3.92.24 \(\int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log (x^2)+e^{e^x} (2+4 x+4 x^2+2 x^3+(1+2 x+e^x (x+2 x^2+2 x^3+x^4)) \log (x^2))}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac {\left (5+e^{e^x}\right ) x \log \left (x^2\right )}{3+\frac {3 x^2}{1+x}} \]

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Rubi [F]  time = 4.68, 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 {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10 + 20*x + 20*x^2 + 10*x^3 + (5 + 10*x)*Log[x^2] + E^E^x*(2 + 4*x + 4*x^2 + 2*x^3 + (1 + 2*x + E^x*(x +
2*x^2 + 2*x^3 + x^4))*Log[x^2]))/(3 + 6*x + 9*x^2 + 6*x^3 + 3*x^4),x]

[Out]

(10*x)/9 - (20*(2 + x))/(9*(1 + x + x^2)) - (20*x*(2 + x))/(9*(1 + x + x^2)) - (10*x^2*(2 + x))/(9*(1 + x + x^
2)) + (10*(1 + 2*x))/(9*(1 + x + x^2)) + (10*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) - (20*Log[I + Sqrt[3] + (2
*I)*x])/9 + (20*Log[I + Sqrt[3] + (2*I)*x])/(9*(1 - I*Sqrt[3])) + (E^E^x*Log[x^2])/3 + (20*x*Log[x^2])/(9*(1 -
I*Sqrt[3] + 2*x)) - (20*x*Log[x^2])/(9*(1 - I*Sqrt[3])*(1 - I*Sqrt[3] + 2*x)) + (20*x*Log[x^2])/(9*(1 + I*Sqr
t[3] + 2*x)) - (20*x*Log[x^2])/(9*(1 + I*Sqrt[3])*(1 + I*Sqrt[3] + 2*x)) - (20*Log[1 + I*Sqrt[3] + 2*x])/9 + (
20*Log[1 + I*Sqrt[3] + 2*x])/(9*(1 + I*Sqrt[3])) + (5*Log[1 + x + x^2])/3 - (4*Log[x^2]*Defer[Int][E^E^x/(-1 +
I*Sqrt[3] - 2*x)^2, x])/9 + (4*(1 - I*Sqrt[3])*Log[x^2]*Defer[Int][E^E^x/(-1 + I*Sqrt[3] - 2*x)^2, x])/9 + ((
(8*I)/3)*Defer[Int][E^E^x/(-1 + I*Sqrt[3] - 2*x), x])/Sqrt[3] - (((2*I)/3)*Log[x^2]*Defer[Int][E^(E^x + x)/(-1
+ I*Sqrt[3] - 2*x), x])/Sqrt[3] - (2*Defer[Int][E^E^x/x, x])/3 + (2*(1 + I*Sqrt[3])*Defer[Int][E^E^x/(1 - I*S
qrt[3] + 2*x), x])/3 - (4*Log[x^2]*Defer[Int][E^E^x/(1 + I*Sqrt[3] + 2*x)^2, x])/9 + (4*(1 + I*Sqrt[3])*Log[x^
2]*Defer[Int][E^E^x/(1 + I*Sqrt[3] + 2*x)^2, x])/9 + (((8*I)/3)*Defer[Int][E^E^x/(1 + I*Sqrt[3] + 2*x), x])/Sq
rt[3] + (2*(1 - I*Sqrt[3])*Defer[Int][E^E^x/(1 + I*Sqrt[3] + 2*x), x])/3 - (((2*I)/3)*Log[x^2]*Defer[Int][E^(E
^x + x)/(1 + I*Sqrt[3] + 2*x), x])/Sqrt[3] + (((4*I)/3)*Defer[Int][Defer[Int][E^(E^x + x)/(-1 + I*Sqrt[3] - 2*
x), x]/x, x])/Sqrt[3] + (8*Defer[Int][Defer[Int][-(E^E^x/(I + Sqrt[3] + (2*I)*x)^2), x]/x, x])/9 - (8*(1 - I*S
qrt[3])*Defer[Int][Defer[Int][-(E^E^x/(I + Sqrt[3] + (2*I)*x)^2), x]/x, x])/9 + (8*Defer[Int][Defer[Int][E^E^x
/(1 + I*Sqrt[3] + 2*x)^2, x]/x, x])/9 - (8*(1 + I*Sqrt[3])*Defer[Int][Defer[Int][E^E^x/(1 + I*Sqrt[3] + 2*x)^2
, x]/x, x])/9 + (((4*I)/3)*Defer[Int][Defer[Int][E^(E^x + x)/(1 + I*Sqrt[3] + 2*x), x]/x, x])/Sqrt[3]

Rubi steps

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

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Mathematica [A]  time = 5.13, size = 34, normalized size = 1.26 \begin {gather*} \frac {1}{3} \left (10 \log (x)+\frac {\left (-5+e^{e^x} x (1+x)\right ) \log \left (x^2\right )}{1+x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10 + 20*x + 20*x^2 + 10*x^3 + (5 + 10*x)*Log[x^2] + E^E^x*(2 + 4*x + 4*x^2 + 2*x^3 + (1 + 2*x + E^x
*(x + 2*x^2 + 2*x^3 + x^4))*Log[x^2]))/(3 + 6*x + 9*x^2 + 6*x^3 + 3*x^4),x]

[Out]

(10*Log[x] + ((-5 + E^E^x*x*(1 + x))*Log[x^2])/(1 + x + x^2))/3

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fricas [A]  time = 0.76, size = 35, normalized size = 1.30 \begin {gather*} \frac {{\left (x^{2} + x\right )} e^{\left (e^{x}\right )} \log \left (x^{2}\right ) + 5 \, {\left (x^{2} + x\right )} \log \left (x^{2}\right )}{3 \, {\left (x^{2} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^4+2*x^3+2*x^2+x)*exp(x)+2*x+1)*log(x^2)+2*x^3+4*x^2+4*x+2)*exp(exp(x))+(10*x+5)*log(x^2)+10*x^
3+20*x^2+20*x+10)/(3*x^4+6*x^3+9*x^2+6*x+3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {10 \, x^{3} + 20 \, x^{2} + {\left (2 \, x^{3} + 4 \, x^{2} + {\left ({\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + x\right )} e^{x} + 2 \, x + 1\right )} \log \left (x^{2}\right ) + 4 \, x + 2\right )} e^{\left (e^{x}\right )} + 5 \, {\left (2 \, x + 1\right )} \log \left (x^{2}\right ) + 20 \, x + 10}{3 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^4+2*x^3+2*x^2+x)*exp(x)+2*x+1)*log(x^2)+2*x^3+4*x^2+4*x+2)*exp(exp(x))+(10*x+5)*log(x^2)+10*x^
3+20*x^2+20*x+10)/(3*x^4+6*x^3+9*x^2+6*x+3),x, algorithm="giac")

[Out]

integrate(1/3*(10*x^3 + 20*x^2 + (2*x^3 + 4*x^2 + ((x^4 + 2*x^3 + 2*x^2 + x)*e^x + 2*x + 1)*log(x^2) + 4*x + 2
)*e^(e^x) + 5*(2*x + 1)*log(x^2) + 20*x + 10)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1), x)

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maple [C]  time = 0.24, size = 215, normalized size = 7.96

method	result	size
risch	\(-\frac {10 \ln \relax (x )}{3 \left (x^{2}+x +1\right )}+\frac {\frac {5 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}}{6}-\frac {5 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{3}+\frac {5 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{6}+\frac {10 x^{2} \ln \relax (x )}{3}+\frac {10 x \ln \relax (x )}{3}+\frac {10 \ln \relax (x )}{3}}{x^{2}+x +1}+\frac {x \left (-i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 x \ln \relax (x )+4 \ln \relax (x )\right ) {\mathrm e}^{{\mathrm e}^{x}}}{6 x^{2}+6 x +6}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x^4+2*x^3+2*x^2+x)*exp(x)+2*x+1)*ln(x^2)+2*x^3+4*x^2+4*x+2)*exp(exp(x))+(10*x+5)*ln(x^2)+10*x^3+20*x^2
+20*x+10)/(3*x^4+6*x^3+9*x^2+6*x+3),x,method=_RETURNVERBOSE)

[Out]

-10/3/(x^2+x+1)*ln(x)+5/6*(I*Pi*csgn(I*x^2)*csgn(I*x)^2-2*I*Pi*csgn(I*x^2)^2*csgn(I*x)+I*Pi*csgn(I*x^2)^3+4*x^
2*ln(x)+4*x*ln(x)+4*ln(x))/(x^2+x+1)+1/6*x*(-I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2-I
*Pi*x*csgn(I*x^2)^3-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I*x^2)^3+4*x*ln(x)+4
*ln(x))/(x^2+x+1)*exp(exp(x))

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maxima [B]  time = 0.50, size = 73, normalized size = 2.70 \begin {gather*} \frac {2 \, {\left ({\left (x^{2} + x\right )} e^{\left (e^{x}\right )} \log \relax (x) + 5 \, {\left (x^{2} + x\right )} \log \relax (x)\right )}}{3 \, {\left (x^{2} + x + 1\right )}} + \frac {20 \, {\left (2 \, x + 1\right )}}{9 \, {\left (x^{2} + x + 1\right )}} - \frac {20 \, {\left (x + 2\right )}}{9 \, {\left (x^{2} + x + 1\right )}} - \frac {20 \, {\left (x - 1\right )}}{9 \, {\left (x^{2} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x^4+2*x^3+2*x^2+x)*exp(x)+2*x+1)*log(x^2)+2*x^3+4*x^2+4*x+2)*exp(exp(x))+(10*x+5)*log(x^2)+10*x^
3+20*x^2+20*x+10)/(3*x^4+6*x^3+9*x^2+6*x+3),x, algorithm="maxima")

[Out]

2/3*((x^2 + x)*e^(e^x)*log(x) + 5*(x^2 + x)*log(x))/(x^2 + x + 1) + 20/9*(2*x + 1)/(x^2 + x + 1) - 20/9*(x + 2
)/(x^2 + x + 1) - 20/9*(x - 1)/(x^2 + x + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {20\,x+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x+\ln \left (x^2\right )\,\left (2\,x+{\mathrm {e}}^x\,\left (x^4+2\,x^3+2\,x^2+x\right )+1\right )+4\,x^2+2\,x^3+2\right )+20\,x^2+10\,x^3+\ln \left (x^2\right )\,\left (10\,x+5\right )+10}{3\,x^4+6\,x^3+9\,x^2+6\,x+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x + exp(exp(x))*(4*x + log(x^2)*(2*x + exp(x)*(x + 2*x^2 + 2*x^3 + x^4) + 1) + 4*x^2 + 2*x^3 + 2) + 20
*x^2 + 10*x^3 + log(x^2)*(10*x + 5) + 10)/(6*x + 9*x^2 + 6*x^3 + 3*x^4 + 3),x)

[Out]

int((20*x + exp(exp(x))*(4*x + log(x^2)*(2*x + exp(x)*(x + 2*x^2 + 2*x^3 + x^4) + 1) + 4*x^2 + 2*x^3 + 2) + 20
*x^2 + 10*x^3 + log(x^2)*(10*x + 5) + 10)/(6*x + 9*x^2 + 6*x^3 + 3*x^4 + 3), x)

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sympy [B]  time = 0.48, size = 53, normalized size = 1.96 \begin {gather*} \frac {\left (x^{2} \log {\left (x^{2} \right )} + x \log {\left (x^{2} \right )}\right ) e^{e^{x}}}{3 x^{2} + 3 x + 3} + \frac {10 \log {\relax (x )}}{3} - \frac {5 \log {\left (x^{2} \right )}}{3 x^{2} + 3 x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x**4+2*x**3+2*x**2+x)*exp(x)+2*x+1)*ln(x**2)+2*x**3+4*x**2+4*x+2)*exp(exp(x))+(10*x+5)*ln(x**2)+
10*x**3+20*x**2+20*x+10)/(3*x**4+6*x**3+9*x**2+6*x+3),x)

[Out]

(x**2*log(x**2) + x*log(x**2))*exp(exp(x))/(3*x**2 + 3*x + 3) + 10*log(x)/3 - 5*log(x**2)/(3*x**2 + 3*x + 3)

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