\(\int \frac {-140+88 x^2-16 x^3-12 x^4+(-40 x+8 x^3) \log (e^{7/x} (25-10 x^2+x^4))}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+(30 x+10 x^2-6 x^3-2 x^4) \log (e^{7/x} (25-10 x^2+x^4))+(-5+x^2) \log ^2(e^{7/x} (25-10 x^2+x^4))} \, dx\) [3277]

Optimal result

Integrand size = 139, antiderivative size = 32 \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=\frac {4 x^2}{-x (3+x)+\log \left (e^{7/x} \left (5-x^2\right )^2\right )} \]

[Out]

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

Rubi [F]

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

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=-\frac {4 x^2}{x (3+x)-\log \left (e^{7/x} \left (-5+x^2\right )^2\right )} \]

[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])

Maple [A] (verified)

Time = 6.90 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(-\frac {4 x^{2}}{x^{2}+3 x -\ln \left (\left (x^{4}-10 x^{2}+25\right ) {\mathrm e}^{\frac {7}{x}}\right )}\)	\(35\)
risch	\(-\frac {8 x^{2}}{i \pi {\operatorname {csgn}\left (i \left (x^{2}-5\right )\right )}^{2} \operatorname {csgn}\left (i \left (x^{2}-5\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (x^{2}-5\right )\right ) {\operatorname {csgn}\left (i \left (x^{2}-5\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (x^{2}-5\right )^{2}\right )}^{3}+i \pi \,\operatorname {csgn}\left (i \left (x^{2}-5\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\frac {7}{x}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\frac {7}{x}} \left (x^{2}-5\right )^{2}\right )-i \pi \,\operatorname {csgn}\left (i \left (x^{2}-5\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\frac {7}{x}} \left (x^{2}-5\right )^{2}\right )^{2}-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{\frac {7}{x}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{\frac {7}{x}} \left (x^{2}-5\right )^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i {\mathrm e}^{\frac {7}{x}} \left (x^{2}-5\right )^{2}\right )^{3}+2 x^{2}+6 x -4 \ln \left (x^{2}-5\right )-2 \ln \left ({\mathrm e}^{\frac {7}{x}}\right )}\)	\(235\)

[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]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=-\frac {4 \, x^{2}}{x^{2} + 3 \, x - \log \left ({\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\frac {7}{x}}\right )} \]

[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)))

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=\frac {4 x^{2}}{- x^{2} - 3 x + \log {\left (\left (x^{4} - 10 x^{2} + 25\right ) e^{\frac {7}{x}} \right )}} \]

[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)))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=-\frac {4 \, x^{3}}{x^{3} + 3 \, x^{2} - 2 \, x \log \left (x^{2} - 5\right ) - 7} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=-\frac {4 \, x^{3}}{x^{3} + 3 \, x^{2} - x \log \left (x^{4} - 10 \, x^{2} + 25\right ) - 7} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {-140+88 x^2-16 x^3-12 x^4+\left (-40 x+8 x^3\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )}{-45 x^2-30 x^3+4 x^4+6 x^5+x^6+\left (30 x+10 x^2-6 x^3-2 x^4\right ) \log \left (e^{7/x} \left (25-10 x^2+x^4\right )\right )+\left (-5+x^2\right ) \log ^2\left (e^{7/x} \left (25-10 x^2+x^4\right )\right )} \, dx=\int -\frac {\ln \left ({\mathrm {e}}^{7/x}\,\left (x^4-10\,x^2+25\right )\right )\,\left (40\,x-8\,x^3\right )-88\,x^2+16\,x^3+12\,x^4+140}{\ln \left ({\mathrm {e}}^{7/x}\,\left (x^4-10\,x^2+25\right )\right )\,\left (-2\,x^4-6\,x^3+10\,x^2+30\,x\right )+{\ln \left ({\mathrm {e}}^{7/x}\,\left (x^4-10\,x^2+25\right )\right )}^2\,\left (x^2-5\right )-45\,x^2-30\,x^3+4\,x^4+6\,x^5+x^6} \,d x \]

[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)