3.85.78 $\int \frac{-4+3x+e^{e^x+x}x}{-3x+e^{e^x}x+3x^2-x\log (x^4)}dx$

Optimal. Leaf size=17 $$\log (3-e^{e^x}-3x+\log \left(x^4\right))$$

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Rubi [F]  time = 0.74, 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 = {} $$\begin{array}{c}\int \frac{-4+3x+e^{e^x+x}x}{-3x+e^{e^x}x+3x^2-x\log \left(x^4\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 17, normalized size = 1.00 $$\begin{array}{c}\log (3-e^{e^x}-3x+\log \left(x^4\right))\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

Log[3 - E^E^x - 3*x + Log[x^4]]

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fricas [A]  time = 1.15, size = 26, normalized size = 1.53 $$\begin{array}{c}-x+\log (3(x-1)e^x-e^x\log \left(x^4\right)+e^{(x+e^x)})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 28, normalized size = 1.65 $$\begin{array}{c}-x+\log (3x{e}^x-e^x\log \left(x^4\right)+e^{(x+e^x)}-3e^x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.15, size = 201, normalized size = 11.82

method	result	size
risch	$\ln ({\mathrm{e}}^{{\mathrm{e}}^x}+\frac{i(\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2+\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}\left(i{x}^2\right)\mathrm{csgn}\left(i{x}^3\right)-\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^3\right)}^2+\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}\left(i{x}^3\right)\mathrm{csgn}\left(i{x}^4\right)-\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^4\right)}^2+\pi \mathrm{csgn}{\left(i{x}^2\right)}^3-\pi \mathrm{csgn}\left(i{x}^2\right)\mathrm{csgn}{\left(i{x}^3\right)}^2+\pi \mathrm{csgn}{\left(i{x}^3\right)}^3-\pi \mathrm{csgn}\left(i{x}^3\right)\mathrm{csgn}{\left(i{x}^4\right)}^2+\pi \mathrm{csgn}{\left(i{x}^4\right)}^3-6ix+8i\ln (x)+6i)}{2})$	$201$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(exp(exp(x))+1/2*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x)*csgn(I*x^2)*csgn(I*
x^3)-Pi*csgn(I*x)*csgn(I*x^3)^2+Pi*csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)-Pi*csgn(I*x)*csgn(I*x^4)^2+Pi*csgn(I*x^2)
^3-Pi*csgn(I*x^2)*csgn(I*x^3)^2+Pi*csgn(I*x^3)^3-Pi*csgn(I*x^3)*csgn(I*x^4)^2+Pi*csgn(I*x^4)^3-6*I*x+8*I*ln(x)
+6*I))

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maxima [A]  time = 0.38, size = 13, normalized size = 0.76 $$\begin{array}{c}\log (3x+e^{\left(e^x\right)}-4\log (x)-3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x + e^(e^x) - 4*log(x) - 3)

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mupad [B]  time = 5.40, size = 15, normalized size = 0.88 $$\begin{array}{c}\ln (3x+{\mathrm{e}}^{{\mathrm{e}}^x}-\ln \left(x^4\right)-3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x + exp(exp(x)) - log(x^4) - 3)

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sympy [A]  time = 0.40, size = 15, normalized size = 0.88 $$\begin{array}{c}\log (3x+e^{e^x}-\log \left(x^4\right)-3)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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