3.36.96 \(\int \frac {(12 x+18 x^2+e^x (12+12 x+6 x^2)+(6 x+6 e^x x) \log (x)) \log (25 e^x x^2+25 x^3+(50 e^x x+50 x^2) \log (x)+(25 e^x+25 x) \log ^2(x))+(3 e^x x+3 x^2+(3 e^x+3 x) \log (x)) \log ^2(25 e^x x^2+25 x^3+(50 e^x x+50 x^2) \log (x)+(25 e^x+25 x) \log ^2(x))}{2 e^x x+2 x^2+(2 e^x+2 x) \log (x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {3}{2} x \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \]

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Rubi [F]  time = 6.31, 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 {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((12*x + 18*x^2 + E^x*(12 + 12*x + 6*x^2) + (6*x + 6*E^x*x)*Log[x])*Log[25*E^x*x^2 + 25*x^3 + (50*E^x*x +
50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2] + (3*E^x*x + 3*x^2 + (3*E^x + 3*x)*Log[x])*Log[25*E^x*x^2 + 25*x^3
+ (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2]^2)/(2*E^x*x + 2*x^2 + (2*E^x + 2*x)*Log[x]),x]

[Out]

3*Log[25*(E^x + x)*(x + Log[x])^2]*Defer[Int][x/(E^x + x), x] - 3*Log[25*(E^x + x)*(x + Log[x])^2]*Defer[Int][
x^2/(E^x + x), x] + 6*Defer[Int][Log[25*(E^x + x)*(x + Log[x])^2]/(x + Log[x]), x] + 6*Defer[Int][(x*Log[25*(E
^x + x)*(x + Log[x])^2])/(x + Log[x]), x] + 3*Defer[Int][(x^2*Log[25*(E^x + x)*(x + Log[x])^2])/(x + Log[x]),
x] + 3*Defer[Int][(x*Log[x]*Log[25*(E^x + x)*(x + Log[x])^2])/(x + Log[x]), x] + (3*Defer[Int][Log[25*(E^x + x
)*(x + Log[x])^2]^2, x])/2 - 3*Defer[Int][Defer[Int][x/(E^x + x), x]/(E^x + x), x] + 3*Defer[Int][(x*Defer[Int
][x/(E^x + x), x])/(E^x + x), x] - 6*Defer[Int][Defer[Int][x/(E^x + x), x]/(x + Log[x]), x] - 6*Defer[Int][Def
er[Int][x/(E^x + x), x]/(x*(x + Log[x])), x] - 3*Defer[Int][(x*Defer[Int][x/(E^x + x), x])/(x + Log[x]), x] -
3*Defer[Int][(Log[x]*Defer[Int][x/(E^x + x), x])/(x + Log[x]), x] + 3*Defer[Int][Defer[Int][x^2/(E^x + x), x]/
(E^x + x), x] - 3*Defer[Int][(x*Defer[Int][x^2/(E^x + x), x])/(E^x + x), x] + 6*Defer[Int][Defer[Int][x^2/(E^x
+ x), x]/(x + Log[x]), x] + 6*Defer[Int][Defer[Int][x^2/(E^x + x), x]/(x*(x + Log[x])), x] + 3*Defer[Int][(x*
Defer[Int][x^2/(E^x + x), x])/(x + Log[x]), x] + 3*Defer[Int][(Log[x]*Defer[Int][x^2/(E^x + x), x])/(x + Log[x
]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \left (4 e^x+4 x+4 e^x x+6 x^2+2 e^x x^2+x \left (e^x+x\right ) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )+\log (x) \left (2 \left (1+e^x\right ) x+\left (e^x+x\right ) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right )\right )}{2 \left (e^x+x\right ) (x+\log (x))} \, dx\\ &=\frac {3}{2} \int \frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \left (4 e^x+4 x+4 e^x x+6 x^2+2 e^x x^2+x \left (e^x+x\right ) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )+\log (x) \left (2 \left (1+e^x\right ) x+\left (e^x+x\right ) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right )\right )}{\left (e^x+x\right ) (x+\log (x))} \, dx\\ &=\frac {3}{2} \int \left (-\frac {2 (-1+x) x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{e^x+x}+\frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \left (4+4 x+2 x^2+2 x \log (x)+x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )+\log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right )}{x+\log (x)}\right ) \, dx\\ &=\frac {3}{2} \int \frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \left (4+4 x+2 x^2+2 x \log (x)+x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )+\log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right )}{x+\log (x)} \, dx-3 \int \frac {(-1+x) x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{e^x+x} \, dx\\ &=\frac {3}{2} \int \left (\frac {2 \left (2+2 x+x^2+x \log (x)\right ) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)}+\log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \, dx+3 \int \frac {\left (x (2+3 x)+e^x \left (2+2 x+x^2\right )+\left (1+e^x\right ) x \log (x)\right ) \left (-\int \frac {x}{e^x+x} \, dx+\int \frac {x^2}{e^x+x} \, dx\right )}{x \left (e^x+x\right ) (x+\log (x))} \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ &=\frac {3}{2} \int \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \, dx+3 \int \frac {\left (2+2 x+x^2+x \log (x)\right ) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+3 \int \left (\frac {(-1+x) \left (\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\right )}{e^x+x}-\frac {\left (2+2 x+x^2+x \log (x)\right ) \left (\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\right )}{x (x+\log (x))}\right ) \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ &=\frac {3}{2} \int \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \, dx+3 \int \left (\frac {2 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)}+\frac {2 x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)}+\frac {x^2 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)}+\frac {x \log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)}\right ) \, dx+3 \int \frac {(-1+x) \left (\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\right )}{e^x+x} \, dx-3 \int \frac {\left (2+2 x+x^2+x \log (x)\right ) \left (\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\right )}{x (x+\log (x))} \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ &=\frac {3}{2} \int \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \, dx+3 \int \frac {x^2 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+3 \int \frac {x \log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+3 \int \left (-\frac {\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx}{e^x+x}+\frac {x \left (\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\right )}{e^x+x}\right ) \, dx-3 \int \left (\frac {\left (2+2 x+x^2+x \log (x)\right ) \int \frac {x}{e^x+x} \, dx}{x (x+\log (x))}-\frac {\left (2+2 x+x^2+x \log (x)\right ) \int \frac {x^2}{e^x+x} \, dx}{x (x+\log (x))}\right ) \, dx+6 \int \frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+6 \int \frac {x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ &=\frac {3}{2} \int \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \, dx+3 \int \frac {x^2 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+3 \int \frac {x \log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx-3 \int \frac {\left (2+2 x+x^2+x \log (x)\right ) \int \frac {x}{e^x+x} \, dx}{x (x+\log (x))} \, dx-3 \int \frac {\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx}{e^x+x} \, dx+3 \int \frac {x \left (\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\right )}{e^x+x} \, dx+3 \int \frac {\left (2+2 x+x^2+x \log (x)\right ) \int \frac {x^2}{e^x+x} \, dx}{x (x+\log (x))} \, dx+6 \int \frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+6 \int \frac {x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ &=\frac {3}{2} \int \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \, dx+3 \int \frac {x^2 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+3 \int \frac {x \log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx-3 \int \left (\frac {2 \int \frac {x}{e^x+x} \, dx}{x+\log (x)}+\frac {2 \int \frac {x}{e^x+x} \, dx}{x (x+\log (x))}+\frac {x \int \frac {x}{e^x+x} \, dx}{x+\log (x)}+\frac {\log (x) \int \frac {x}{e^x+x} \, dx}{x+\log (x)}\right ) \, dx-3 \int \left (\frac {\int \frac {x}{e^x+x} \, dx}{e^x+x}-\frac {\int \frac {x^2}{e^x+x} \, dx}{e^x+x}\right ) \, dx+3 \int \left (\frac {x \int \frac {x}{e^x+x} \, dx}{e^x+x}-\frac {x \int \frac {x^2}{e^x+x} \, dx}{e^x+x}\right ) \, dx+3 \int \left (\frac {2 \int \frac {x^2}{e^x+x} \, dx}{x+\log (x)}+\frac {2 \int \frac {x^2}{e^x+x} \, dx}{x (x+\log (x))}+\frac {x \int \frac {x^2}{e^x+x} \, dx}{x+\log (x)}+\frac {\log (x) \int \frac {x^2}{e^x+x} \, dx}{x+\log (x)}\right ) \, dx+6 \int \frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+6 \int \frac {x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ &=\frac {3}{2} \int \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \, dx+3 \int \frac {x^2 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+3 \int \frac {x \log (x) \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx-3 \int \frac {\int \frac {x}{e^x+x} \, dx}{e^x+x} \, dx+3 \int \frac {x \int \frac {x}{e^x+x} \, dx}{e^x+x} \, dx-3 \int \frac {x \int \frac {x}{e^x+x} \, dx}{x+\log (x)} \, dx-3 \int \frac {\log (x) \int \frac {x}{e^x+x} \, dx}{x+\log (x)} \, dx+3 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{e^x+x} \, dx-3 \int \frac {x \int \frac {x^2}{e^x+x} \, dx}{e^x+x} \, dx+3 \int \frac {x \int \frac {x^2}{e^x+x} \, dx}{x+\log (x)} \, dx+3 \int \frac {\log (x) \int \frac {x^2}{e^x+x} \, dx}{x+\log (x)} \, dx+6 \int \frac {\log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx+6 \int \frac {x \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )}{x+\log (x)} \, dx-6 \int \frac {\int \frac {x}{e^x+x} \, dx}{x+\log (x)} \, dx-6 \int \frac {\int \frac {x}{e^x+x} \, dx}{x (x+\log (x))} \, dx+6 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x+\log (x)} \, dx+6 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x (x+\log (x))} \, dx+\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x}{e^x+x} \, dx-\left (3 \log \left (25 \left (e^x+x\right ) (x+\log (x))^2\right )\right ) \int \frac {x^2}{e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((12*x + 18*x^2 + E^x*(12 + 12*x + 6*x^2) + (6*x + 6*E^x*x)*Log[x])*Log[25*E^x*x^2 + 25*x^3 + (50*E^
x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2] + (3*E^x*x + 3*x^2 + (3*E^x + 3*x)*Log[x])*Log[25*E^x*x^2 + 2
5*x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2]^2)/(2*E^x*x + 2*x^2 + (2*E^x + 2*x)*Log[x]),x]

[Out]

Integrate[((12*x + 18*x^2 + E^x*(12 + 12*x + 6*x^2) + (6*x + 6*E^x*x)*Log[x])*Log[25*E^x*x^2 + 25*x^3 + (50*E^
x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2] + (3*E^x*x + 3*x^2 + (3*E^x + 3*x)*Log[x])*Log[25*E^x*x^2 + 2
5*x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2]^2)/(2*E^x*x + 2*x^2 + (2*E^x + 2*x)*Log[x]), x]

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fricas [B]  time = 0.49, size = 41, normalized size = 1.95 \begin {gather*} \frac {3}{2} \, x \log \left (25 \, x^{3} + 25 \, x^{2} e^{x} + 25 \, {\left (x + e^{x}\right )} \log \relax (x)^{2} + 50 \, {\left (x^{2} + x e^{x}\right )} \log \relax (x)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)+3*x)*log(x)+3*exp(x)*x+3*x^2)*log((25*exp(x)+25*x)*log(x)^2+(50*exp(x)*x+50*x^2)*log(x)+
25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6*x)*log(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*log((25*exp(x)+25*x)*log(
x)^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*log(x)+2*exp(x)*x+2*x^2),x, algorithm=
"fricas")

[Out]

3/2*x*log(25*x^3 + 25*x^2*e^x + 25*(x + e^x)*log(x)^2 + 50*(x^2 + x*e^x)*log(x))^2

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giac [B]  time = 2.85, size = 48, normalized size = 2.29 \begin {gather*} \frac {3}{2} \, x \log \left (25 \, x^{3} + 25 \, x^{2} e^{x} + 50 \, x^{2} \log \relax (x) + 50 \, x e^{x} \log \relax (x) + 25 \, x \log \relax (x)^{2} + 25 \, e^{x} \log \relax (x)^{2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)+3*x)*log(x)+3*exp(x)*x+3*x^2)*log((25*exp(x)+25*x)*log(x)^2+(50*exp(x)*x+50*x^2)*log(x)+
25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6*x)*log(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*log((25*exp(x)+25*x)*log(
x)^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*log(x)+2*exp(x)*x+2*x^2),x, algorithm=
"giac")

[Out]

3/2*x*log(25*x^3 + 25*x^2*e^x + 50*x^2*log(x) + 50*x*e^x*log(x) + 25*x*log(x)^2 + 25*e^x*log(x)^2)^2

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maple [C]  time = 0.40, size = 1627, normalized size = 77.48

method	result	size
risch	\(\text {Expression too large to display}\)	\(1627\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*exp(x)+3*x)*ln(x)+3*exp(x)*x+3*x^2)*ln((25*exp(x)+25*x)*ln(x)^2+(50*exp(x)*x+50*x^2)*ln(x)+25*exp(x)*
x^2+25*x^3)^2+((6*exp(x)*x+6*x)*ln(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*ln((25*exp(x)+25*x)*ln(x)^2+(50*exp(
x)*x+50*x^2)*ln(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*ln(x)+2*exp(x)*x+2*x^2),x,method=_RETURNVERBOSE)

[Out]

12*ln(5)*x*ln(x+ln(x))+3/2*Pi^2*x*csgn(I*(x+ln(x)))*csgn(I*(x+ln(x))^2)^2*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3-3/4
*Pi^2*x*csgn(I*(x+ln(x))^2)^4*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)+3/4*Pi^2*x*csgn(I*(x+ln(x))^2)
^3*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2-3/4*Pi^2*x*csgn(I*(x+ln(x))^2)^3*csgn(I*(exp(x)+x)*(x+l
n(x))^2)^3-3/8*Pi^2*x*csgn(I*(x+ln(x))^2)^2*csgn(I*(exp(x)+x))^2*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2+3/4*Pi^2*x*c
sgn(I*(x+ln(x))^2)^2*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3+3/4*Pi^2*x*csgn(I*(x+ln(x))^2)*csgn(I
*(exp(x)+x))^2*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3-3/2*Pi^2*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x))*csgn(I*(exp(
x)+x)*(x+ln(x))^2)^4+3/4*Pi^2*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x)*(x+ln(x))^2)^5+3/4*Pi^2*x*csgn(I*(exp(x)
+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^5-3*I*ln(5)*Pi*x*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3-3*I*ln(5)*Pi*x*csgn(I*(x
+ln(x))^2)^3-3*I*Pi*x*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3*ln(x+ln(x))-3*I*Pi*x*csgn(I*(x+ln(x))^2)^3*ln(x+ln(x))+
3/4*Pi^2*x*csgn(I*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)^2*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2-3/4*Pi^2*x*csgn(I*(x+ln(
x)))^2*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3-3/2*Pi^2*x*csgn(I*(x+ln(x)))*csgn(I*(x+ln(x))^2)^3
*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2-3*I*ln(5)*Pi*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+l
n(x))^2)-3*I*Pi*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)*ln(x+ln(x))+3/4*Pi^2*x
*csgn(I*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2+3/2*Pi^2*x*csgn(I
*(x+ln(x)))*csgn(I*(x+ln(x))^2)^3*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)-3/2*Pi^2*x*csgn(I*(x+ln(x)
))*csgn(I*(x+ln(x))^2)^2*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2-3/4*Pi^2*x*csgn(I*(x+ln(x)))^2*cs
gn(I*(x+ln(x))^2)^2*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)-3*I*ln(5)*Pi*x*csgn(I*(x+ln(x)))^2*csgn(
I*(x+ln(x))^2)+6*I*ln(5)*Pi*x*csgn(I*(x+ln(x)))*csgn(I*(x+ln(x))^2)^2+3*I*ln(5)*Pi*x*csgn(I*(x+ln(x))^2)*csgn(
I*(exp(x)+x)*(x+ln(x))^2)^2+3*I*ln(5)*Pi*x*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2-3*I*Pi*x*csgn(I
*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)*ln(x+ln(x))+3*I*Pi*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2*ln
(x+ln(x))+3*I*Pi*x*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2*ln(x+ln(x))+6*I*Pi*x*csgn(I*(x+ln(x)))*
csgn(I*(x+ln(x))^2)^2*ln(x+ln(x))-3/8*Pi^2*x*csgn(I*(exp(x)+x)*(x+ln(x))^2)^6-3/8*Pi^2*x*csgn(I*(x+ln(x))^2)^6
+(6*x*ln(x+ln(x))-3/2*I*Pi*x*csgn(I*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)+3*I*Pi*x*csgn(I*(x+ln(x)))*csgn(I*(x+ln(x
))^2)^2-3/2*I*Pi*x*csgn(I*(x+ln(x))^2)^3-3/2*I*Pi*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x))*csgn(I*(exp(x)+x)*(
x+ln(x))^2)+3/2*I*Pi*x*csgn(I*(x+ln(x))^2)*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2+3/2*I*Pi*x*csgn(I*(exp(x)+x))*csgn
(I*(exp(x)+x)*(x+ln(x))^2)^2-3/2*I*Pi*x*csgn(I*(exp(x)+x)*(x+ln(x))^2)^3+6*x*ln(5))*ln(exp(x)+x)+6*x*ln(x+ln(x
))^2+3/2*x*ln(exp(x)+x)^2-3/8*Pi^2*x*csgn(I*(x+ln(x)))^4*csgn(I*(x+ln(x))^2)^2+3/2*Pi^2*x*csgn(I*(x+ln(x)))^3*
csgn(I*(x+ln(x))^2)^3-9/4*Pi^2*x*csgn(I*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)^4+3/2*Pi^2*x*csgn(I*(x+ln(x)))*csgn(I
*(x+ln(x))^2)^5+3/4*Pi^2*x*csgn(I*(x+ln(x))^2)^4*csgn(I*(exp(x)+x)*(x+ln(x))^2)^2-3/8*Pi^2*x*csgn(I*(x+ln(x))^
2)^2*csgn(I*(exp(x)+x)*(x+ln(x))^2)^4-3/8*Pi^2*x*csgn(I*(exp(x)+x))^2*csgn(I*(exp(x)+x)*(x+ln(x))^2)^4+6*x*ln(
5)^2

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maxima [B]  time = 0.61, size = 57, normalized size = 2.71 \begin {gather*} 6 \, x \log \relax (5)^{2} + \frac {3}{2} \, x \log \left (x + e^{x}\right )^{2} + 12 \, x \log \relax (5) \log \left (x + \log \relax (x)\right ) + 6 \, x \log \left (x + \log \relax (x)\right )^{2} + 6 \, {\left (x \log \relax (5) + x \log \left (x + \log \relax (x)\right )\right )} \log \left (x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)+3*x)*log(x)+3*exp(x)*x+3*x^2)*log((25*exp(x)+25*x)*log(x)^2+(50*exp(x)*x+50*x^2)*log(x)+
25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6*x)*log(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*log((25*exp(x)+25*x)*log(
x)^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*log(x)+2*exp(x)*x+2*x^2),x, algorithm=
"maxima")

[Out]

6*x*log(5)^2 + 3/2*x*log(x + e^x)^2 + 12*x*log(5)*log(x + log(x)) + 6*x*log(x + log(x))^2 + 6*(x*log(5) + x*lo
g(x + log(x)))*log(x + e^x)

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mupad [B]  time = 2.87, size = 46, normalized size = 2.19 \begin {gather*} \frac {3\,x\,{\ln \left (25\,x^2\,{\mathrm {e}}^x+{\ln \relax (x)}^2\,\left (25\,x+25\,{\mathrm {e}}^x\right )+\ln \relax (x)\,\left (50\,x\,{\mathrm {e}}^x+50\,x^2\right )+25\,x^3\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(25*x^2*exp(x) + log(x)^2*(25*x + 25*exp(x)) + log(x)*(50*x*exp(x) + 50*x^2) + 25*x^3)*(12*x + log(x)*
(6*x + 6*x*exp(x)) + exp(x)*(12*x + 6*x^2 + 12) + 18*x^2) + log(25*x^2*exp(x) + log(x)^2*(25*x + 25*exp(x)) +
log(x)*(50*x*exp(x) + 50*x^2) + 25*x^3)^2*(3*x*exp(x) + log(x)*(3*x + 3*exp(x)) + 3*x^2))/(2*x*exp(x) + log(x)
*(2*x + 2*exp(x)) + 2*x^2),x)

[Out]

(3*x*log(25*x^2*exp(x) + log(x)^2*(25*x + 25*exp(x)) + log(x)*(50*x*exp(x) + 50*x^2) + 25*x^3)^2)/2

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sympy [B]  time = 1.79, size = 49, normalized size = 2.33 \begin {gather*} \frac {3 x \log {\left (25 x^{3} + 25 x^{2} e^{x} + \left (25 x + 25 e^{x}\right ) \log {\relax (x )}^{2} + \left (50 x^{2} + 50 x e^{x}\right ) \log {\relax (x )} \right )}^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*exp(x)+3*x)*ln(x)+3*exp(x)*x+3*x**2)*ln((25*exp(x)+25*x)*ln(x)**2+(50*exp(x)*x+50*x**2)*ln(x)+2
5*exp(x)*x**2+25*x**3)**2+((6*exp(x)*x+6*x)*ln(x)+(6*x**2+12*x+12)*exp(x)+18*x**2+12*x)*ln((25*exp(x)+25*x)*ln
(x)**2+(50*exp(x)*x+50*x**2)*ln(x)+25*exp(x)*x**2+25*x**3))/((2*exp(x)+2*x)*ln(x)+2*exp(x)*x+2*x**2),x)

[Out]

3*x*log(25*x**3 + 25*x**2*exp(x) + (25*x + 25*exp(x))*log(x)**2 + (50*x**2 + 50*x*exp(x))*log(x))**2/2

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