3.33.77 $\int \frac{-140+88x^2-16x^3-12x^4+(-40x+8x^3)\log (e^{7/x}(25-10x^2+x^4))}{-45x^2-30x^3+4x^4+6x^5+x^6+(30x+10x^2-6x^3-2x^4)\log (e^{7/x}(25-10x^2+x^4))+(-5+x^2){\log }^2(e^{7/x}(25-10x^2+x^4))}dx$

Optimal. Leaf size=32 $$\frac{4x^2}{-x(3+x)+\log \left(e^{7/x}{(5-x^2)}^2\right)}$$

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Rubi [F]  time = 1.36, 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 = {} $$\begin{array}{c}\int \frac{-140+88x^2-16x^3-12x^4+(-40x+8x^3)\log \left(e^{7/x}(25-10x^2+x^4)\right)}{-45x^2-30x^3+4x^4+6x^5+x^6+(30x+10x^2-6x^3-2x^4)\log \left(e^{7/x}(25-10x^2+x^4)\right)+(-5+x^2){\log }^2\left(e^{7/x}(25-10x^2+x^4)\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-140 + 88*x^2 - 16*x^3 - 12*x^4 + (-40*x + 8*x^3)*Log[E^(7/x)*(25 - 10*x^2 + x^4)])/(-45*x^2 - 30*x^3 + 4
*x^4 + 6*x^5 + x^6 + (30*x + 10*x^2 - 6*x^3 - 2*x^4)*Log[E^(7/x)*(25 - 10*x^2 + x^4)] + (-5 + x^2)*Log[E^(7/x)
*(25 - 10*x^2 + x^4)]^2),x]

[Out]

28*Defer[Int][(3*x + x^2 - Log[E^(7/x)*(-5 + x^2)^2])^(-2), x] + 40*Defer[Int][1/((Sqrt[5] - x)*(3*x + x^2 - L
og[E^(7/x)*(-5 + x^2)^2])^2), x] - 16*Defer[Int][x/(3*x + x^2 - Log[E^(7/x)*(-5 + x^2)^2])^2, x] + 12*Defer[In
t][x^2/(3*x + x^2 - Log[E^(7/x)*(-5 + x^2)^2])^2, x] + 8*Defer[Int][x^3/(3*x + x^2 - Log[E^(7/x)*(-5 + x^2)^2]
)^2, x] - 40*Defer[Int][1/((Sqrt[5] + x)*(3*x + x^2 - Log[E^(7/x)*(-5 + x^2)^2])^2), x] - 8*Defer[Int][x/(3*x
+ x^2 - Log[E^(7/x)*(-5 + x^2)^2]), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{4(35-22x^2+4x^3+3x^4-2x(-5+x^2)\log \left(e^{7/x}{(-5+x^2)}^2\right))}{(5-x^2){(x(3+x)-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx\\ & =4\int \frac{35-22x^2+4x^3+3x^4-2x(-5+x^2)\log \left(e^{7/x}{(-5+x^2)}^2\right)}{(5-x^2){(x(3+x)-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx\\ & =4\int (\frac{-35-8x^2-14x^3+3x^4+2x^5}{(-5+x^2){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}-\frac{2x}{3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right)})dx\\ & =4\int \frac{-35-8x^2-14x^3+3x^4+2x^5}{(-5+x^2){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-8\int \frac{x}{3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right)}dx\\ & =4\int (\frac{7}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}-\frac{4x}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}+\frac{3x^2}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}+\frac{2x^3}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}-\frac{20x}{(-5+x^2){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2})dx-8\int \frac{x}{3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right)}dx\\ & =8\int \frac{x^3}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-8\int \frac{x}{3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right)}dx+12\int \frac{x^2}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-16\int \frac{x}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx+28\int \frac{1}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-80\int \frac{x}{(-5+x^2){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx\\ & =8\int \frac{x^3}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-8\int \frac{x}{3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right)}dx+12\int \frac{x^2}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-16\int \frac{x}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx+28\int \frac{1}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-80\int (-\frac{1}{2(\sqrt{5}-x){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}+\frac{1}{2(\sqrt{5}+x){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2})dx\\ & =8\int \frac{x^3}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-8\int \frac{x}{3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right)}dx+12\int \frac{x^2}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-16\int \frac{x}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx+28\int \frac{1}{{(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx+40\int \frac{1}{(\sqrt{5}-x){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx-40\int \frac{1}{(\sqrt{5}+x){(3x+x^2-\log \left(e^{7/x}{(-5+x^2)}^2\right))}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 1.02, size = 31, normalized size = 0.97 $$\begin{array}{c}-\frac{4x^2}{x(3+x)-\log \left(e^{7/x}{(-5+x^2)}^2\right)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-140 + 88*x^2 - 16*x^3 - 12*x^4 + (-40*x + 8*x^3)*Log[E^(7/x)*(25 - 10*x^2 + x^4)])/(-45*x^2 - 30*x
^3 + 4*x^4 + 6*x^5 + x^6 + (30*x + 10*x^2 - 6*x^3 - 2*x^4)*Log[E^(7/x)*(25 - 10*x^2 + x^4)] + (-5 + x^2)*Log[E
^(7/x)*(25 - 10*x^2 + x^4)]^2),x]

[Out]

(-4*x^2)/(x*(3 + x) - Log[E^(7/x)*(-5 + x^2)^2])

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fricas [A]  time = 0.58, size = 34, normalized size = 1.06 $$\begin{array}{c}-\frac{4x^2}{x^2+3x-\log \left((x^4-10x^2+25)e^{\frac{7}{x}}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-40*x)*log((x^4-10*x^2+25)*exp(7/x))-12*x^4-16*x^3+88*x^2-140)/((x^2-5)*log((x^4-10*x^2+25)*e
xp(7/x))^2+(-2*x^4-6*x^3+10*x^2+30*x)*log((x^4-10*x^2+25)*exp(7/x))+x^6+6*x^5+4*x^4-30*x^3-45*x^2),x, algorith
m="fricas")

[Out]

-4*x^2/(x^2 + 3*x - log((x^4 - 10*x^2 + 25)*e^(7/x)))

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giac [A]  time = 0.76, size = 31, normalized size = 0.97 $$\begin{array}{c}-\frac{4x^3}{x^3+3x^2-x\log (x^4-10x^2+25)-7}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-40*x)*log((x^4-10*x^2+25)*exp(7/x))-12*x^4-16*x^3+88*x^2-140)/((x^2-5)*log((x^4-10*x^2+25)*e
xp(7/x))^2+(-2*x^4-6*x^3+10*x^2+30*x)*log((x^4-10*x^2+25)*exp(7/x))+x^6+6*x^5+4*x^4-30*x^3-45*x^2),x, algorith
m="giac")

[Out]

-4*x^3/(x^3 + 3*x^2 - x*log(x^4 - 10*x^2 + 25) - 7)

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maple [C]  time = 0.14, size = 235, normalized size = 7.34

method	result	size
risch	$-\frac{8x^2}{i\pi \mathrm{csgn}{\left(i(x^2-5)\right)}^2\mathrm{csgn}\left(i{(x^2-5)}^2\right)-2i\pi \mathrm{csgn}\left(i(x^2-5)\right)\mathrm{csgn}{\left(i{(x^2-5)}^2\right)}^2+i\pi \mathrm{csgn}{\left(i{(x^2-5)}^2\right)}^3+i\pi \mathrm{csgn}\left(i{(x^2-5)}^2\right)\mathrm{csgn}\left(i{\mathrm{e}}^{\frac{7}{x}}\right)\mathrm{csgn}\left(i{\mathrm{e}}^{\frac{7}{x}}{(x^2-5)}^2\right)-i\pi \mathrm{csgn}\left(i{(x^2-5)}^2\right)\mathrm{csgn}{\left(i{\mathrm{e}}^{\frac{7}{x}}{(x^2-5)}^2\right)}^2-i\pi \mathrm{csgn}\left(i{\mathrm{e}}^{\frac{7}{x}}\right)\mathrm{csgn}{\left(i{\mathrm{e}}^{\frac{7}{x}}{(x^2-5)}^2\right)}^2+i\pi \mathrm{csgn}{\left(i{\mathrm{e}}^{\frac{7}{x}}{(x^2-5)}^2\right)}^3+2x^2+6x-4\ln (x^2-5)-2\ln \left({\mathrm{e}}^{\frac{7}{x}}\right)}$	$235$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^3-40*x)*ln((x^4-10*x^2+25)*exp(7/x))-12*x^4-16*x^3+88*x^2-140)/((x^2-5)*ln((x^4-10*x^2+25)*exp(7/x))
^2+(-2*x^4-6*x^3+10*x^2+30*x)*ln((x^4-10*x^2+25)*exp(7/x))+x^6+6*x^5+4*x^4-30*x^3-45*x^2),x,method=_RETURNVERB
OSE)

[Out]

-8*x^2/(I*Pi*csgn(I*(x^2-5))^2*csgn(I*(x^2-5)^2)-2*I*Pi*csgn(I*(x^2-5))*csgn(I*(x^2-5)^2)^2+I*Pi*csgn(I*(x^2-5
)^2)^3+I*Pi*csgn(I*(x^2-5)^2)*csgn(I*exp(7/x))*csgn(I*exp(7/x)*(x^2-5)^2)-I*Pi*csgn(I*(x^2-5)^2)*csgn(I*exp(7/
x)*(x^2-5)^2)^2-I*Pi*csgn(I*exp(7/x))*csgn(I*exp(7/x)*(x^2-5)^2)^2+I*Pi*csgn(I*exp(7/x)*(x^2-5)^2)^3+2*x^2+6*x
-4*ln(x^2-5)-2*ln(exp(7/x)))

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maxima [A]  time = 0.94, size = 26, normalized size = 0.81 $$\begin{array}{c}-\frac{4x^3}{x^3+3x^2-2x\log (x^2-5)-7}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-40*x)*log((x^4-10*x^2+25)*exp(7/x))-12*x^4-16*x^3+88*x^2-140)/((x^2-5)*log((x^4-10*x^2+25)*e
xp(7/x))^2+(-2*x^4-6*x^3+10*x^2+30*x)*log((x^4-10*x^2+25)*exp(7/x))+x^6+6*x^5+4*x^4-30*x^3-45*x^2),x, algorith
m="maxima")

[Out]

-4*x^3/(x^3 + 3*x^2 - 2*x*log(x^2 - 5) - 7)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 $$\begin{array}{c}\int -\frac{\ln \left({\mathrm{e}}^{7/x}(x^4-10x^2+25)\right)(40x-8x^3)-88x^2+16x^3+12x^4+140}{\ln \left({\mathrm{e}}^{7/x}(x^4-10x^2+25)\right)(-2x^4-6x^3+10x^2+30x)+{\ln \left({\mathrm{e}}^{7/x}(x^4-10x^2+25)\right)}^2(x^2-5)-45x^2-30x^3+4x^4+6x^5+x^6}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(exp(7/x)*(x^4 - 10*x^2 + 25))*(40*x - 8*x^3) - 88*x^2 + 16*x^3 + 12*x^4 + 140)/(log(exp(7/x)*(x^4 -
10*x^2 + 25))*(30*x + 10*x^2 - 6*x^3 - 2*x^4) + log(exp(7/x)*(x^4 - 10*x^2 + 25))^2*(x^2 - 5) - 45*x^2 - 30*x^
3 + 4*x^4 + 6*x^5 + x^6),x)

[Out]

int(-(log(exp(7/x)*(x^4 - 10*x^2 + 25))*(40*x - 8*x^3) - 88*x^2 + 16*x^3 + 12*x^4 + 140)/(log(exp(7/x)*(x^4 -
10*x^2 + 25))*(30*x + 10*x^2 - 6*x^3 - 2*x^4) + log(exp(7/x)*(x^4 - 10*x^2 + 25))^2*(x^2 - 5) - 45*x^2 - 30*x^
3 + 4*x^4 + 6*x^5 + x^6), x)

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sympy [A]  time = 0.46, size = 27, normalized size = 0.84 $$\begin{array}{c}\frac{4x^2}{-x^2-3x+\log \left((x^4-10x^2+25)e^{\frac{7}{x}}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**3-40*x)*ln((x**4-10*x**2+25)*exp(7/x))-12*x**4-16*x**3+88*x**2-140)/((x**2-5)*ln((x**4-10*x**
2+25)*exp(7/x))**2+(-2*x**4-6*x**3+10*x**2+30*x)*ln((x**4-10*x**2+25)*exp(7/x))+x**6+6*x**5+4*x**4-30*x**3-45*
x**2),x)

[Out]

4*x**2/(-x**2 - 3*x + log((x**4 - 10*x**2 + 25)*exp(7/x)))

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