

[Out]

Timed out

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