3.83.63 \(\int \frac {e^{\frac {1}{3} (9-x \log (\frac {-5 x^2+\log (5)-x^2 \log (-4+2 x+x^2)}{5 x+x \log (-4+2 x+x^2)}))} (100 x^2-50 x^3-25 x^4+(20-12 x-7 x^2) \log (5)+(40 x^2-20 x^3-10 x^4+(4-2 x-x^2) \log (5)) \log (-4+2 x+x^2)+(4 x^2-2 x^3-x^4) \log ^2(-4+2 x+x^2)+(100 x^2-50 x^3-25 x^4+(-20+10 x+5 x^2) \log (5)+(40 x^2-20 x^3-10 x^4+(-4+2 x+x^2) \log (5)) \log (-4+2 x+x^2)+(4 x^2-2 x^3-x^4) \log ^2(-4+2 x+x^2)) \log (\frac {-5 x^2+\log (5)-x^2 \log (-4+2 x+x^2)}{5 x+x \log (-4+2 x+x^2)}))}{-300 x^2+150 x^3+75 x^4+(60-30 x-15 x^2) \log (5)+(-120 x^2+60 x^3+30 x^4+(12-6 x-3 x^2) \log (5)) \log (-4+2 x+x^2)+(-12 x^2+6 x^3+3 x^4) \log ^2(-4+2 x+x^2)} \, dx\)

Optimal. Leaf size=33 \[ e^{3-\frac {1}{3} x \log \left (-x+\frac {\log (5)}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )} \]

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Rubi [F]  time = 63.39, 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 {\exp \left (\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )\right ) \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((9 - x*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x*Log[-4 + 2*x + x^2])])/3)*(100*x^2 - 5
0*x^3 - 25*x^4 + (20 - 12*x - 7*x^2)*Log[5] + (40*x^2 - 20*x^3 - 10*x^4 + (4 - 2*x - x^2)*Log[5])*Log[-4 + 2*x
+ x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 + 2*x + x^2]^2 + (100*x^2 - 50*x^3 - 25*x^4 + (-20 + 10*x + 5*x^2)*Log[
5] + (40*x^2 - 20*x^3 - 10*x^4 + (-4 + 2*x + x^2)*Log[5])*Log[-4 + 2*x + x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 +
2*x + x^2]^2)*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x*Log[-4 + 2*x + x^2])]))/(-300*x^2 + 15
0*x^3 + 75*x^4 + (60 - 30*x - 15*x^2)*Log[5] + (-120*x^2 + 60*x^3 + 30*x^4 + (12 - 6*x - 3*x^2)*Log[5])*Log[-4
+ 2*x + x^2] + (-12*x^2 + 6*x^3 + 3*x^4)*Log[-4 + 2*x + x^2]^2),x]

[Out]

-1/3*(E^3*Defer[Int][((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(-1/3*x), x])
+ (2*E^3*Defer[Int][1/((5 + Log[-4 + 2*x + x^2])*((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4
+ 2*x + x^2])))^(x/3)), x])/3 - (8*E^3*Defer[Int][1/((-2 + 2*Sqrt[5] - 2*x)*(5 + Log[-4 + 2*x + x^2])*((-5*x^2
+ Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)), x])/(3*Sqrt[5]) - (2*(5 - Sqrt[5])
*E^3*Defer[Int][1/((2 - 2*Sqrt[5] + 2*x)*(5 + Log[-4 + 2*x + x^2])*((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2]
)/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)), x])/15 - (8*E^3*Defer[Int][1/((2 + 2*Sqrt[5] + 2*x)*(5 + Log[-4 + 2*x
+ x^2])*((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)), x])/(3*Sqrt[5]) -
(2*(5 + Sqrt[5])*E^3*Defer[Int][1/((2 + 2*Sqrt[5] + 2*x)*(5 + Log[-4 + 2*x + x^2])*((-5*x^2 + Log[5] - x^2*Lo
g[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)), x])/15 + (2*E^3*Log[5]*Defer[Int][1/((-5*x^2 + Log[5
] - x^2*Log[-4 + 2*x + x^2])*((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)
), x])/3 - 4*E^3*Defer[Int][1/(((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/
3)*(5*x^2 - Log[5] + x^2*Log[-4 + 2*x + x^2])), x] + (16*E^3*Defer[Int][1/((-2 + 2*Sqrt[5] - 2*x)*((-5*x^2 + L
og[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)*(5*x^2 - Log[5] + x^2*Log[-4 + 2*x + x^2
])), x])/Sqrt[5] + (2*E^3*Defer[Int][x/(((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^
2])))^(x/3)*(5*x^2 - Log[5] + x^2*Log[-4 + 2*x + x^2])), x])/3 - (2*E^3*Defer[Int][x^2/(((-5*x^2 + Log[5] - x^
2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)*(5*x^2 - Log[5] + x^2*Log[-4 + 2*x + x^2])), x])/3
+ (32*(5 - Sqrt[5])*E^3*Defer[Int][1/((2 - 2*Sqrt[5] + 2*x)*((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(
5 + Log[-4 + 2*x + x^2])))^(x/3)*(5*x^2 - Log[5] + x^2*Log[-4 + 2*x + x^2])), x])/15 + (16*E^3*Defer[Int][1/((
2 + 2*Sqrt[5] + 2*x)*((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)*(5*x^2
- Log[5] + x^2*Log[-4 + 2*x + x^2])), x])/Sqrt[5] + (32*(5 + Sqrt[5])*E^3*Defer[Int][1/((2 + 2*Sqrt[5] + 2*x)*
((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3)*(5*x^2 - Log[5] + x^2*Log[-4
+ 2*x + x^2])), x])/15 - (E^3*Defer[Int][Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x
+ x^2]))]/((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^2])))^(x/3), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^3 \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3} \left (-100 x^2+50 x^3+25 x^4-\left (20-12 x-7 x^2\right ) \log (5)+\left (-4+2 x+x^2\right ) \left (10 x^2+\log (5)\right ) \log \left (-4+2 x+x^2\right )+x^2 \left (-4+2 x+x^2\right ) \log ^2\left (-4+2 x+x^2\right )+\left (-4+2 x+x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )\right )}{3 \left (4-2 x-x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )} \, dx\\ &=\frac {1}{3} e^3 \int \frac {\left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3} \left (-100 x^2+50 x^3+25 x^4-\left (20-12 x-7 x^2\right ) \log (5)+\left (-4+2 x+x^2\right ) \left (10 x^2+\log (5)\right ) \log \left (-4+2 x+x^2\right )+x^2 \left (-4+2 x+x^2\right ) \log ^2\left (-4+2 x+x^2\right )+\left (-4+2 x+x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )\right )}{\left (4-2 x-x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )} \, dx\\ &=\frac {1}{3} e^3 \int \left (\frac {100 x^2 \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3}}{\left (-4+2 x+x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )}-\frac {50 x^3 \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3}}{\left (-4+2 x+x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )}-\frac {25 x^4 \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3}}{\left (-4+2 x+x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )}-\frac {\left (-20+12 x+7 x^2\right ) \log (5) \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3}}{\left (-4+2 x+x^2\right ) \left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )}-\frac {\left (10 x^2+\log (5)\right ) \log \left (-4+2 x+x^2\right ) \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3}}{\left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )}-\frac {x^2 \log ^2\left (-4+2 x+x^2\right ) \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3}}{\left (5+\log \left (-4+2 x+x^2\right )\right ) \left (5 x^2-\log (5)+x^2 \log \left (-4+2 x+x^2\right )\right )}-\left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3} \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 49, normalized size = 1.48 \begin {gather*} e^3 \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((9 - x*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x*Log[-4 + 2*x + x^2])])/3)*(100*x
^2 - 50*x^3 - 25*x^4 + (20 - 12*x - 7*x^2)*Log[5] + (40*x^2 - 20*x^3 - 10*x^4 + (4 - 2*x - x^2)*Log[5])*Log[-4
+ 2*x + x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 + 2*x + x^2]^2 + (100*x^2 - 50*x^3 - 25*x^4 + (-20 + 10*x + 5*x^2
)*Log[5] + (40*x^2 - 20*x^3 - 10*x^4 + (-4 + 2*x + x^2)*Log[5])*Log[-4 + 2*x + x^2] + (4*x^2 - 2*x^3 - x^4)*Lo
g[-4 + 2*x + x^2]^2)*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x*Log[-4 + 2*x + x^2])]))/(-300*x^
2 + 150*x^3 + 75*x^4 + (60 - 30*x - 15*x^2)*Log[5] + (-120*x^2 + 60*x^3 + 30*x^4 + (12 - 6*x - 3*x^2)*Log[5])*
Log[-4 + 2*x + x^2] + (-12*x^2 + 6*x^3 + 3*x^4)*Log[-4 + 2*x + x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.68, size = 49, normalized size = 1.48 \begin {gather*} e^{\left (-\frac {1}{3} \, x \log \left (-\frac {x^{2} \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x^{2} - \log \relax (5)}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x}\right ) + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2*x-4)+(5*x^
2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^2)*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-
2*x^3+4*x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2*x-4)+(-7*x^2-12*x+20)*log(5
)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+
6*x^3-12*x^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)*log(x^2+2*x-4)+(-15*x^2-30*x+60)
*log(5)+75*x^4+150*x^3-300*x^2),x, algorithm="fricas")

[Out]

e^(-1/3*x*log(-(x^2*log(x^2 + 2*x - 4) + 5*x^2 - log(5))/(x*log(x^2 + 2*x - 4) + 5*x)) + 3)

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giac [B]  time = 19.05, size = 81, normalized size = 2.45 \begin {gather*} e^{\left (-\frac {1}{3} \, x \log \left (-\frac {x^{2} \log \left (x^{2} + 2 \, x - 4\right )}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x} - \frac {5 \, x^{2}}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x} + \frac {\log \relax (5)}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x}\right ) + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2*x-4)+(5*x^
2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^2)*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-
2*x^3+4*x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2*x-4)+(-7*x^2-12*x+20)*log(5
)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+
6*x^3-12*x^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)*log(x^2+2*x-4)+(-15*x^2-30*x+60)
*log(5)+75*x^4+150*x^3-300*x^2),x, algorithm="giac")

[Out]

e^(-1/3*x*log(-x^2*log(x^2 + 2*x - 4)/(x*log(x^2 + 2*x - 4) + 5*x) - 5*x^2/(x*log(x^2 + 2*x - 4) + 5*x) + log(
5)/(x*log(x^2 + 2*x - 4) + 5*x)) + 3)

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maple [C]  time = 0.99, size = 579, normalized size = 17.55

method	result	size
risch	\(x^{\frac {x}{3}} \left (\ln \left (x^{2}+2 x -4\right )+5\right )^{\frac {x}{3}} \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )^{-\frac {x}{3}} {\mathrm e}^{3+\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right )^{3}}{6}-\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (x^{2}+2 x -4\right )+5}\right )}{6}-\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right )^{2} \mathrm {csgn}\left (i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )\right )}{6}+\frac {i x \pi \,\mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \mathrm {csgn}\left (i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )\right )}{6}-\frac {i x \pi \,\mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right )^{2}}{6}+\frac {i x \pi \,\mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )}{6}+\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right )^{3}}{6}-\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\left (-\ln \left (x^{2}+2 x -4\right )-5\right ) x^{2}+\ln \relax (5)\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )}{6}}\)	\(579\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^4-2*x^3+4*x^2)*ln(x^2+2*x-4)^2+((x^2+2*x-4)*ln(5)-10*x^4-20*x^3+40*x^2)*ln(x^2+2*x-4)+(5*x^2+10*x-20
)*ln(5)-25*x^4-50*x^3+100*x^2)*ln((-x^2*ln(x^2+2*x-4)+ln(5)-5*x^2)/(x*ln(x^2+2*x-4)+5*x))+(-x^4-2*x^3+4*x^2)*l
n(x^2+2*x-4)^2+((-x^2-2*x+4)*ln(5)-10*x^4-20*x^3+40*x^2)*ln(x^2+2*x-4)+(-7*x^2-12*x+20)*ln(5)-25*x^4-50*x^3+10
0*x^2)*exp(-1/3*x*ln((-x^2*ln(x^2+2*x-4)+ln(5)-5*x^2)/(x*ln(x^2+2*x-4)+5*x))+3)/((3*x^4+6*x^3-12*x^2)*ln(x^2+2
*x-4)^2+((-3*x^2-6*x+12)*ln(5)+30*x^4+60*x^3-120*x^2)*ln(x^2+2*x-4)+(-15*x^2-30*x+60)*ln(5)+75*x^4+150*x^3-300
*x^2),x,method=_RETURNVERBOSE)

[Out]

x^(1/3*x)*(ln(x^2+2*x-4)+5)^(1/3*x)*((-ln(x^2+2*x-4)-5)*x^2+ln(5))^(-1/3*x)*exp(3+1/6*I*x*Pi*csgn(I/(ln(x^2+2*
x-4)+5)*((-ln(x^2+2*x-4)-5)*x^2+ln(5)))^3-1/6*I*x*Pi*csgn(I/(ln(x^2+2*x-4)+5)*((-ln(x^2+2*x-4)-5)*x^2+ln(5)))^
2*csgn(I/(ln(x^2+2*x-4)+5))-1/6*I*x*Pi*csgn(I/(ln(x^2+2*x-4)+5)*((-ln(x^2+2*x-4)-5)*x^2+ln(5)))^2*csgn(I*((-ln
(x^2+2*x-4)-5)*x^2+ln(5)))+1/6*I*x*Pi*csgn(I/(ln(x^2+2*x-4)+5)*((-ln(x^2+2*x-4)-5)*x^2+ln(5)))*csgn(I/(ln(x^2+
2*x-4)+5))*csgn(I*((-ln(x^2+2*x-4)-5)*x^2+ln(5)))-1/6*I*x*Pi*csgn(I/(ln(x^2+2*x-4)+5)*((-ln(x^2+2*x-4)-5)*x^2+
ln(5)))*csgn(I*((-ln(x^2+2*x-4)-5)*x^2+ln(5))/x/(ln(x^2+2*x-4)+5))^2+1/6*I*x*Pi*csgn(I/(ln(x^2+2*x-4)+5)*((-ln
(x^2+2*x-4)-5)*x^2+ln(5)))*csgn(I*((-ln(x^2+2*x-4)-5)*x^2+ln(5))/x/(ln(x^2+2*x-4)+5))*csgn(I/x)+1/6*I*x*Pi*csg
n(I*((-ln(x^2+2*x-4)-5)*x^2+ln(5))/x/(ln(x^2+2*x-4)+5))^3-1/6*I*x*Pi*csgn(I*((-ln(x^2+2*x-4)-5)*x^2+ln(5))/x/(
ln(x^2+2*x-4)+5))^2*csgn(I/x))

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maxima [A]  time = 0.68, size = 46, normalized size = 1.39 \begin {gather*} e^{\left (-\frac {1}{3} \, x \log \left (-x^{2} {\left (\log \left (x^{2} + 2 \, x - 4\right ) + 5\right )} + \log \relax (5)\right ) + \frac {1}{3} \, x \log \relax (x) + \frac {1}{3} \, x \log \left (\log \left (x^{2} + 2 \, x - 4\right ) + 5\right ) + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2*x-4)+(5*x^
2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^2)*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-
2*x^3+4*x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2*x-4)+(-7*x^2-12*x+20)*log(5
)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+
6*x^3-12*x^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)*log(x^2+2*x-4)+(-15*x^2-30*x+60)
*log(5)+75*x^4+150*x^3-300*x^2),x, algorithm="maxima")

[Out]

e^(-1/3*x*log(-x^2*(log(x^2 + 2*x - 4) + 5) + log(5)) + 1/3*x*log(x) + 1/3*x*log(log(x^2 + 2*x - 4) + 5) + 3)

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mupad [B]  time = 7.37, size = 51, normalized size = 1.55 \begin {gather*} \frac {{\mathrm {e}}^3}{{\left (-\frac {x^2\,\ln \left (x^2+2\,x-4\right )-\ln \relax (5)+5\,x^2}{5\,x+x\,\ln \left (x^2+2\,x-4\right )}\right )}^{x/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3 - (x*log(-(x^2*log(2*x + x^2 - 4) - log(5) + 5*x^2)/(5*x + x*log(2*x + x^2 - 4))))/3)*(log(5)*(12*x
+ 7*x^2 - 20) + log(2*x + x^2 - 4)*(20*x^3 - 40*x^2 + 10*x^4 + log(5)*(2*x + x^2 - 4)) - 100*x^2 + 50*x^3 + 2
5*x^4 + log(2*x + x^2 - 4)^2*(2*x^3 - 4*x^2 + x^4) - log(-(x^2*log(2*x + x^2 - 4) - log(5) + 5*x^2)/(5*x + x*l
og(2*x + x^2 - 4)))*(log(5)*(10*x + 5*x^2 - 20) + log(2*x + x^2 - 4)*(40*x^2 - 20*x^3 - 10*x^4 + log(5)*(2*x +
x^2 - 4)) + 100*x^2 - 50*x^3 - 25*x^4 - log(2*x + x^2 - 4)^2*(2*x^3 - 4*x^2 + x^4))))/(log(2*x + x^2 - 4)*(lo
g(5)*(6*x + 3*x^2 - 12) + 120*x^2 - 60*x^3 - 30*x^4) + log(5)*(30*x + 15*x^2 - 60) - log(2*x + x^2 - 4)^2*(6*x
^3 - 12*x^2 + 3*x^4) + 300*x^2 - 150*x^3 - 75*x^4),x)

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**4-2*x**3+4*x**2)*ln(x**2+2*x-4)**2+((x**2+2*x-4)*ln(5)-10*x**4-20*x**3+40*x**2)*ln(x**2+2*x-4
)+(5*x**2+10*x-20)*ln(5)-25*x**4-50*x**3+100*x**2)*ln((-x**2*ln(x**2+2*x-4)+ln(5)-5*x**2)/(x*ln(x**2+2*x-4)+5*
x))+(-x**4-2*x**3+4*x**2)*ln(x**2+2*x-4)**2+((-x**2-2*x+4)*ln(5)-10*x**4-20*x**3+40*x**2)*ln(x**2+2*x-4)+(-7*x
**2-12*x+20)*ln(5)-25*x**4-50*x**3+100*x**2)*exp(-1/3*x*ln((-x**2*ln(x**2+2*x-4)+ln(5)-5*x**2)/(x*ln(x**2+2*x-
4)+5*x))+3)/((3*x**4+6*x**3-12*x**2)*ln(x**2+2*x-4)**2+((-3*x**2-6*x+12)*ln(5)+30*x**4+60*x**3-120*x**2)*ln(x*
*2+2*x-4)+(-15*x**2-30*x+60)*ln(5)+75*x**4+150*x**3-300*x**2),x)

[Out]

Exception raised: PolynomialError

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