3.66.31 \(\int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 (16 x+10 x^2+6 x^3)+e^2 (14+46 x+30 x^2+16 x^3+6 x^4)+(-26 x+2 e^4 x-2 x^2+e^2 (2+6 x+2 x^2)) \log (25 e^x x)-2 x \log ^2(25 e^x x)}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx\)

Optimal. Leaf size=24 \[ \left (x+\frac {7+2 x+\log \left (25 e^x x\right )}{e^2+x}\right )^2 \]

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Rubi [F]  time = 1.59, 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 {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-84*x - 10*x^2 + 2*E^6*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + E^4*(16*x + 10*x^2 + 6*x^3) + E^2*(14 + 46*x + 30*x^
2 + 16*x^3 + 6*x^4) + (-26*x + 2*E^4*x - 2*x^2 + E^2*(2 + 6*x + 2*x^2))*Log[25*E^x*x] - 2*x*Log[25*E^x*x]^2)/(
E^6*x + 3*E^4*x^2 + 3*E^2*x^3 + x^4),x]

[Out]

6*x + x^2 + 42/(E^2 + x)^2 - (3*E^4)/(E^2 + x)^2 + (3*E^6)/(E^2 + x)^2 - E^8/(E^2 + x)^2 - (E^4*(8 - 5*E^2 + 3
*E^4))/(E^2 + x)^2 + (7 - 23*E^2 + 15*E^4 - 8*E^6 + 3*E^8)/(E^2 + x)^2 - ((5 - E^6)*x^2)/(E^2*(E^2 + x)^2) + (
12*E^2)/(E^2 + x) - (18*E^4)/(E^2 + x) + (8*E^6)/(E^2 + x) - (2*E^4*(5 - 6*E^2))/(E^2 + x) + (2*(1 - E^(-2))*(
7 - 2*E^2))/(E^2 + x) + (2*(7 - 15*E^4 + 16*E^6 - 9*E^8))/(E^2*(E^2 + x)) + (14*Log[x])/E^4 - (2*(7 - 2*E^2)*L
og[x])/E^4 - (2*(1 + E^2 - E^4)*Log[x])/E^4 + (2*(7 - 2*E^2)*Log[25*E^x*x])/(E^2 + x)^2 + (2*(1 + E^2 - E^4)*L
og[25*E^x*x])/(E^2*(E^2 + x)) + 6*Log[E^2 + x] - 18*E^2*Log[E^2 + x] + 18*E^4*Log[E^2 + x] + (2*(7 - 2*E^2)*Lo
g[E^2 + x])/E^4 + (2*(1 - 2*E^4 + E^6)*Log[E^2 + x])/E^4 - (2*(7 - 8*E^6 + 9*E^8)*Log[E^2 + x])/E^4 + (2*Defer
[Int][Log[25*E^x*x]/x, x])/E^4 - (2*Defer[Int][Log[25*E^x*x]/(E^2 + x), x])/E^4 - 2*Defer[Int][Log[25*E^x*x]^2
/(E^2 + x)^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-84 x+\left (-10+2 e^6\right ) x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx\\ &=\int \frac {-84 x+\left (-10+2 e^6\right ) x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{x \left (e^6+3 e^4 x+3 e^2 x^2+x^3\right )} \, dx\\ &=\int \left (-\frac {84}{\left (e^2+x\right )^3}+\frac {2 \left (-5+e^6\right ) x}{\left (e^2+x\right )^3}+\frac {6 x^2}{\left (e^2+x\right )^3}+\frac {6 x^3}{\left (e^2+x\right )^3}+\frac {2 x^4}{\left (e^2+x\right )^3}+\frac {2 e^4 \left (8+5 x+3 x^2\right )}{\left (e^2+x\right )^3}+\frac {2 e^2 \left (7+23 x+15 x^2+8 x^3+3 x^4\right )}{x \left (e^2+x\right )^3}+\frac {2 \left (e^2-\left (13-3 e^2-e^4\right ) x-\left (1-e^2\right ) x^2\right ) \log \left (25 e^x x\right )}{x \left (e^2+x\right )^3}-\frac {2 \log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3}\right ) \, dx\\ &=\frac {42}{\left (e^2+x\right )^2}+2 \int \frac {x^4}{\left (e^2+x\right )^3} \, dx+2 \int \frac {\left (e^2-\left (13-3 e^2-e^4\right ) x-\left (1-e^2\right ) x^2\right ) \log \left (25 e^x x\right )}{x \left (e^2+x\right )^3} \, dx-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+6 \int \frac {x^2}{\left (e^2+x\right )^3} \, dx+6 \int \frac {x^3}{\left (e^2+x\right )^3} \, dx+\left (2 e^2\right ) \int \frac {7+23 x+15 x^2+8 x^3+3 x^4}{x \left (e^2+x\right )^3} \, dx+\left (2 e^4\right ) \int \frac {8+5 x+3 x^2}{\left (e^2+x\right )^3} \, dx-\left (2 \left (5-e^6\right )\right ) \int \frac {x}{\left (e^2+x\right )^3} \, dx\\ &=\frac {42}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+2 \int \left (-3 e^2+x+\frac {e^8}{\left (e^2+x\right )^3}-\frac {4 e^6}{\left (e^2+x\right )^2}+\frac {6 e^4}{e^2+x}\right ) \, dx-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+2 \int \left (\frac {\log \left (25 e^x x\right )}{e^4 x}+\frac {2 \left (-7+2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^3}+\frac {\left (-1-e^2+e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )^2}-\frac {\log \left (25 e^x x\right )}{e^4 \left (e^2+x\right )}\right ) \, dx+6 \int \left (\frac {e^4}{\left (e^2+x\right )^3}-\frac {2 e^2}{\left (e^2+x\right )^2}+\frac {1}{e^2+x}\right ) \, dx+6 \int \left (1-\frac {e^6}{\left (e^2+x\right )^3}+\frac {3 e^4}{\left (e^2+x\right )^2}-\frac {3 e^2}{e^2+x}\right ) \, dx+\left (2 e^2\right ) \int \left (3+\frac {7}{e^6 x}+\frac {-7+23 e^2-15 e^4+8 e^6-3 e^8}{e^2 \left (e^2+x\right )^3}+\frac {-7+15 e^4-16 e^6+9 e^8}{e^4 \left (e^2+x\right )^2}+\frac {-7+8 e^6-9 e^8}{e^6 \left (e^2+x\right )}\right ) \, dx+\left (2 e^4\right ) \int \left (\frac {8-5 e^2+3 e^4}{\left (e^2+x\right )^3}+\frac {5-6 e^2}{\left (e^2+x\right )^2}+\frac {3}{e^2+x}\right ) \, dx\\ &=6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}-\left (4 \left (7-2 e^2\right )\right ) \int \frac {\log \left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {\left (2 \left (-1-e^2+e^4\right )\right ) \int \frac {\log \left (25 e^x x\right )}{\left (e^2+x\right )^2} \, dx}{e^2}\\ &=6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^2}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}-\left (2 \left (7-2 e^2\right )\right ) \int \frac {1+x}{x \left (e^2+x\right )^2} \, dx+\frac {\left (2 \left (-1-e^2+e^4\right )\right ) \int \frac {1+x}{x \left (e^2+x\right )} \, dx}{e^2}\\ &=6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^2}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}-\left (2 \left (7-2 e^2\right )\right ) \int \left (\frac {1}{e^4 x}+\frac {-1+e^2}{e^2 \left (e^2+x\right )^2}-\frac {1}{e^4 \left (e^2+x\right )}\right ) \, dx+\frac {\left (2 \left (-1-e^2+e^4\right )\right ) \int \left (\frac {1}{e^2 x}+\frac {-1+e^2}{e^2 \left (e^2+x\right )}\right ) \, dx}{e^2}\\ &=6 x+x^2+\frac {42}{\left (e^2+x\right )^2}-\frac {3 e^4}{\left (e^2+x\right )^2}+\frac {3 e^6}{\left (e^2+x\right )^2}-\frac {e^8}{\left (e^2+x\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (e^2+x\right )^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (e^2+x\right )^2}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (e^2+x\right )^2}+\frac {12 e^2}{e^2+x}-\frac {18 e^4}{e^2+x}+\frac {8 e^6}{e^2+x}-\frac {2 e^4 \left (5-6 e^2\right )}{e^2+x}+\frac {2 \left (1-\frac {1}{e^2}\right ) \left (7-2 e^2\right )}{e^2+x}+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (e^2+x\right )}+\frac {14 \log (x)}{e^4}-\frac {2 \left (7-2 e^2\right ) \log (x)}{e^4}-\frac {2 \left (1+e^2-e^4\right ) \log (x)}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (e^2+x\right )^2}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (e^2+x\right )}+6 \log \left (e^2+x\right )-18 e^2 \log \left (e^2+x\right )+18 e^4 \log \left (e^2+x\right )+\frac {2 \left (7-2 e^2\right ) \log \left (e^2+x\right )}{e^4}+\frac {2 \left (1-2 e^4+e^6\right ) \log \left (e^2+x\right )}{e^4}-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (e^2+x\right )}{e^4}-2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^3} \, dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x} \, dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{e^2+x} \, dx}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.10, size = 113, normalized size = 4.71 \begin {gather*} \frac {49-23 e^4+6 e^6+28 x-32 e^2 x+14 e^4 x-5 x^2+14 e^2 x^2+e^4 x^2+6 x^3+2 e^2 x^3+x^4+2 \left (e^2+x\right )^2 \log (x)-2 \left (-7+e^4-2 x+e^2 x\right ) \log \left (25 e^x x\right )+\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-84*x - 10*x^2 + 2*E^6*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + E^4*(16*x + 10*x^2 + 6*x^3) + E^2*(14 + 46*x +
30*x^2 + 16*x^3 + 6*x^4) + (-26*x + 2*E^4*x - 2*x^2 + E^2*(2 + 6*x + 2*x^2))*Log[25*E^x*x] - 2*x*Log[25*E^x*x
]^2)/(E^6*x + 3*E^4*x^2 + 3*E^2*x^3 + x^4),x]

[Out]

(49 - 23*E^4 + 6*E^6 + 28*x - 32*E^2*x + 14*E^4*x - 5*x^2 + 14*E^2*x^2 + E^4*x^2 + 6*x^3 + 2*E^2*x^3 + x^4 + 2
*(E^2 + x)^2*Log[x] - 2*(-7 + E^4 - 2*x + E^2*x)*Log[25*E^x*x] + Log[25*E^x*x]^2)/(E^2 + x)^2

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fricas [B]  time = 0.90, size = 86, normalized size = 3.58 \begin {gather*} \frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 8 \, x - 18\right )} e^{4} + 2 \, {\left (x^{3} + 4 \, x^{2} - 11 \, x\right )} e^{2} + 2 \, {\left (x^{2} + x e^{2} + 2 \, x + 7\right )} \log \left (25 \, x e^{x}\right ) + \log \left (25 \, x e^{x}\right )^{2} + 28 \, x + 4 \, e^{6} + 49}{x^{2} + 2 \, x e^{2} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^2-26*x)*log(25*exp(x)*x)+2*x^2*exp(2
)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)
^3+3*x^2*exp(2)^2+3*x^3*exp(2)+x^4),x, algorithm="fricas")

[Out]

(x^4 + 4*x^3 + (x^2 + 8*x - 18)*e^4 + 2*(x^3 + 4*x^2 - 11*x)*e^2 + 2*(x^2 + x*e^2 + 2*x + 7)*log(25*x*e^x) + l
og(25*x*e^x)^2 + 28*x + 4*e^6 + 49)/(x^2 + 2*x*e^2 + e^4)

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giac [B]  time = 0.15, size = 126, normalized size = 5.25 \begin {gather*} \frac {x^{4} + 2 \, x^{3} e^{2} + 6 \, x^{3} + x^{2} e^{4} + 12 \, x^{2} e^{2} - 4 \, x e^{2} \log \relax (5) + 2 \, x^{2} \log \relax (x) + 2 \, x e^{2} \log \relax (x) + 12 \, x e^{4} - 32 \, x e^{2} + 12 \, x \log \relax (5) - 4 \, e^{4} \log \relax (5) + 4 \, \log \relax (5)^{2} + 6 \, x \log \relax (x) + 4 \, \log \relax (5) \log \relax (x) + \log \relax (x)^{2} + 42 \, x + 6 \, e^{6} - 23 \, e^{4} + 28 \, \log \relax (5) + 14 \, \log \relax (x) + 49}{x^{2} + 2 \, x e^{2} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^2-26*x)*log(25*exp(x)*x)+2*x^2*exp(2
)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)
^3+3*x^2*exp(2)^2+3*x^3*exp(2)+x^4),x, algorithm="giac")

[Out]

(x^4 + 2*x^3*e^2 + 6*x^3 + x^2*e^4 + 12*x^2*e^2 - 4*x*e^2*log(5) + 2*x^2*log(x) + 2*x*e^2*log(x) + 12*x*e^4 -
32*x*e^2 + 12*x*log(5) - 4*e^4*log(5) + 4*log(5)^2 + 6*x*log(x) + 4*log(5)*log(x) + log(x)^2 + 42*x + 6*e^6 -
23*e^4 + 28*log(5) + 14*log(x) + 49)/(x^2 + 2*x*e^2 + e^4)

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maple [C]  time = 0.44, size = 941, normalized size = 39.21

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{x}\right )^{2}}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}}-\frac {\left (i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}-14-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}+2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x -4 \ln \relax (5)-4 x -2 \ln \relax (x )\right ) \ln \left ({\mathrm e}^{x}\right )}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}}+\frac {196+112 x -88 \,{\mathrm e}^{2} x -16 x \,{\mathrm e}^{2} \ln \relax (5)+8 x^{2} \ln \relax (x )+8 x \,{\mathrm e}^{2} \ln \relax (x )+40 x \,{\mathrm e}^{4}+16 \,{\mathrm e}^{6}+112 \ln \relax (5)+16 \ln \relax (5)^{2}-72 \,{\mathrm e}^{4}+56 \ln \relax (x )+4 \ln \relax (x )^{2}+4 x^{4}+24 x^{3}+4 i \pi \,{\mathrm e}^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )-4 i \pi \,{\mathrm e}^{2} x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}-4 i \pi \,{\mathrm e}^{2} x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}-8 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )-8 i \ln \relax (5) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )-4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )+4 i \pi \,{\mathrm e}^{2} x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )-16 \,{\mathrm e}^{4} \ln \relax (5)+4 x^{2} {\mathrm e}^{4}+32 x \ln \relax (5)+8 x^{3} {\mathrm e}^{2}+48 x^{2} {\mathrm e}^{2}+16 x \ln \relax (x )+28 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+28 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+4 i \pi \,{\mathrm e}^{4} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}-8 i \ln \relax (5) \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}-4 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}-8 i \pi x \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}-\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{4}+16 \ln \relax (5) \ln \relax (x )-28 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{5}+2 \pi ^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{5}-\pi ^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{4}-\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{4}-4 i \pi \,{\mathrm e}^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}-\pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{6}-4 i \pi \,{\mathrm e}^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+4 i \pi \,{\mathrm e}^{2} x \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}+8 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+8 i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+8 i \ln \relax (5) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+8 i \ln \relax (5) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}+4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2}-28 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )}{4 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{2} x +4 x^{2}}\)	\(941\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^2-26*x)*ln(25*exp(x)*x)+2*x^2*exp(2)^3+(6*x
^3+10*x^2+16*x)*exp(2)^2+(6*x^4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)^3+3*x^2
*exp(2)^2+3*x^3*exp(2)+x^4),x,method=_RETURNVERBOSE)

[Out]

1/(exp(4)+2*exp(2)*x+x^2)*ln(exp(x))^2-(I*Pi*csgn(I*x)*csgn(I*exp(x))*csgn(I*x*exp(x))-I*Pi*csgn(I*x)*csgn(I*x
*exp(x))^2-14-I*Pi*csgn(I*exp(x))*csgn(I*x*exp(x))^2+I*Pi*csgn(I*x*exp(x))^3+2*exp(4)+2*exp(2)*x-4*ln(5)-4*x-2
*ln(x))/(exp(4)+2*exp(2)*x+x^2)*ln(exp(x))+1/4*(196+112*x-88*exp(2)*x-16*x*exp(2)*ln(5)+8*x^2*ln(x)+8*x*exp(2)
*ln(x)+40*x*exp(4)+16*exp(6)+112*ln(5)+16*ln(5)^2-72*exp(4)+56*ln(x)+4*ln(x)^2+4*x^4+24*x^3-16*exp(4)*ln(5)+4*
x^2*exp(4)+32*x*ln(5)+8*x^3*exp(2)+48*x^2*exp(2)+16*x*ln(x)+4*I*Pi*exp(4)*csgn(I*x*exp(x))^3-8*I*ln(5)*Pi*csgn
(I*x*exp(x))^3-4*I*ln(x)*Pi*csgn(I*x*exp(x))^3+28*I*Pi*csgn(I*x)*csgn(I*x*exp(x))^2+28*I*Pi*csgn(I*exp(x))*csg
n(I*x*exp(x))^2-8*I*Pi*x*csgn(I*x*exp(x))^3-Pi^2*csgn(I*x)^2*csgn(I*exp(x))^2*csgn(I*x*exp(x))^2+2*Pi^2*csgn(I
*x)^2*csgn(I*exp(x))*csgn(I*x*exp(x))^3+2*Pi^2*csgn(I*x)*csgn(I*exp(x))^2*csgn(I*x*exp(x))^3-4*Pi^2*csgn(I*x)*
csgn(I*exp(x))*csgn(I*x*exp(x))^4+2*Pi^2*csgn(I*x)*csgn(I*x*exp(x))^5+2*Pi^2*csgn(I*exp(x))*csgn(I*x*exp(x))^5
+16*ln(5)*ln(x)+4*I*Pi*exp(4)*csgn(I*x)*csgn(I*exp(x))*csgn(I*x*exp(x))-Pi^2*csgn(I*x*exp(x))^6-Pi^2*csgn(I*ex
p(x))^2*csgn(I*x*exp(x))^4-Pi^2*csgn(I*x)^2*csgn(I*x*exp(x))^4-28*I*Pi*csgn(I*x*exp(x))^3+4*I*Pi*exp(2)*x*csgn
(I*x)*csgn(I*exp(x))*csgn(I*x*exp(x))-4*I*Pi*exp(2)*x*csgn(I*x)*csgn(I*x*exp(x))^2-4*I*Pi*exp(2)*x*csgn(I*exp(
x))*csgn(I*x*exp(x))^2-8*I*Pi*x*csgn(I*x)*csgn(I*exp(x))*csgn(I*x*exp(x))-8*I*ln(5)*Pi*csgn(I*x)*csgn(I*exp(x)
)*csgn(I*x*exp(x))-4*I*ln(x)*Pi*csgn(I*x)*csgn(I*exp(x))*csgn(I*x*exp(x))-4*I*Pi*exp(4)*csgn(I*x)*csgn(I*x*exp
(x))^2+4*I*Pi*exp(2)*x*csgn(I*x*exp(x))^3+8*I*Pi*x*csgn(I*x)*csgn(I*x*exp(x))^2+8*I*Pi*x*csgn(I*exp(x))*csgn(I
*x*exp(x))^2+8*I*ln(5)*Pi*csgn(I*x)*csgn(I*x*exp(x))^2+8*I*ln(5)*Pi*csgn(I*exp(x))*csgn(I*x*exp(x))^2-28*I*Pi*
csgn(I*x)*csgn(I*exp(x))*csgn(I*x*exp(x))+4*I*ln(x)*Pi*csgn(I*x)*csgn(I*x*exp(x))^2+4*I*ln(x)*Pi*csgn(I*exp(x)
)*csgn(I*x*exp(x))^2-4*I*Pi*exp(4)*csgn(I*exp(x))*csgn(I*x*exp(x))^2)/(exp(4)+2*exp(2)*x+x^2)

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maxima [B]  time = 0.58, size = 548, normalized size = 22.83 \begin {gather*} 2 \, {\left (e^{6} - 3 \, e^{4} + 7\right )} e^{\left (-4\right )} \log \left (x + e^{2}\right ) + x^{2} + 3 \, {\left (\frac {4 \, x e^{2} + 3 \, e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} + 2 \, \log \left (x + e^{2}\right )\right )} e^{4} - 3 \, {\left (6 \, e^{2} \log \left (x + e^{2}\right ) - 2 \, x + \frac {6 \, x e^{4} + 5 \, e^{6}}{x^{2} + 2 \, x e^{2} + e^{4}}\right )} e^{2} - 7 \, {\left (2 \, e^{\left (-6\right )} \log \left (x + e^{2}\right ) - 2 \, e^{\left (-6\right )} \log \relax (x) - \frac {2 \, x + 3 \, e^{2}}{x^{2} e^{4} + 2 \, x e^{6} + e^{8}}\right )} e^{2} - 6 \, x e^{2} + 8 \, {\left (\frac {4 \, x e^{2} + 3 \, e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} + 2 \, \log \left (x + e^{2}\right )\right )} e^{2} + 12 \, e^{4} \log \left (x + e^{2}\right ) - 18 \, e^{2} \log \left (x + e^{2}\right ) + 6 \, x - \frac {{\left (2 \, x + e^{2}\right )} e^{6}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {5 \, {\left (2 \, x + e^{2}\right )} e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {15 \, {\left (2 \, x + e^{2}\right )} e^{2}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {e^{4} \log \relax (x)^{2} - 2 \, {\left ({\left (2 \, \log \relax (5) + 7\right )} e^{6} - 2 \, {\left (3 \, \log \relax (5) + 8\right )} e^{4} - e^{8} + 7 \, e^{2}\right )} x - {\left (4 \, \log \relax (5) + 9\right )} e^{8} + 2 \, {\left (2 \, \log \relax (5)^{2} + 14 \, \log \relax (5) - 7\right )} e^{4} + 2 \, {\left (x^{2} {\left (e^{4} - 7\right )} + x {\left (e^{6} + 3 \, e^{4} - 14 \, e^{2}\right )} + 2 \, e^{4} \log \relax (5)\right )} \log \relax (x) + 2 \, e^{10} + 18 \, e^{6}}{x^{2} e^{4} + 2 \, x e^{6} + e^{8}} + \frac {8 \, x e^{6} + 7 \, e^{8}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {3 \, {\left (6 \, x e^{4} + 5 \, e^{6}\right )}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {3 \, {\left (4 \, x e^{2} + 3 \, e^{4}\right )}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {5 \, {\left (2 \, x + e^{2}\right )}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {8 \, e^{4}}{x^{2} + 2 \, x e^{2} + e^{4}} - \frac {23 \, e^{2}}{x^{2} + 2 \, x e^{2} + e^{4}} + \frac {42}{x^{2} + 2 \, x e^{2} + e^{4}} + 6 \, \log \left (x + e^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^2-26*x)*log(25*exp(x)*x)+2*x^2*exp(2
)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)
^3+3*x^2*exp(2)^2+3*x^3*exp(2)+x^4),x, algorithm="maxima")

[Out]

2*(e^6 - 3*e^4 + 7)*e^(-4)*log(x + e^2) + x^2 + 3*((4*x*e^2 + 3*e^4)/(x^2 + 2*x*e^2 + e^4) + 2*log(x + e^2))*e
^4 - 3*(6*e^2*log(x + e^2) - 2*x + (6*x*e^4 + 5*e^6)/(x^2 + 2*x*e^2 + e^4))*e^2 - 7*(2*e^(-6)*log(x + e^2) - 2
*e^(-6)*log(x) - (2*x + 3*e^2)/(x^2*e^4 + 2*x*e^6 + e^8))*e^2 - 6*x*e^2 + 8*((4*x*e^2 + 3*e^4)/(x^2 + 2*x*e^2
+ e^4) + 2*log(x + e^2))*e^2 + 12*e^4*log(x + e^2) - 18*e^2*log(x + e^2) + 6*x - (2*x + e^2)*e^6/(x^2 + 2*x*e^
2 + e^4) - 5*(2*x + e^2)*e^4/(x^2 + 2*x*e^2 + e^4) - 15*(2*x + e^2)*e^2/(x^2 + 2*x*e^2 + e^4) + (e^4*log(x)^2
- 2*((2*log(5) + 7)*e^6 - 2*(3*log(5) + 8)*e^4 - e^8 + 7*e^2)*x - (4*log(5) + 9)*e^8 + 2*(2*log(5)^2 + 14*log(
5) - 7)*e^4 + 2*(x^2*(e^4 - 7) + x*(e^6 + 3*e^4 - 14*e^2) + 2*e^4*log(5))*log(x) + 2*e^10 + 18*e^6)/(x^2*e^4 +
2*x*e^6 + e^8) + (8*x*e^6 + 7*e^8)/(x^2 + 2*x*e^2 + e^4) - 3*(6*x*e^4 + 5*e^6)/(x^2 + 2*x*e^2 + e^4) + 3*(4*x
*e^2 + 3*e^4)/(x^2 + 2*x*e^2 + e^4) + 5*(2*x + e^2)/(x^2 + 2*x*e^2 + e^4) - 8*e^4/(x^2 + 2*x*e^2 + e^4) - 23*e
^2/(x^2 + 2*x*e^2 + e^4) + 42/(x^2 + 2*x*e^2 + e^4) + 6*log(x + e^2)

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mupad [B]  time = 4.96, size = 113, normalized size = 4.71 \begin {gather*} 6\,x+2\,\ln \relax (x)+\frac {{\ln \left (25\,x\,{\mathrm {e}}^x\right )}^2}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}+\frac {4\,{\mathrm {e}}^6-18\,{\mathrm {e}}^4+x\,\left (4\,{\mathrm {e}}^4-22\,{\mathrm {e}}^2+28\right )+49}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}+x^2-\frac {\ln \left (25\,x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^2+{\mathrm {e}}^4+{\mathrm {e}}^2\,\left ({\mathrm {e}}^2-2\right )+x\,\left (2\,{\mathrm {e}}^2-4\right )-14\right )}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*(16*x + 10*x^2 + 6*x^3) - 2*x*log(25*x*exp(x))^2 - log(25*x*exp(x))*(26*x - exp(2)*(6*x + 2*x^2 +
2) - 2*x*exp(4) + 2*x^2) - 84*x + 2*x^2*exp(6) + exp(2)*(46*x + 30*x^2 + 16*x^3 + 6*x^4 + 14) - 10*x^2 + 6*x^3
+ 6*x^4 + 2*x^5)/(x*exp(6) + 3*x^3*exp(2) + 3*x^2*exp(4) + x^4),x)

[Out]

6*x + 2*log(x) + log(25*x*exp(x))^2/(exp(4) + 2*x*exp(2) + x^2) + (4*exp(6) - 18*exp(4) + x*(4*exp(4) - 22*exp
(2) + 28) + 49)/(exp(4) + 2*x*exp(2) + x^2) + x^2 - (log(25*x*exp(x))*(2*exp(2) + exp(4) + exp(2)*(exp(2) - 2)
+ x*(2*exp(2) - 4) - 14))/(exp(4) + 2*x*exp(2) + x^2)

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sympy [B]  time = 1.17, size = 112, normalized size = 4.67 \begin {gather*} x^{2} + 6 x + 2 \log {\relax (x )} + \frac {x \left (- 22 e^{2} + 28 + 4 e^{4}\right ) - 18 e^{4} + 49 + 4 e^{6}}{x^{2} + 2 x e^{2} + e^{4}} + \frac {\left (- 2 x e^{2} + 4 x - 2 e^{4} + 14\right ) \log {\left (25 x e^{x} \right )}}{x^{2} + 2 x e^{2} + e^{4}} + \frac {\log {\left (25 x e^{x} \right )}^{2}}{x^{2} + 2 x e^{2} + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(25*exp(x)*x)**2+(2*x*exp(2)**2+(2*x**2+6*x+2)*exp(2)-2*x**2-26*x)*ln(25*exp(x)*x)+2*x**2*ex
p(2)**3+(6*x**3+10*x**2+16*x)*exp(2)**2+(6*x**4+16*x**3+30*x**2+46*x+14)*exp(2)+2*x**5+6*x**4+6*x**3-10*x**2-8
4*x)/(x*exp(2)**3+3*x**2*exp(2)**2+3*x**3*exp(2)+x**4),x)

[Out]

x**2 + 6*x + 2*log(x) + (x*(-22*exp(2) + 28 + 4*exp(4)) - 18*exp(4) + 49 + 4*exp(6))/(x**2 + 2*x*exp(2) + exp(
4)) + (-2*x*exp(2) + 4*x - 2*exp(4) + 14)*log(25*x*exp(x))/(x**2 + 2*x*exp(2) + exp(4)) + log(25*x*exp(x))**2/
(x**2 + 2*x*exp(2) + exp(4))

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