3.85.98 $\int \frac{75x^3-30x^4+3x^5+e^{3/x}(15+2x+24x^2-10x^3+x^4)+(-e^{3/x}{x}^2-3x^3)\log (\frac{e^{3/x}+3x}{x})}{75x^5-30x^6+3x^7+e^{3/x}(25x^4-10x^5+x^6)+(-30x^4+6x^5+e^{3/x}(-10x^3+2x^4))\log (\frac{e^{3/x}+3x}{x})+(e^{3/x}{x}^2+3x^3){\log }^2(\frac{e^{3/x}+3x}{x})}dx$

Optimal. Leaf size=30 $$4+\frac{1}{-x+\frac{\log (3+\frac{e^{3/x}}{x})}{5-x}}$$

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Rubi [F]  time = 3.93, 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 = {} $$\begin{array}{c}\int \frac{75x^3-30x^4+3x^5+e^{3/x}(15+2x+24x^2-10x^3+x^4)+(-e^{3/x}{x}^2-3x^3)\log \left(\frac{e^{3/x}+3x}{x}\right)}{75x^5-30x^6+3x^7+e^{3/x}(25x^4-10x^5+x^6)+(-30x^4+6x^5+e^{3/x}(-10x^3+2x^4))\log \left(\frac{e^{3/x}+3x}{x}\right)+(e^{3/x}{x}^2+3x^3){\log }^2\left(\frac{e^{3/x}+3x}{x}\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(75*x^3 - 30*x^4 + 3*x^5 + E^(3/x)*(15 + 2*x + 24*x^2 - 10*x^3 + x^4) + (-(E^(3/x)*x^2) - 3*x^3)*Log[(E^(3
/x) + 3*x)/x])/(75*x^5 - 30*x^6 + 3*x^7 + E^(3/x)*(25*x^4 - 10*x^5 + x^6) + (-30*x^4 + 6*x^5 + E^(3/x)*(-10*x^
3 + 2*x^4))*Log[(E^(3/x) + 3*x)/x] + (E^(3/x)*x^2 + 3*x^3)*Log[(E^(3/x) + 3*x)/x]^2),x]

[Out]

24*Defer[Int][(-5*x + x^2 + Log[3 + E^(3/x)/x])^(-2), x] + 15*Defer[Int][1/(x^2*(-5*x + x^2 + Log[3 + E^(3/x)/
x])^2), x] + 2*Defer[Int][1/(x*(-5*x + x^2 + Log[3 + E^(3/x)/x])^2), x] - 15*Defer[Int][x/(-5*x + x^2 + Log[3
+ E^(3/x)/x])^2, x] + 2*Defer[Int][x^2/(-5*x + x^2 + Log[3 + E^(3/x)/x])^2, x] - 6*Defer[Int][1/((E^(3/x) + 3*
x)*(-5*x + x^2 + Log[3 + E^(3/x)/x])^2), x] - 45*Defer[Int][1/(x*(E^(3/x) + 3*x)*(-5*x + x^2 + Log[3 + E^(3/x)
/x])^2), x] + 3*Defer[Int][x/((E^(3/x) + 3*x)*(-5*x + x^2 + Log[3 + E^(3/x)/x])^2), x] - Defer[Int][(-5*x + x^
2 + Log[3 + E^(3/x)/x])^(-1), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{(-5+x)(3(-5+x)x^3+e^{3/x}(-3-x-5x^2+x^3))-x^2(e^{3/x}+3x)\log (3+\frac{e^{3/x}}{x})}{x^2(e^{3/x}+3x){((-5+x)x+\log (3+\frac{e^{3/x}}{x}))}^2}dx\\ & =\int (\frac{3(-15-2x+x^2)}{x(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}+\frac{15+2x+24x^2-10x^3+x^4-x^2\log (3+\frac{e^{3/x}}{x})}{x^2{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2})dx\\ & =3\int \frac{-15-2x+x^2}{x(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx+\int \frac{15+2x+24x^2-10x^3+x^4-x^2\log (3+\frac{e^{3/x}}{x})}{x^2{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx\\ & =3\int (-\frac{2}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}-\frac{15}{x(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}+\frac{x}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2})dx+\int (\frac{15+2x+24x^2-15x^3+2x^4}{x^2{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}-\frac{1}{-5x+x^2+\log (3+\frac{e^{3/x}}{x})})dx\\ & =3\int \frac{x}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-6\int \frac{1}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-45\int \frac{1}{x(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx+\int \frac{15+2x+24x^2-15x^3+2x^4}{x^2{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-\int \frac{1}{-5x+x^2+\log (3+\frac{e^{3/x}}{x})}dx\\ & =3\int \frac{x}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-6\int \frac{1}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-45\int \frac{1}{x(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-\int \frac{1}{-5x+x^2+\log (3+\frac{e^{3/x}}{x})}dx+\int (\frac{24}{{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}+\frac{15}{x^2{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}+\frac{2}{x{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}-\frac{15x}{{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}+\frac{2x^2}{{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2})dx\\ & =2\int \frac{1}{x{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx+2\int \frac{x^2}{{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx+3\int \frac{x}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-6\int \frac{1}{(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx+15\int \frac{1}{x^2{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-15\int \frac{x}{{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx+24\int \frac{1}{{(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-45\int \frac{1}{x(e^{3/x}+3x){(-5x+x^2+\log (3+\frac{e^{3/x}}{x}))}^2}dx-\int \frac{1}{-5x+x^2+\log (3+\frac{e^{3/x}}{x})}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.10, size = 29, normalized size = 0.97 $$\begin{array}{c}\frac{5-x}{-5x+x^2+\log (3+\frac{e^{3/x}}{x})}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(75*x^3 - 30*x^4 + 3*x^5 + E^(3/x)*(15 + 2*x + 24*x^2 - 10*x^3 + x^4) + (-(E^(3/x)*x^2) - 3*x^3)*Log
[(E^(3/x) + 3*x)/x])/(75*x^5 - 30*x^6 + 3*x^7 + E^(3/x)*(25*x^4 - 10*x^5 + x^6) + (-30*x^4 + 6*x^5 + E^(3/x)*(
-10*x^3 + 2*x^4))*Log[(E^(3/x) + 3*x)/x] + (E^(3/x)*x^2 + 3*x^3)*Log[(E^(3/x) + 3*x)/x]^2),x]

[Out]

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

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fricas [A]  time = 0.49, size = 29, normalized size = 0.97 $$\begin{array}{c}-\frac{x-5}{x^2-5x+\log \left(\frac{3x+e^{\frac{3}{x}}}{x}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3
)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(
x^6-10*x^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x, algorithm="fricas")

[Out]

-(x - 5)/(x^2 - 5*x + log((3*x + e^(3/x))/x))

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giac [A]  time = 0.39, size = 29, normalized size = 0.97 $$\begin{array}{c}-\frac{x-5}{x^2-5x+\log \left(\frac{3x+e^{\frac{3}{x}}}{x}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3
)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(
x^6-10*x^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x, algorithm="giac")

[Out]

-(x - 5)/(x^2 - 5*x + log((3*x + e^(3/x))/x))

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maple [C]  time = 0.15, size = 170, normalized size = 5.67

method	result	size
risch	$-\frac{2(x-5)}{-i\pi \mathrm{csgn}\left(\frac{i}{x}\right)\mathrm{csgn}\left(i(\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)\right)\mathrm{csgn}\left(\frac{i(\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)}{x}\right)+i\pi \mathrm{csgn}\left(\frac{i}{x}\right)\mathrm{csgn}{\left(\frac{i(\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)}{x}\right)}^2+i\pi \mathrm{csgn}\left(i(\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)\right)\mathrm{csgn}{\left(\frac{i(\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)}{x}\right)}^2-i\pi \mathrm{csgn}{\left(\frac{i(\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)}{x}\right)}^3+2x^2+2\ln (3)-10x-2\ln (x)+2\ln (\frac{{\mathrm{e}}^{\frac{3}{x}}}{3}+x)}$	$170$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*exp(3/x)-3*x^3)*ln((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3)/((x^2
*exp(3/x)+3*x^3)*ln((exp(3/x)+3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*ln((exp(3/x)+3*x)/x)+(x^6-10*x^
5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x,method=_RETURNVERBOSE)

[Out]

-2*(x-5)/(-I*Pi*csgn(I/x)*csgn(I*(1/3*exp(3/x)+x))*csgn(I/x*(1/3*exp(3/x)+x))+I*Pi*csgn(I/x)*csgn(I/x*(1/3*exp
(3/x)+x))^2+I*Pi*csgn(I*(1/3*exp(3/x)+x))*csgn(I/x*(1/3*exp(3/x)+x))^2-I*Pi*csgn(I/x*(1/3*exp(3/x)+x))^3+2*x^2
+2*ln(3)-10*x-2*ln(x)+2*ln(1/3*exp(3/x)+x))

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maxima [A]  time = 0.42, size = 29, normalized size = 0.97 $$\begin{array}{c}-\frac{x-5}{x^2-5x+\log (3x+e^{\frac{3}{x}})-\log (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(3/x)-3*x^3)*log((exp(3/x)+3*x)/x)+(x^4-10*x^3+24*x^2+2*x+15)*exp(3/x)+3*x^5-30*x^4+75*x^3
)/((x^2*exp(3/x)+3*x^3)*log((exp(3/x)+3*x)/x)^2+((2*x^4-10*x^3)*exp(3/x)+6*x^5-30*x^4)*log((exp(3/x)+3*x)/x)+(
x^6-10*x^5+25*x^4)*exp(3/x)+3*x^7-30*x^6+75*x^5),x, algorithm="maxima")

[Out]

-(x - 5)/(x^2 - 5*x + log(3*x + e^(3/x)) - log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 $$\begin{array}{c}\int \frac{{\mathrm{e}}^{3/x}(x^4-10x^3+24x^2+2x+15)-\ln \left(\frac{3x+{\mathrm{e}}^{3/x}}{x}\right)(x^2{\mathrm{e}}^{3/x}+3x^3)+75x^3-30x^4+3x^5}{{\ln \left(\frac{3x+{\mathrm{e}}^{3/x}}{x}\right)}^2(x^2{\mathrm{e}}^{3/x}+3x^3)-\ln \left(\frac{3x+{\mathrm{e}}^{3/x}}{x}\right)({\mathrm{e}}^{3/x}(10x^3-2x^4)+30x^4-6x^5)+{\mathrm{e}}^{3/x}(x^6-10x^5+25x^4)+75x^5-30x^6+3x^7}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3/x)*(2*x + 24*x^2 - 10*x^3 + x^4 + 15) - log((3*x + exp(3/x))/x)*(x^2*exp(3/x) + 3*x^3) + 75*x^3 - 3
0*x^4 + 3*x^5)/(log((3*x + exp(3/x))/x)^2*(x^2*exp(3/x) + 3*x^3) - log((3*x + exp(3/x))/x)*(exp(3/x)*(10*x^3 -
2*x^4) + 30*x^4 - 6*x^5) + exp(3/x)*(25*x^4 - 10*x^5 + x^6) + 75*x^5 - 30*x^6 + 3*x^7),x)

[Out]

int((exp(3/x)*(2*x + 24*x^2 - 10*x^3 + x^4 + 15) - log((3*x + exp(3/x))/x)*(x^2*exp(3/x) + 3*x^3) + 75*x^3 - 3
0*x^4 + 3*x^5)/(log((3*x + exp(3/x))/x)^2*(x^2*exp(3/x) + 3*x^3) - log((3*x + exp(3/x))/x)*(exp(3/x)*(10*x^3 -
2*x^4) + 30*x^4 - 6*x^5) + exp(3/x)*(25*x^4 - 10*x^5 + x^6) + 75*x^5 - 30*x^6 + 3*x^7), x)

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sympy [A]  time = 0.55, size = 20, normalized size = 0.67 $$\begin{array}{c}\frac{5-x}{x^2-5x+\log \left(\frac{3x+e^{\frac{3}{x}}}{x}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2*exp(3/x)-3*x**3)*ln((exp(3/x)+3*x)/x)+(x**4-10*x**3+24*x**2+2*x+15)*exp(3/x)+3*x**5-30*x**4+
75*x**3)/((x**2*exp(3/x)+3*x**3)*ln((exp(3/x)+3*x)/x)**2+((2*x**4-10*x**3)*exp(3/x)+6*x**5-30*x**4)*ln((exp(3/
x)+3*x)/x)+(x**6-10*x**5+25*x**4)*exp(3/x)+3*x**7-30*x**6+75*x**5),x)

[Out]

(5 - x)/(x**2 - 5*x + log((3*x + exp(3/x))/x))

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