3.89.11 $\int \frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}(3+7x+36x^3+36x^4+(2+2x-24x^3-24x^4)\log (4+4x)+(-1-x+4x^3+4x^4){\log }^2(4+4x))}{9+9x+(-6-6x)\log (4+4x)+(1+x){\log }^2(4+4x)}dx$

Optimal. Leaf size=22 $$e^{-x+x^4-\frac{4x}{-3+\log (4+4x)}}$$

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Rubi [F]  time = 22.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(3+7x+36x^3+36x^4+(2+2x-24x^3-24x^4)\log (4+4x)+(-1-x+4x^3+4x^4){\log }^2(4+4x))}{9+9x+(-6-6x)\log (4+4x)+(1+x){\log }^2(4+4x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(3 + 7*x + 36*x^3 + 36*x^4 + (2 + 2*x - 24
*x^3 - 24*x^4)*Log[4 + 4*x] + (-1 - x + 4*x^3 + 4*x^4)*Log[4 + 4*x]^2))/(9 + 9*x + (-6 - 6*x)*Log[4 + 4*x] + (
1 + x)*Log[4 + 4*x]^2),x]

[Out]

-29*Defer[Int][E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))/(-3 + Log[4*(1 + x)])^2, x] - 4*
Defer[Int][E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))/((1 + x)*(-3 + Log[4*(1 + x)])^2), x
] + 108*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(1 + x))/(-3 + Log[4*(1 + x
)])^2, x] - 108*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(1 + x)^2)/(-3 + Lo
g[4*(1 + x)])^2, x] + 36*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(1 + x)^3)
/(-3 + Log[4*(1 + x)])^2, x] + 2*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*Lo
g[4 + 4*x])/(3 - Log[4*(1 + x)])^2, x] - 24*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4
+ 4*x]))*x^3*Log[4 + 4*x])/(3 - Log[4*(1 + x)])^2, x] - Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/
(-3 + Log[4 + 4*x]))*Log[4 + 4*x]^2)/(3 - Log[4*(1 + x)])^2, x] + 4*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Lo
g[4 + 4*x])/(-3 + Log[4 + 4*x]))*x^3*Log[4 + 4*x]^2)/(3 - Log[4*(1 + x)])^2, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(3+7x+36x^3+36x^4+(2+2x-24x^3-24x^4)\log (4+4x)+(-1-x+4x^3+4x^4){\log }^2(4+4x))}{(1+x)(3-\log (4(1+x)){)}^2}dx\\ & =\int (\frac{3\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}+\frac{7\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x}{(1+x)(-3+\log (4(1+x)){)}^2}+\frac{36\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3}{(1+x)(-3+\log (4(1+x)){)}^2}+\frac{36\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^4}{(1+x)(-3+\log (4(1+x)){)}^2}+\frac{2\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1-12x^3)\log (4+4x)}{(3-\log (4(1+x)){)}^2}+\frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(-1+4x^3){\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2})dx\\ & =2\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1-12x^3)\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx+3\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx+7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x}{(1+x)(-3+\log (4(1+x)){)}^2}dx+36\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3}{(1+x)(-3+\log (4(1+x)){)}^2}dx+36\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^4}{(1+x)(-3+\log (4(1+x)){)}^2}dx+\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(-1+4x^3){\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx\\ & =2\int (\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}\log (4+4x)}{(3-\log (4(1+x)){)}^2}-\frac{12e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}{x}^3\log (4+4x)}{(3-\log (4(1+x)){)}^2})dx+3\int \frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(1+x)(-3+\log (4(1+x)){)}^2}dx+7\int (\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(-3+\log (4(1+x)){)}^2}-\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(1+x)(-3+\log (4(1+x)){)}^2})dx+36\int (\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(-3+\log (4(1+x)){)}^2}-\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}x}{(-3+\log (4(1+x)){)}^2}+\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}{x}^2}{(-3+\log (4(1+x)){)}^2}-\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(1+x)(-3+\log (4(1+x)){)}^2})dx+36\int (-\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(-3+\log (4(1+x)){)}^2}+\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}x}{(-3+\log (4(1+x)){)}^2}-\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}{x}^2}{(-3+\log (4(1+x)){)}^2}+\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}{x}^3}{(-3+\log (4(1+x)){)}^2}+\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}}{(1+x)(-3+\log (4(1+x)){)}^2})dx+\int (-\frac{e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}{\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}+\frac{4e^{\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}}{x}^3{\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2})dx\\ & =2\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx+3\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx+4\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3{\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx+7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(-3+\log (4(1+x)){)}^2}dx-7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx-24\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx+36\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3}{(-3+\log (4(1+x)){)}^2}dx-\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right){\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx\\ & =2\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx+3\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx+4\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3{\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx+7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(-3+\log (4(1+x)){)}^2}dx-7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx-24\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx+36\int (-\frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(-3+\log (4(1+x)){)}^2}+\frac{3\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1+x)}{(-3+\log (4(1+x)){)}^2}-\frac{3\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1+x{)}^2}{(-3+\log (4(1+x)){)}^2}+\frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1+x{)}^3}{(-3+\log (4(1+x)){)}^2})dx-\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right){\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx\\ & =2\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx+3\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx+4\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3{\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx+7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(-3+\log (4(1+x)){)}^2}dx-7\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(1+x)(-3+\log (4(1+x)){)}^2}dx-24\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)x^3\log (4+4x)}{(3-\log (4(1+x)){)}^2}dx-36\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)}{(-3+\log (4(1+x)){)}^2}dx+36\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1+x{)}^3}{(-3+\log (4(1+x)){)}^2}dx+108\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1+x)}{(-3+\log (4(1+x)){)}^2}dx-108\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right)(1+x{)}^2}{(-3+\log (4(1+x)){)}^2}dx-\int \frac{\exp \left(\frac{-x-3x^4+(-x+x^4)\log (4+4x)}{-3+\log (4+4x)}\right){\log }^2(4+4x)}{(3-\log (4(1+x)){)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.25, size = 33, normalized size = 1.50 $$\begin{array}{c}{e}^{\frac{x(-1-3x^3+(-1+x^3)\log (4(1+x)))}{-3+\log (4(1+x))}}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(3 + 7*x + 36*x^3 + 36*x^4 + (2 + 2*
x - 24*x^3 - 24*x^4)*Log[4 + 4*x] + (-1 - x + 4*x^3 + 4*x^4)*Log[4 + 4*x]^2))/(9 + 9*x + (-6 - 6*x)*Log[4 + 4*
x] + (1 + x)*Log[4 + 4*x]^2),x]

[Out]

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

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fricas [A]  time = 0.55, size = 35, normalized size = 1.59 $$\begin{array}{c}{e}^{(-\frac{3x^4-(x^4-x)\log (4x+4)+x}{\log (4x+4)-3})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+4*x^3-x-1)*log(4*x+4)^2+(-24*x^4-24*x^3+2*x+2)*log(4*x+4)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*
log(4*x+4)-3*x^4-x)/(log(4*x+4)-3))/((x+1)*log(4*x+4)^2+(-6*x-6)*log(4*x+4)+9*x+9),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 5.31, size = 69, normalized size = 3.14 $$\begin{array}{c}{e}^{(\frac{x^4\log (4x+4)}{\log (4x+4)-3}-\frac{3x^4}{\log (4x+4)-3}-\frac{x\log (4x+4)}{\log (4x+4)-3}-\frac{x}{\log (4x+4)-3})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+4*x^3-x-1)*log(4*x+4)^2+(-24*x^4-24*x^3+2*x+2)*log(4*x+4)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*
log(4*x+4)-3*x^4-x)/(log(4*x+4)-3))/((x+1)*log(4*x+4)^2+(-6*x-6)*log(4*x+4)+9*x+9),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 39, normalized size = 1.77

method	result	size
risch	${\mathrm{e}}^{\frac{x(\ln (4x+4)x^3-3x^3-\ln (4x+4)-1)}{\ln (4x+4)-3}}$	$39$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+4*x^3-x-1)*ln(4*x+4)^2+(-24*x^4-24*x^3+2*x+2)*ln(4*x+4)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*ln(4*x+4
)-3*x^4-x)/(ln(4*x+4)-3))/((x+1)*ln(4*x+4)^2+(-6*x-6)*ln(4*x+4)+9*x+9),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.64, size = 109, normalized size = 4.95 $$\begin{array}{c}{e}^{(\frac{2x^4\log (2)}{2\log (2)+\log (x+1)-3}+\frac{x^4\log (x+1)}{2\log (2)+\log (x+1)-3}-\frac{3x^4}{2\log (2)+\log (x+1)-3}-\frac{2x\log (2)}{2\log (2)+\log (x+1)-3}-\frac{x\log (x+1)}{2\log (2)+\log (x+1)-3}-\frac{x}{2\log (2)+\log (x+1)-3})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+4*x^3-x-1)*log(4*x+4)^2+(-24*x^4-24*x^3+2*x+2)*log(4*x+4)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*
log(4*x+4)-3*x^4-x)/(log(4*x+4)-3))/((x+1)*log(4*x+4)^2+(-6*x-6)*log(4*x+4)+9*x+9),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.99, size = 57, normalized size = 2.59 $$\begin{array}{c}\frac{{\mathrm{e}}^{-\frac{x}{\ln (4x+4)-3}}{\mathrm{e}}^{-\frac{3x^4}{\ln (4x+4)-3}}}{{(4x+4)}^{\frac{x-x^4}{\ln (4x+4)-3}}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x + log(4*x + 4)*(x - x^4) + 3*x^4)/(log(4*x + 4) - 3))*(7*x + log(4*x + 4)*(2*x - 24*x^3 - 24*x^4
+ 2) - log(4*x + 4)^2*(x - 4*x^3 - 4*x^4 + 1) + 36*x^3 + 36*x^4 + 3))/(9*x - log(4*x + 4)*(6*x + 6) + log(4*x
+ 4)^2*(x + 1) + 9),x)

[Out]

(exp(-x/(log(4*x + 4) - 3))*exp(-(3*x^4)/(log(4*x + 4) - 3)))/(4*x + 4)^((x - x^4)/(log(4*x + 4) - 3))

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sympy [A]  time = 0.66, size = 27, normalized size = 1.23 $$\begin{array}{c}{e}^{\frac{-3x^4-x+(x^4-x)\log (4x+4)}{\log (4x+4)-3}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+4*x**3-x-1)*ln(4*x+4)**2+(-24*x**4-24*x**3+2*x+2)*ln(4*x+4)+36*x**4+36*x**3+7*x+3)*exp(((x*
*4-x)*ln(4*x+4)-3*x**4-x)/(ln(4*x+4)-3))/((x+1)*ln(4*x+4)**2+(-6*x-6)*ln(4*x+4)+9*x+9),x)

[Out]

exp((-3*x**4 - x + (x**4 - x)*log(4*x + 4))/(log(4*x + 4) - 3))

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