3.33.75 $\int \frac{e^4(x^3+x^4)+e^{4+x^2}(2x^2+2x^3-2x^4-2x^5)+(e^{4+x^2}(1+x)+e^4(x+x^2))\log (1+x)+\log (3x)(e^{4+x^2}x+e^4{x}^2+(e^4(-x-x^2)+e^{4+x^2}(-2x^2-2x^3))\log (1+x))}{x^3+x^4+e^{2x^2}(x+x^2)+e^{x^2}(2x^2+2x^3)}dx$

Optimal. Leaf size=26 $$\frac{e^4(x^2+\log (3x)\log (1+x))}{e^{x^2}+x}$$

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Rubi [F]  time = 16.86, 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 = {} $$\begin{array}{c}\int \frac{e^4(x^3+x^4)+e^{4+x^2}(2x^2+2x^3-2x^4-2x^5)+(e^{4+x^2}(1+x)+e^4(x+x^2))\log (1+x)+\log (3x)(e^{4+x^2}x+e^4{x}^2+(e^4(-x-x^2)+e^{4+x^2}(-2x^2-2x^3))\log (1+x))}{x^3+x^4+e^{2x^2}(x+x^2)+e^{x^2}(2x^2+2x^3)}dx\end{array}$$

Verification is not applicable to the result.

[In]

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

[Out]

-(E^4*Log[3*x]*Log[1 + x]*Defer[Int][(E^x^2 + x)^(-2), x]) - E^4*Defer[Int][x^2/(E^x^2 + x)^2, x] + 2*E^4*Log[
3*x]*Log[1 + x]*Defer[Int][x^2/(E^x^2 + x)^2, x] + 2*E^4*Defer[Int][x^4/(E^x^2 + x)^2, x] + E^4*Log[1 + x]*Def
er[Int][1/(x*(E^x^2 + x)), x] + 2*E^4*Defer[Int][x/(E^x^2 + x), x] - 2*E^4*Log[3*x]*Log[1 + x]*Defer[Int][x/(E
^x^2 + x), x] - 2*E^4*Defer[Int][x^3/(E^x^2 + x), x] + E^4*Log[3*x]*Defer[Int][1/((1 + x)*(E^x^2 + x)), x] + E
^4*Log[1 + x]*Defer[Int][Defer[Int][(E^x^2 + x)^(-2), x]/x, x] + E^4*Log[3*x]*Defer[Int][Defer[Int][(E^x^2 + x
)^(-2), x]/(1 + x), x] - 2*E^4*Log[1 + x]*Defer[Int][Defer[Int][x^2/(E^x^2 + x)^2, x]/x, x] - 2*E^4*Log[3*x]*D
efer[Int][Defer[Int][x^2/(E^x^2 + x)^2, x]/(1 + x), x] + 2*E^4*Log[1 + x]*Defer[Int][Defer[Int][x/(E^x^2 + x),
x]/x, x] + 2*E^4*Log[3*x]*Defer[Int][Defer[Int][x/(E^x^2 + x), x]/(1 + x), x] - E^4*Defer[Int][Defer[Int][1/(
(1 + x)*(E^x^2 + x)), x]/x, x] - E^4*Defer[Int][Defer[Int][(E^x^2*x + x^2)^(-1), x]/(1 + x), x] - E^4*Defer[In
t][Defer[Int][Defer[Int][(E^x^2 + x)^(-2), x]/x, x]/(1 + x), x] - E^4*Defer[Int][Defer[Int][Defer[Int][(E^x^2
+ x)^(-2), x]/(1 + x), x]/x, x] + 2*E^4*Defer[Int][Defer[Int][Defer[Int][x^2/(E^x^2 + x)^2, x]/x, x]/(1 + x),
x] + 2*E^4*Defer[Int][Defer[Int][Defer[Int][x^2/(E^x^2 + x)^2, x]/(1 + x), x]/x, x] - 2*E^4*Defer[Int][Defer[I
nt][Defer[Int][x/(E^x^2 + x), x]/x, x]/(1 + x), x] - 2*E^4*Defer[Int][Defer[Int][Defer[Int][x/(E^x^2 + x), x]/
(1 + x), x]/x, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^4(-((1+x)(x^2(-x+2e^{x^2}(-1+x^2))-(e^{x^2}+x)\log (1+x)))-x\log (3x)(-e^{x^2}-x+(1+x)(1+2e^{x^2}x)\log (1+x)))}{x(1+x){(e^{x^2}+x)}^2}dx\\ & =e^4\int \frac{-((1+x)(x^2(-x+2e^{x^2}(-1+x^2))-(e^{x^2}+x)\log (1+x)))-x\log (3x)(-e^{x^2}-x+(1+x)(1+2e^{x^2}x)\log (1+x))}{x(1+x){(e^{x^2}+x)}^2}dx\\ & =e^4\int (\frac{(-1+2x^2)(x^2+\log (3x)\log (1+x))}{{(e^{x^2}+x)}^2}-\frac{-2x^2-2x^3+2x^4+2x^5-x\log (3x)-\log (1+x)-x\log (1+x)+2x^2\log (3x)\log (1+x)+2x^3\log (3x)\log (1+x)}{x(1+x)(e^{x^2}+x)})dx\\ & =e^4\int \frac{(-1+2x^2)(x^2+\log (3x)\log (1+x))}{{(e^{x^2}+x)}^2}dx-e^4\int \frac{-2x^2-2x^3+2x^4+2x^5-x\log (3x)-\log (1+x)-x\log (1+x)+2x^2\log (3x)\log (1+x)+2x^3\log (3x)\log (1+x)}{x(1+x)(e^{x^2}+x)}dx\\ & =-(e^4\int \frac{(1+x)(2x^2(-1+x^2)-\log (1+x))+x\log (3x)(-1+2x(1+x)\log (1+x))}{x(1+x)(e^{x^2}+x)}dx)+e^4\int (-\frac{x^2+\log (3x)\log (1+x)}{{(e^{x^2}+x)}^2}+\frac{2x^2(x^2+\log (3x)\log (1+x))}{{(e^{x^2}+x)}^2})dx\\ & =-(e^4\int \frac{x^2+\log (3x)\log (1+x)}{{(e^{x^2}+x)}^2}dx)-e^4\int (\frac{-2x^2-2x^3+2x^4+2x^5-x\log (3x)-\log (1+x)-x\log (1+x)+2x^2\log (3x)\log (1+x)+2x^3\log (3x)\log (1+x)}{x(e^{x^2}+x)}-\frac{-2x^2-2x^3+2x^4+2x^5-x\log (3x)-\log (1+x)-x\log (1+x)+2x^2\log (3x)\log (1+x)+2x^3\log (3x)\log (1+x)}{(1+x)(e^{x^2}+x)})dx+\left(2e^4\right)\int \frac{x^2(x^2+\log (3x)\log (1+x))}{{(e^{x^2}+x)}^2}dx\\ & =-(e^4\int \frac{-2x^2-2x^3+2x^4+2x^5-x\log (3x)-\log (1+x)-x\log (1+x)+2x^2\log (3x)\log (1+x)+2x^3\log (3x)\log (1+x)}{x(e^{x^2}+x)}dx)+e^4\int \frac{-2x^2-2x^3+2x^4+2x^5-x\log (3x)-\log (1+x)-x\log (1+x)+2x^2\log (3x)\log (1+x)+2x^3\log (3x)\log (1+x)}{(1+x)(e^{x^2}+x)}dx-e^4\int (\frac{x^2}{{(e^{x^2}+x)}^2}+\frac{\log (3x)\log (1+x)}{{(e^{x^2}+x)}^2})dx+\left(2e^4\right)\int (\frac{x^4}{{(e^{x^2}+x)}^2}+\frac{x^2\log (3x)\log (1+x)}{{(e^{x^2}+x)}^2})dx\\ & =-(e^4\int \frac{x^2}{{(e^{x^2}+x)}^2}dx)-e^4\int \frac{\log (3x)\log (1+x)}{{(e^{x^2}+x)}^2}dx-e^4\int \frac{(1+x)(2x^2(-1+x^2)-\log (1+x))+x\log (3x)(-1+2x(1+x)\log (1+x))}{x(e^{x^2}+x)}dx+e^4\int \frac{(1+x)(2x^2(-1+x^2)-\log (1+x))+x\log (3x)(-1+2x(1+x)\log (1+x))}{(1+x)(e^{x^2}+x)}dx+\left(2e^4\right)\int \frac{x^4}{{(e^{x^2}+x)}^2}dx+\left(2e^4\right)\int \frac{x^2\log (3x)\log (1+x)}{{(e^{x^2}+x)}^2}dx\\ & =-(e^4\int \frac{x^2}{{(e^{x^2}+x)}^2}dx)-e^4\int (-\frac{2x}{e^{x^2}+x}-\frac{2x^2}{e^{x^2}+x}+\frac{2x^3}{e^{x^2}+x}+\frac{2x^4}{e^{x^2}+x}-\frac{\log (3x)}{e^{x^2}+x}-\frac{\log (1+x)}{e^{x^2}+x}-\frac{\log (1+x)}{x(e^{x^2}+x)}+\frac{2x\log (3x)\log (1+x)}{e^{x^2}+x}+\frac{2x^2\log (3x)\log (1+x)}{e^{x^2}+x})dx+e^4\int (-\frac{2x^2}{(1+x)(e^{x^2}+x)}-\frac{2x^3}{(1+x)(e^{x^2}+x)}+\frac{2x^4}{(1+x)(e^{x^2}+x)}+\frac{2x^5}{(1+x)(e^{x^2}+x)}-\frac{x\log (3x)}{(1+x)(e^{x^2}+x)}-\frac{\log (1+x)}{(1+x)(e^{x^2}+x)}-\frac{x\log (1+x)}{(1+x)(e^{x^2}+x)}+\frac{2x^2\log (3x)\log (1+x)}{(1+x)(e^{x^2}+x)}+\frac{2x^3\log (3x)\log (1+x)}{(1+x)(e^{x^2}+x)})dx+e^4\int \frac{\log (3x)\int \frac{1}{{(e^{x^2}+x)}^2}dx}{1+x}dx+e^4\int \frac{\log (1+x)\int \frac{1}{{(e^{x^2}+x)}^2}dx}{x}dx+\left(2e^4\right)\int \frac{x^4}{{(e^{x^2}+x)}^2}dx-\left(2e^4\right)\int \frac{\log (3x)\int \frac{x^2}{{(e^{x^2}+x)}^2}dx}{1+x}dx-\left(2e^4\right)\int \frac{\log (1+x)\int \frac{x^2}{{(e^{x^2}+x)}^2}dx}{x}dx-(e^4\log (3x)\log (1+x))\int \frac{1}{{(e^{x^2}+x)}^2}dx+(2e^4\log (3x)\log (1+x))\int \frac{x^2}{{(e^{x^2}+x)}^2}dx\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

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Mathematica [A]  time = 0.15, size = 26, normalized size = 1.00 $$\begin{array}{c}\frac{e^4(x^2+\log (3x)\log (1+x))}{e^{x^2}+x}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.90, size = 32, normalized size = 1.23 $$\begin{array}{c}\frac{x^2{e}^8+e^8\log \left(3x\right)\log (x+1)}{x{e}^4+e^{(x^2+4)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*e^8 + e^8*log(3*x)*log(x + 1))/(x*e^4 + e^(x^2 + 4))

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giac [A]  time = 0.25, size = 34, normalized size = 1.31 $$\begin{array}{c}\frac{x^2{e}^4+e^4\log (3)\log (x+1)+e^4\log (x+1)\log (x)}{x+e^{\left(x^2\right)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 35, normalized size = 1.35

method	result	size
risch	$\frac{{\mathrm{e}}^4\ln \left(3x\right)\ln (x+1)}{{\mathrm{e}}^{x^2}+x}+\frac{x^2{\mathrm{e}}^4}{{\mathrm{e}}^{x^2}+x}$	$35$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.85, size = 32, normalized size = 1.23 $$\begin{array}{c}\frac{x^2{e}^4+(e^4\log (3)+e^4\log (x))\log (x+1)}{x+e^{\left(x^2\right)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.33, size = 24, normalized size = 0.92 $$\begin{array}{c}\frac{{\mathrm{e}}^4(x^2+\ln \left(3x\right)\ln (x+1))}{x+{\mathrm{e}}^{x^2}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.76, size = 26, normalized size = 1.00 $$\begin{array}{c}\frac{x^2{e}^4+e^4\log \left(3x\right)\log (x+1)}{x+e^{x^2}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-2*x**3-2*x**2)*exp(4)*exp(x**2)+(-x**2-x)*exp(4))*ln(x+1)+x*exp(4)*exp(x**2)+x**2*exp(4))*ln(3*
x)+((x+1)*exp(4)*exp(x**2)+(x**2+x)*exp(4))*ln(x+1)+(-2*x**5-2*x**4+2*x**3+2*x**2)*exp(4)*exp(x**2)+(x**4+x**3
)*exp(4))/((x**2+x)*exp(x**2)**2+(2*x**3+2*x**2)*exp(x**2)+x**4+x**3),x)

[Out]

(x**2*exp(4) + exp(4)*log(3*x)*log(x + 1))/(x + exp(x**2))

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