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3.86.89 \(\int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} (e^5 (-200 x+50 x^2+x^4)+2 e^5 x^2 \log (4)+e^5 \log ^2(4))}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} (-2 x^4-4 x^2 \log (4)-2 \log ^2(4))+e^{\frac {50}{x^2+\log (4)}} (x^4+2 x^2 \log (4)+\log ^2(4))} \, dx\)

Optimal. Leaf size=27 \[ e^5+\frac {e^5 (-4+x)}{-1+e^{\frac {25}{x^2+\log (4)}}} \]

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Rubi [F]  time = 125.90, 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 {gather*} \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(E^5*x^4) - 2*E^5*x^2*Log[4] - E^5*Log[4]^2 + E^(25/(x^2 + Log[4]))*(E^5*(-200*x + 50*x^2 + x^4) + 2*E^5
*x^2*Log[4] + E^5*Log[4]^2))/(x^4 + 2*x^2*Log[4] + Log[4]^2 + E^(25/(x^2 + Log[4]))*(-2*x^4 - 4*x^2*Log[4] - 2
*Log[4]^2) + E^(50/(x^2 + Log[4]))*(x^4 + 2*x^2*Log[4] + Log[4]^2)),x]

[Out]

(4*E^5)/(25*(1 - E^(x^2 + Log[4])^(-1))) + (16*E^5)/(25*(1 - E^(5/(x^2 + Log[4])))) - (4*E^(5 + (x^2 + Log[4])
^(-1)))/(25*(1 - E^(5/(x^2 + Log[4])))) - (4*E^(5 + 2/(x^2 + Log[4])))/(25*(1 - E^(5/(x^2 + Log[4])))) - (4*E^
(5 + 3/(x^2 + Log[4])))/(25*(1 - E^(5/(x^2 + Log[4])))) - (4*E^(5 + 4/(x^2 + Log[4])))/(25*(1 - E^(5/(x^2 + Lo
g[4])))) + (16*E^5)/(5*(1 - E^(25/(x^2 + Log[4])))) - (4*E^(5 + 5/(x^2 + Log[4])))/(5*(1 - E^(25/(x^2 + Log[4]
)))) - (4*E^(5 + 10/(x^2 + Log[4])))/(5*(1 - E^(25/(x^2 + Log[4])))) - (4*E^(5 + 15/(x^2 + Log[4])))/(5*(1 - E
^(25/(x^2 + Log[4])))) - (4*E^(5 + 20/(x^2 + Log[4])))/(5*(1 - E^(25/(x^2 + Log[4])))) - (12*Sqrt[2*(5 - Sqrt[
5])]*E^5*ArcTan[(1 - Sqrt[5] + 4*E^(x^2 + Log[4])^(-1))/Sqrt[2*(5 + Sqrt[5])]])/125 + (12*Sqrt[2*(5 + Sqrt[5])
]*E^5*ArcTan[(1 - Sqrt[5] + 4*E^(x^2 + Log[4])^(-1))/Sqrt[2*(5 + Sqrt[5])]])/125 + (12*Sqrt[2*(5 - Sqrt[5])]*E
^5*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*E^(x^2 + Log[4])^(-1)))/2])/125 + (12*Sqrt[2*(5 + Sqrt[5])]
*E^5*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*E^(x^2 + Log[4])^(-1)))/2])/125 - (12*Sqrt[2*(5 - Sqrt[5]
)]*E^5*ArcTan[(1 - Sqrt[5] + 4*E^(5/(x^2 + Log[4])))/Sqrt[2*(5 + Sqrt[5])]])/25 + (12*Sqrt[2*(5 + Sqrt[5])]*E^
5*ArcTan[(1 - Sqrt[5] + 4*E^(5/(x^2 + Log[4])))/Sqrt[2*(5 + Sqrt[5])]])/25 + (12*Sqrt[2*(5 - Sqrt[5])]*E^5*Arc
Tan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*E^(5/(x^2 + Log[4]))))/2])/25 + (12*Sqrt[2*(5 + Sqrt[5])]*E^5*Arc
Tan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*E^(5/(x^2 + Log[4]))))/2])/25 + (24*(-1)^(2/5)*(1 - (-1)^(2/5))*E
^5*Log[(-1)^(2/5) - E^(x^2 + Log[4])^(-1)])/125 - (24*(-1)^(3/5)*(1 + (-1)^(3/5))*E^5*Log[-(-1)^(3/5) - E^(x^2
 + Log[4])^(-1)])/125 - (24*(-1)^(1/5)*(1 + (-1)^(1/5))*E^5*Log[(-1)^(1/5) + E^(x^2 + Log[4])^(-1)])/125 + (24
*(-1)^(4/5)*(1 - (-1)^(4/5))*E^5*Log[-(-1)^(4/5) + E^(x^2 + Log[4])^(-1)])/125 + (6*(1 - Sqrt[5])*E^5*Log[2 +
E^(x^2 + Log[4])^(-1) - Sqrt[5]*E^(x^2 + Log[4])^(-1) + 2*E^(2/(x^2 + Log[4]))])/125 - (6*(1 + Sqrt[5])*E^5*Lo
g[2 + E^(x^2 + Log[4])^(-1) - Sqrt[5]*E^(x^2 + Log[4])^(-1) + 2*E^(2/(x^2 + Log[4]))])/125 - (6*(1 - Sqrt[5])*
E^5*Log[2 + E^(x^2 + Log[4])^(-1) + Sqrt[5]*E^(x^2 + Log[4])^(-1) + 2*E^(2/(x^2 + Log[4]))])/125 + (6*(1 + Sqr
t[5])*E^5*Log[2 + E^(x^2 + Log[4])^(-1) + Sqrt[5]*E^(x^2 + Log[4])^(-1) + 2*E^(2/(x^2 + Log[4]))])/125 + (24*(
-1)^(2/5)*(1 - (-1)^(2/5))*E^5*Log[(-1)^(2/5) - E^(5/(x^2 + Log[4]))])/25 - (24*(-1)^(3/5)*(1 + (-1)^(3/5))*E^
5*Log[-(-1)^(3/5) - E^(5/(x^2 + Log[4]))])/25 - (24*(-1)^(1/5)*(1 + (-1)^(1/5))*E^5*Log[(-1)^(1/5) + E^(5/(x^2
 + Log[4]))])/25 + (24*(-1)^(4/5)*(1 - (-1)^(4/5))*E^5*Log[-(-1)^(4/5) + E^(5/(x^2 + Log[4]))])/25 + (6*(1 - S
qrt[5])*E^5*Log[2 + E^(5/(x^2 + Log[4])) - Sqrt[5]*E^(5/(x^2 + Log[4])) + 2*E^(10/(x^2 + Log[4]))])/25 - (6*(1
 + Sqrt[5])*E^5*Log[2 + E^(5/(x^2 + Log[4])) - Sqrt[5]*E^(5/(x^2 + Log[4])) + 2*E^(10/(x^2 + Log[4]))])/25 - (
6*(1 - Sqrt[5])*E^5*Log[2 + E^(5/(x^2 + Log[4])) + Sqrt[5]*E^(5/(x^2 + Log[4])) + 2*E^(10/(x^2 + Log[4]))])/25
 + (6*(1 + Sqrt[5])*E^5*Log[2 + E^(5/(x^2 + Log[4])) + Sqrt[5]*E^(5/(x^2 + Log[4])) + 2*E^(10/(x^2 + Log[4]))]
)/25 + (E^5*Defer[Int][(-1 + E^(x^2 + Log[4])^(-1))^(-1), x])/25 - (4*E^5*Defer[Int][(1 + E^(x^2 + Log[4])^(-1
) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^(-1), x])/25 - (3*E^5*Defer[Int][E^(x^
2 + Log[4])^(-1)/(1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4]
))), x])/25 - (2*E^5*Defer[Int][E^(2/(x^2 + Log[4]))/(1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/
(x^2 + Log[4])) + E^(4/(x^2 + Log[4]))), x])/25 - (E^5*Defer[Int][E^(3/(x^2 + Log[4]))/(1 + E^(x^2 + Log[4])^(
-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4]))), x])/25 - (4*E^5*Defer[Int][(1 + E^(
5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^(-1), x])/5 - (3*E^
5*Defer[Int][E^(5/(x^2 + Log[4]))/(1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) +
E^(20/(x^2 + Log[4]))), x])/5 - (2*E^5*Defer[Int][E^(10/(x^2 + Log[4]))/(1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2
 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4]))), x])/5 - (E^5*Defer[Int][E^(15/(x^2 + Log[4]))/(1
 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4]))), x])/5 + ((I/
25)*E^5*Defer[Int][1/((-1 + E^(x^2 + Log[4])^(-1))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((I/25)*E^5*De
fer[Int][1/((-1 + E^(x^2 + Log[4])^(-1))*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (((2*I)/5)*E^5*Defer[Int][
1/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(-x + I*
Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (((3*I)/5)*E^5*Defer[Int][E^(x^2 + Log[4])^(-1)/((1 + E^(x^2 + Log[4])^(-1)
 + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[
4]] + (((3*I)/5)*E^5*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3
/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (((2*I)/5)*E^5*Defer[Int
][E^(3/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 +
 Log[4])))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((4*I)/25)*E^5*Sqrt[Log[4]]*Defer[Int][1/((1 + E^(x^2
+ Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x
] - (((2*I)/25)*E^5*(5 + Log[16])*Defer[Int][1/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2
+ Log[4])) + E^(4/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] - (((6*I)/25)*E^5*(1 + Log[2])*Def
er[Int][E^(x^2 + Log[4])^(-1)/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4
/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((3*I)/25)*E^5*Sqrt[Log[4]]*Defer[Int][E^(x^2 + L
og[4])^(-1)/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*
(-x + I*Sqrt[Log[4]])), x] + ((2*I)/25)*E^5*Sqrt[Log[4]]*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4]
)^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x] - ((I/
25)*E^5*(3 + Log[16])*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(
3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (I/25)*E^5*Sqrt[Log[4]]*D
efer[Int][E^(3/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(
4/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x] - ((I/25)*E^5*(1 + Log[4])*Defer[Int][E^(3/(x^2 + Log[4]))/((1 +
 E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(-x + I*Sqrt[Log[
4]])), x])/Sqrt[Log[4]] + ((10*I)*E^5*Defer[Int][1/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/
(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((15*I)*E^5*Defer[Int][E
^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 +
Log[4])))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((15*I)*E^5*Defer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5
/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(-x + I*Sqrt[Log[4
]])), x])/Sqrt[Log[4]] + ((10*I)*E^5*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2
+ Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((4*I
)/5)*E^5*Sqrt[Log[4]]*Defer[Int][1/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4]))
+ E^(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x] - (((2*I)/5)*E^5*(25 + Log[16])*Defer[Int][1/((1 + E^(5/(x
^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])),
 x])/Sqrt[Log[4]] - (((6*I)/5)*E^5*(5 + Log[2])*Defer[Int][E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E
^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]]
 + ((3*I)/5)*E^5*Sqrt[Log[4]]*Defer[Int][E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4]
)) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x] + ((2*I)/5)*E^5*Sqrt[Log[4]]*De
fer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^
(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x] - ((I/5)*E^5*(15 + Log[16])*Defer[Int][E^(10/(x^2 + Log[4]))/(
(1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(-x + I*Sqr
t[Log[4]])), x])/Sqrt[Log[4]] + (I/5)*E^5*Sqrt[Log[4]]*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[
4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x] - ((I
/5)*E^5*(5 + Log[4])*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(
15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(-x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((I/25)*E^5*Defer[Int][
1/((-1 + E^(x^2 + Log[4])^(-1))^2*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((I/25)*E^5*Defer[Int][1/((-1 + E^
(x^2 + Log[4])^(-1))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (((2*I)/5)*E^5*Defer[Int][1/((1 + E^(x^2 + Log[
4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(x + I*Sqrt[Log[4]])), x])/Sq
rt[Log[4]] + (((3*I)/5)*E^5*Defer[Int][E^(x^2 + Log[4])^(-1)/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])
) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (((3*I)/5)*E^5*De
fer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4
/(x^2 + Log[4])))^2*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (((2*I)/5)*E^5*Defer[Int][E^(3/(x^2 + Log[4]))/(
(1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(x + I*Sqrt
[Log[4]])), x])/Sqrt[Log[4]] + ((4*I)/25)*E^5*Sqrt[Log[4]]*Defer[Int][1/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^
2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x] - (((2*I)/25)*E^5*(5 + L
og[16])*Defer[Int][1/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + L
og[4])))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] - (((6*I)/25)*E^5*(1 + Log[2])*Defer[Int][E^(x^2 + Log[4])^(-
1)/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x + I*Sq
rt[Log[4]])), x])/Sqrt[Log[4]] + ((3*I)/25)*E^5*Sqrt[Log[4]]*Defer[Int][E^(x^2 + Log[4])^(-1)/((1 + E^(x^2 + L
og[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x] +
((2*I)/25)*E^5*Sqrt[Log[4]]*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4]))
 + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x] - ((I/25)*E^5*(3 + Log[16])*Defer[In
t][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2
+ Log[4])))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (I/25)*E^5*Sqrt[Log[4]]*Defer[Int][E^(3/(x^2 + Log[4]))/
((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x + I*Sqrt[
Log[4]])), x] - ((I/25)*E^5*(1 + Log[4])*Defer[Int][E^(3/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x
^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((10*I)
*E^5*Defer[Int][1/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Lo
g[4])))^2*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((15*I)*E^5*Defer[Int][E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^
2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(x + I*Sqrt[Log[4]])),
 x])/Sqrt[Log[4]] + ((15*I)*E^5*Defer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log
[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((10*I)*E^5
*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) +
 E^(20/(x^2 + Log[4])))^2*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((4*I)/5)*E^5*Sqrt[Log[4]]*Defer[Int][1/((
1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x + I*Sqrt[
Log[4]])), x] - (((2*I)/5)*E^5*(25 + Log[16])*Defer[Int][1/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4]))
+ E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] - (((6*I)/5)*E^5*(5 +
 Log[2])*Defer[Int][E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log
[4])) + E^(20/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + ((3*I)/5)*E^5*Sqrt[Log[4]]*Defer[Int]
[E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2
+ Log[4])))*(x + I*Sqrt[Log[4]])), x] + ((2*I)/5)*E^5*Sqrt[Log[4]]*Defer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5
/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])
), x] - ((I/5)*E^5*(15 + Log[16])*Defer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + L
og[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] + (I/5)*E^5*S
qrt[Log[4]]*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 +
 Log[4])) + E^(20/(x^2 + Log[4])))*(x + I*Sqrt[Log[4]])), x] - ((I/5)*E^5*(5 + Log[4])*Defer[Int][E^(15/(x^2 +
 Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*
(x + I*Sqrt[Log[4]])), x])/Sqrt[Log[4]] - (2*E^5*Log[4]*Defer[Int][1/((-1 + E^(x^2 + Log[4])^(-1))^2*(x^2 + Lo
g[4])^2), x])/25 - (2*E^5*Log[4]*Defer[Int][1/((-1 + E^(x^2 + Log[4])^(-1))*(x^2 + Log[4])^2), x])/25 - (4*E^5
*Log[4]*Defer[Int][1/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + L
og[4])))^2*(x^2 + Log[4])^2), x])/5 - (6*E^5*Log[4]*Defer[Int][E^(x^2 + Log[4])^(-1)/((1 + E^(x^2 + Log[4])^(-
1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(x^2 + Log[4])^2), x])/5 - (6*E^5*L
og[4]*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])
) + E^(4/(x^2 + Log[4])))^2*(x^2 + Log[4])^2), x])/5 - (4*E^5*Log[4]*Defer[Int][E^(3/(x^2 + Log[4]))/((1 + E^(
x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))^2*(x^2 + Log[4])^2),
x])/5 - (8*E^5*Log[4]^2*Defer[Int][1/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4]))
 + E^(4/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/25 + (4*E^5*Log[4]*(5 + Log[16])*Defer[Int][1/((1 + E^(x^2 + L
og[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/25 +
(12*E^5*(1 + Log[2])*Log[4]*Defer[Int][E^(x^2 + Log[4])^(-1)/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])
) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/25 - (6*E^5*Log[4]^2*Defer[Int][E^(x^2
 + Log[4])^(-1)/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4]
)))*(x^2 + Log[4])^2), x])/25 - (4*E^5*Log[4]^2*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) +
E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/25 + (2*E^5*Log[4]*
(3 + Log[16])*Defer[Int][E^(2/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 +
 Log[4])) + E^(4/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/25 - (2*E^5*Log[4]^2*Defer[Int][E^(3/(x^2 + Log[4]))/
((1 + E^(x^2 + Log[4])^(-1) + E^(2/(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x^2 + Log[4
])^2), x])/25 + (2*E^5*Log[4]*(1 + Log[4])*Defer[Int][E^(3/(x^2 + Log[4]))/((1 + E^(x^2 + Log[4])^(-1) + E^(2/
(x^2 + Log[4])) + E^(3/(x^2 + Log[4])) + E^(4/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/25 - 20*E^5*Log[4]*Defer
[Int][1/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*
(x^2 + Log[4])^2), x] - 30*E^5*Log[4]*Defer[Int][E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2
+ Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(x^2 + Log[4])^2), x] - 30*E^5*Log[4]*Defer[Int]
[E^(10/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2
 + Log[4])))^2*(x^2 + Log[4])^2), x] - 20*E^5*Log[4]*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4]
)) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))^2*(x^2 + Log[4])^2), x] - (8*E^5*L
og[4]^2*Defer[Int][1/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 +
 Log[4])))*(x^2 + Log[4])^2), x])/5 + (4*E^5*Log[4]*(25 + Log[16])*Defer[Int][1/((1 + E^(5/(x^2 + Log[4])) + E
^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/5 + (12*E^5*(5 +
Log[2])*Log[4]*Defer[Int][E^(5/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2
 + Log[4])) + E^(20/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/5 - (6*E^5*Log[4]^2*Defer[Int][E^(5/(x^2 + Log[4])
)/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x^2 + L
og[4])^2), x])/5 - (4*E^5*Log[4]^2*Defer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 +
Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/5 + (2*E^5*Log[4]*(15 + Log[1
6])*Defer[Int][E^(10/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])
) + E^(20/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/5 - (2*E^5*Log[4]^2*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E
^(5/(x^2 + Log[4])) + E^(10/(x^2 + Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x^2 + Log[4])^2)
, x])/5 + (2*E^5*Log[4]*(5 + Log[4])*Defer[Int][E^(15/(x^2 + Log[4]))/((1 + E^(5/(x^2 + Log[4])) + E^(10/(x^2
+ Log[4])) + E^(15/(x^2 + Log[4])) + E^(20/(x^2 + Log[4])))*(x^2 + Log[4])^2), x])/5

Rubi steps

Aborted

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Mathematica [A]  time = 5.08, size = 23, normalized size = 0.85 \begin {gather*} \frac {e^5 (-4+x)}{-1+e^{\frac {25}{x^2+\log (4)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^5*x^4) - 2*E^5*x^2*Log[4] - E^5*Log[4]^2 + E^(25/(x^2 + Log[4]))*(E^5*(-200*x + 50*x^2 + x^4) +
 2*E^5*x^2*Log[4] + E^5*Log[4]^2))/(x^4 + 2*x^2*Log[4] + Log[4]^2 + E^(25/(x^2 + Log[4]))*(-2*x^4 - 4*x^2*Log[
4] - 2*Log[4]^2) + E^(50/(x^2 + Log[4]))*(x^4 + 2*x^2*Log[4] + Log[4]^2)),x]

[Out]

(E^5*(-4 + x))/(-1 + E^(25/(x^2 + Log[4])))

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fricas [A]  time = 1.08, size = 23, normalized size = 0.85 \begin {gather*} \frac {{\left (x - 4\right )} e^{5}}{e^{\left (\frac {25}{x^{2} + 2 \, \log \relax (2)}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5))*exp(25/(x^2+2*log(2)))-4*exp(5)*l
og(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5))/((4*log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8
*x^2*log(2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x, algorithm="fricas")

[Out]

(x - 4)*e^5/(e^(25/(x^2 + 2*log(2))) - 1)

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giac [B]  time = 0.24, size = 181, normalized size = 6.70 \begin {gather*} \frac {x^{4} e^{\left (\frac {5 \, {\left (2 \, \log \relax (2) + 5\right )}}{2 \, \log \relax (2)}\right )} - x^{4} e^{\left (\frac {25}{2 \, \log \relax (2)} + 5\right )} + 4 \, x^{2} e^{\left (\frac {5 \, {\left (2 \, \log \relax (2) + 5\right )}}{2 \, \log \relax (2)}\right )} \log \relax (2) - 4 \, x^{2} e^{\left (\frac {25}{2 \, \log \relax (2)} + 5\right )} \log \relax (2) + 50 \, x^{2} e^{\left (\frac {5 \, {\left (2 \, \log \relax (2) + 5\right )}}{2 \, \log \relax (2)}\right )} + 4 \, e^{\left (\frac {5 \, {\left (2 \, \log \relax (2) + 5\right )}}{2 \, \log \relax (2)}\right )} \log \relax (2)^{2} - 4 \, e^{\left (\frac {25}{2 \, \log \relax (2)} + 5\right )} \log \relax (2)^{2} - 200 \, x e^{\left (\frac {5 \, {\left (2 \, \log \relax (2) + 5\right )}}{2 \, \log \relax (2)}\right )}}{50 \, {\left (x e^{\left (-\frac {25 \, x^{2}}{2 \, {\left (x^{2} \log \relax (2) + 2 \, \log \relax (2)^{2}\right )}} + \frac {25}{\log \relax (2)}\right )} - x e^{\left (\frac {25}{2 \, \log \relax (2)}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5))*exp(25/(x^2+2*log(2)))-4*exp(5)*l
og(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5))/((4*log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8
*x^2*log(2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x, algorithm="giac")

[Out]

1/50*(x^4*e^(5/2*(2*log(2) + 5)/log(2)) - x^4*e^(25/2/log(2) + 5) + 4*x^2*e^(5/2*(2*log(2) + 5)/log(2))*log(2)
 - 4*x^2*e^(25/2/log(2) + 5)*log(2) + 50*x^2*e^(5/2*(2*log(2) + 5)/log(2)) + 4*e^(5/2*(2*log(2) + 5)/log(2))*l
og(2)^2 - 4*e^(25/2/log(2) + 5)*log(2)^2 - 200*x*e^(5/2*(2*log(2) + 5)/log(2)))/(x*e^(-25/2*x^2/(x^2*log(2) +
2*log(2)^2) + 25/log(2)) - x*e^(25/2/log(2)))

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maple [A]  time = 0.30, size = 24, normalized size = 0.89

method result size
risch \(\frac {\left (x -4\right ) {\mathrm e}^{5}}{{\mathrm e}^{\frac {25}{x^{2}+2 \ln \relax (2)}}-1}\) \(24\)
norman \(\frac {x^{3} {\mathrm e}^{5}-4 x^{2} {\mathrm e}^{5}-8 \,{\mathrm e}^{5} \ln \relax (2)+2 x \,{\mathrm e}^{5} \ln \relax (2)}{\left ({\mathrm e}^{\frac {25}{x^{2}+2 \ln \relax (2)}}-1\right ) \left (x^{2}+2 \ln \relax (2)\right )}\) \(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*exp(5)*ln(2)^2+4*x^2*exp(5)*ln(2)+(x^4+50*x^2-200*x)*exp(5))*exp(25/(x^2+2*ln(2)))-4*exp(5)*ln(2)^2-4*
x^2*exp(5)*ln(2)-x^4*exp(5))/((4*ln(2)^2+4*x^2*ln(2)+x^4)*exp(25/(x^2+2*ln(2)))^2+(-8*ln(2)^2-8*x^2*ln(2)-2*x^
4)*exp(25/(x^2+2*ln(2)))+4*ln(2)^2+4*x^2*ln(2)+x^4),x,method=_RETURNVERBOSE)

[Out]

(x-4)*exp(5)/(exp(25/(x^2+2*ln(2)))-1)

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maxima [A]  time = 0.50, size = 27, normalized size = 1.00 \begin {gather*} \frac {x e^{5} - 4 \, e^{5}}{e^{\left (\frac {25}{x^{2} + 2 \, \log \relax (2)}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5))*exp(25/(x^2+2*log(2)))-4*exp(5)*l
og(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5))/((4*log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8
*x^2*log(2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x, algorithm="maxima")

[Out]

(x*e^5 - 4*e^5)/(e^(25/(x^2 + 2*log(2))) - 1)

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mupad [B]  time = 0.56, size = 41, normalized size = 1.52 \begin {gather*} \frac {{\mathrm {e}}^5\,{\left (x^2+\ln \relax (4)\right )}^2\,\left (x-4\right )}{{\left (x^2+2\,\ln \relax (2)\right )}^2\,\left ({\mathrm {e}}^{\frac {25}{x^2+2\,\ln \relax (2)}}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*exp(5)*log(2)^2 - exp(25/(2*log(2) + x^2))*(exp(5)*(50*x^2 - 200*x + x^4) + 4*exp(5)*log(2)^2 + 4*x^2*
exp(5)*log(2)) + x^4*exp(5) + 4*x^2*exp(5)*log(2))/(exp(50/(2*log(2) + x^2))*(4*x^2*log(2) + 4*log(2)^2 + x^4)
 + 4*x^2*log(2) + 4*log(2)^2 + x^4 - exp(25/(2*log(2) + x^2))*(8*x^2*log(2) + 8*log(2)^2 + 2*x^4)),x)

[Out]

(exp(5)*(log(4) + x^2)^2*(x - 4))/((2*log(2) + x^2)^2*(exp(25/(2*log(2) + x^2)) - 1))

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sympy [A]  time = 0.20, size = 22, normalized size = 0.81 \begin {gather*} \frac {x e^{5} - 4 e^{5}}{e^{\frac {25}{x^{2} + 2 \log {\relax (2 )}}} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(5)*ln(2)**2+4*x**2*exp(5)*ln(2)+(x**4+50*x**2-200*x)*exp(5))*exp(25/(x**2+2*ln(2)))-4*exp(5)
*ln(2)**2-4*x**2*exp(5)*ln(2)-x**4*exp(5))/((4*ln(2)**2+4*x**2*ln(2)+x**4)*exp(25/(x**2+2*ln(2)))**2+(-8*ln(2)
**2-8*x**2*ln(2)-2*x**4)*exp(25/(x**2+2*ln(2)))+4*ln(2)**2+4*x**2*ln(2)+x**4),x)

[Out]

(x*exp(5) - 4*exp(5))/(exp(25/(x**2 + 2*log(2))) - 1)

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