3.13.37 \(\int \frac {(160+160 x+40 x^2+e^x (-8 x^2-4 x^3)) \log (2 x) \log (x^2)+(40+40 x+10 x^2+e^x (-2 x^2-x^3)+e^x (-4 x^2-3 x^3-x^4) \log (2 x)) \log ^2(x^2)}{4 x+4 x^2+x^3} \, dx\)

Optimal. Leaf size=26 \[ \left (10-\frac {e^x x^2}{2+x}\right ) \log (2 x) \log ^2\left (x^2\right ) \]

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Rubi [F]  time = 38.95, 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 (160+160 x+40 x^2+e^x \left (-8 x^2-4 x^3\right )\right ) \log (2 x) \log \left (x^2\right )+\left (40+40 x+10 x^2+e^x \left (-2 x^2-x^3\right )+e^x \left (-4 x^2-3 x^3-x^4\right ) \log (2 x)\right ) \log ^2\left (x^2\right )}{4 x+4 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((160 + 160*x + 40*x^2 + E^x*(-8*x^2 - 4*x^3))*Log[2*x]*Log[x^2] + (40 + 40*x + 10*x^2 + E^x*(-2*x^2 - x^3
) + E^x*(-4*x^2 - 3*x^3 - x^4)*Log[2*x])*Log[x^2]^2)/(4*x + 4*x^2 + x^3),x]

[Out]

-16*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, x] - 8*Log[-x]^2 - 16*EulerGamma*Log[x] - 16*(ExpIntegralE[1, -x
] + ExpIntegralEi[x])*Log[x] + 8*ExpIntegralEi[x]*Log[2*x] - (40*Log[2*x]^3)/3 + 4*ExpIntegralEi[x]*Log[x^2] -
4*E^x*Log[2*x]*Log[x^2] + (8*ExpIntegralEi[2 + x]*Log[2*x]*Log[x^2])/E^2 + 20*Log[2*x]^2*Log[x^2] + (5*Log[x^
2]^3)/3 - (16*Log[2*x]*Defer[Int][ExpIntegralEi[2 + x]/x, x])/E^2 - (8*Log[x^2]*Defer[Int][ExpIntegralEi[2 + x
]/x, x])/E^2 - Defer[Int][E^x*Log[x^2]^2, x] + 2*Defer[Int][(E^x*Log[x^2]^2)/(2 + x), x] + Defer[Int][E^x*Log[
2*x]*Log[x^2]^2, x] - Defer[Int][E^x*x*Log[2*x]*Log[x^2]^2, x] + 4*Defer[Int][(E^x*Log[2*x]*Log[x^2]^2)/(2 + x
)^2, x] - 4*Defer[Int][(E^x*Log[2*x]*Log[x^2]^2)/(2 + x), x] + (32*Defer[Int][Defer[Int][ExpIntegralEi[2 + x]/
x, x]/x, x])/E^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (160+160 x+40 x^2+e^x \left (-8 x^2-4 x^3\right )\right ) \log (2 x) \log \left (x^2\right )+\left (40+40 x+10 x^2+e^x \left (-2 x^2-x^3\right )+e^x \left (-4 x^2-3 x^3-x^4\right ) \log (2 x)\right ) \log ^2\left (x^2\right )}{x \left (4+4 x+x^2\right )} \, dx\\ &=\int \frac {\left (160+160 x+40 x^2+e^x \left (-8 x^2-4 x^3\right )\right ) \log (2 x) \log \left (x^2\right )+\left (40+40 x+10 x^2+e^x \left (-2 x^2-x^3\right )+e^x \left (-4 x^2-3 x^3-x^4\right ) \log (2 x)\right ) \log ^2\left (x^2\right )}{x (2+x)^2} \, dx\\ &=\int \left (\frac {160 \log (2 x) \log \left (x^2\right )}{(2+x)^2}+\frac {160 \log (2 x) \log \left (x^2\right )}{x (2+x)^2}+\frac {40 x \log (2 x) \log \left (x^2\right )}{(2+x)^2}+\frac {40 \log ^2\left (x^2\right )}{(2+x)^2}+\frac {40 \log ^2\left (x^2\right )}{x (2+x)^2}+\frac {10 x \log ^2\left (x^2\right )}{(2+x)^2}-\frac {e^x x \log \left (x^2\right ) \left (8 \log (2 x)+4 x \log (2 x)+2 \log \left (x^2\right )+x \log \left (x^2\right )+4 \log (2 x) \log \left (x^2\right )+3 x \log (2 x) \log \left (x^2\right )+x^2 \log (2 x) \log \left (x^2\right )\right )}{(2+x)^2}\right ) \, dx\\ &=10 \int \frac {x \log ^2\left (x^2\right )}{(2+x)^2} \, dx+40 \int \frac {x \log (2 x) \log \left (x^2\right )}{(2+x)^2} \, dx+40 \int \frac {\log ^2\left (x^2\right )}{(2+x)^2} \, dx+40 \int \frac {\log ^2\left (x^2\right )}{x (2+x)^2} \, dx+160 \int \frac {\log (2 x) \log \left (x^2\right )}{(2+x)^2} \, dx+160 \int \frac {\log (2 x) \log \left (x^2\right )}{x (2+x)^2} \, dx-\int \frac {e^x x \log \left (x^2\right ) \left (8 \log (2 x)+4 x \log (2 x)+2 \log \left (x^2\right )+x \log \left (x^2\right )+4 \log (2 x) \log \left (x^2\right )+3 x \log (2 x) \log \left (x^2\right )+x^2 \log (2 x) \log \left (x^2\right )\right )}{(2+x)^2} \, dx\\ &=-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {20 x \log ^2\left (x^2\right )}{2+x}+10 \int \left (-\frac {2 \log ^2\left (x^2\right )}{(2+x)^2}+\frac {\log ^2\left (x^2\right )}{2+x}\right ) \, dx-20 \int \frac {\log ^2\left (x^2\right )}{(2+x)^2} \, dx+20 \int \frac {\log ^2\left (x^2\right )}{x (2+x)} \, dx+40 \int \left (-\frac {2 \log (2 x) \log \left (x^2\right )}{(2+x)^2}+\frac {\log (2 x) \log \left (x^2\right )}{2+x}\right ) \, dx-80 \int \frac {\log \left (x^2\right )}{2+x} \, dx-160 \int -\frac {2 \log (2 x)}{x (2+x)} \, dx+160 \int \frac {\log \left (x^2\right )}{x (2+x)} \, dx+160 \int \left (\frac {\log (2 x) \log \left (x^2\right )}{4 x}-\frac {\log (2 x) \log \left (x^2\right )}{2 (2+x)^2}-\frac {\log (2 x) \log \left (x^2\right )}{4 (2+x)}\right ) \, dx-\int \frac {e^x x \log \left (x^2\right ) \left ((2+x) \log \left (x^2\right )+\log (2 x) \left (4 (2+x)+\left (4+3 x+x^2\right ) \log \left (x^2\right )\right )\right )}{(2+x)^2} \, dx\\ &=-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {10 x \log ^2\left (x^2\right )}{2+x}+10 \int \frac {\log ^2\left (x^2\right )}{x} \, dx-20 \int \frac {\log ^2\left (x^2\right )}{(2+x)^2} \, dx+40 \int \frac {\log \left (x^2\right )}{2+x} \, dx+40 \int \frac {\log (2 x) \log \left (x^2\right )}{x} \, dx+80 \int \frac {\log \left (x^2\right )}{x} \, dx-80 \int \frac {\log \left (x^2\right )}{2+x} \, dx-2 \left (80 \int \frac {\log (2 x) \log \left (x^2\right )}{(2+x)^2} \, dx\right )+160 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx+320 \int \frac {\log (2 x)}{x (2+x)} \, dx-\int \left (\frac {4 e^x x \log (2 x) \log \left (x^2\right )}{2+x}+\frac {e^x x \left (2+x+4 \log (2 x)+3 x \log (2 x)+x^2 \log (2 x)\right ) \log ^2\left (x^2\right )}{(2+x)^2}\right ) \, dx\\ &=-120 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )-160 \text {Li}_2\left (-\frac {x}{2}\right )-4 \int \frac {e^x x \log (2 x) \log \left (x^2\right )}{2+x} \, dx+5 \operatorname {Subst}\left (\int x^2 \, dx,x,\log \left (x^2\right )\right )+40 \int \frac {\log \left (x^2\right )}{2+x} \, dx-80 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx-80 \int \frac {\log ^2(2 x)}{2 x} \, dx-2 \left (-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}-80 \int -\frac {2 \log (2 x)}{x (2+x)} \, dx+80 \int \frac {\log \left (x^2\right )}{x (2+x)} \, dx\right )+160 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx+160 \int \frac {\log (2 x)}{x} \, dx-160 \int \frac {\log (2 x)}{2+x} \, dx-\int \frac {e^x x \left (2+x+4 \log (2 x)+3 x \log (2 x)+x^2 \log (2 x)\right ) \log ^2\left (x^2\right )}{(2+x)^2} \, dx\\ &=-160 \log \left (1+\frac {x}{2}\right ) \log (2 x)+80 \log ^2(2 x)-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-4 e^x \log (2 x) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {8 \text {Ei}(2+x) \log (2 x) \log \left (x^2\right )}{e^2}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )+\frac {5}{3} \log ^3\left (x^2\right )-240 \text {Li}_2\left (-\frac {x}{2}\right )+4 \int \frac {2 \left (e^x-\frac {2 \text {Ei}(2+x)}{e^2}\right ) \log (2 x)}{x} \, dx+4 \int \frac {\left (e^x-\frac {2 \text {Ei}(2+x)}{e^2}\right ) \log \left (x^2\right )}{x} \, dx-40 \int \frac {\log ^2(2 x)}{x} \, dx-80 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx+160 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx-2 \left (-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}+40 \int \frac {\log \left (x^2\right )}{x} \, dx-40 \int \frac {\log \left (x^2\right )}{2+x} \, dx+160 \int \frac {\log (2 x)}{x (2+x)} \, dx\right )-\int \frac {e^x x \left (2+x+\left (4+3 x+x^2\right ) \log (2 x)\right ) \log ^2\left (x^2\right )}{(2+x)^2} \, dx\\ &=-160 \log \left (1+\frac {x}{2}\right ) \log (2 x)+80 \log ^2(2 x)+4 \text {Ei}(x) \log \left (x^2\right )-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-4 e^x \log (2 x) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {8 \text {Ei}(2+x) \log (2 x) \log \left (x^2\right )}{e^2}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )+\frac {5}{3} \log ^3\left (x^2\right )-320 \text {Li}_2\left (-\frac {x}{2}\right )-4 \int \frac {2 \left (\text {Ei}(x)-\frac {2 \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}\right )}{x} \, dx+8 \int \frac {\left (e^x-\frac {2 \text {Ei}(2+x)}{e^2}\right ) \log (2 x)}{x} \, dx-40 \operatorname {Subst}\left (\int x^2 \, dx,x,\log (2 x)\right )-2 \left (-40 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}+10 \log ^2\left (x^2\right )+80 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx+80 \int \frac {\log (2 x)}{x} \, dx-80 \int \frac {\log (2 x)}{2+x} \, dx\right )-\frac {\left (8 \log \left (x^2\right )\right ) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\int \left (-\frac {2 e^x \left (2+x+4 \log (2 x)+3 x \log (2 x)+x^2 \log (2 x)\right ) \log ^2\left (x^2\right )}{(2+x)^2}+\frac {e^x \left (2+x+4 \log (2 x)+3 x \log (2 x)+x^2 \log (2 x)\right ) \log ^2\left (x^2\right )}{2+x}\right ) \, dx\\ &=8 \text {Ei}(x) \log (2 x)-160 \log \left (1+\frac {x}{2}\right ) \log (2 x)+80 \log ^2(2 x)-\frac {40}{3} \log ^3(2 x)+4 \text {Ei}(x) \log \left (x^2\right )-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-4 e^x \log (2 x) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {8 \text {Ei}(2+x) \log (2 x) \log \left (x^2\right )}{e^2}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )+\frac {5}{3} \log ^3\left (x^2\right )-320 \text {Li}_2\left (-\frac {x}{2}\right )+2 \int \frac {e^x \left (2+x+4 \log (2 x)+3 x \log (2 x)+x^2 \log (2 x)\right ) \log ^2\left (x^2\right )}{(2+x)^2} \, dx-2 \left (8 \int \frac {\text {Ei}(x)-\frac {2 \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}}{x} \, dx\right )-2 \left (-80 \log \left (1+\frac {x}{2}\right ) \log (2 x)+40 \log ^2(2 x)-40 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}+10 \log ^2\left (x^2\right )-80 \text {Li}_2\left (-\frac {x}{2}\right )+80 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx\right )-\frac {(16 \log (2 x)) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\frac {\left (8 \log \left (x^2\right )\right ) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\int \frac {e^x \left (2+x+4 \log (2 x)+3 x \log (2 x)+x^2 \log (2 x)\right ) \log ^2\left (x^2\right )}{2+x} \, dx\\ &=8 \text {Ei}(x) \log (2 x)-160 \log \left (1+\frac {x}{2}\right ) \log (2 x)+80 \log ^2(2 x)-\frac {40}{3} \log ^3(2 x)+4 \text {Ei}(x) \log \left (x^2\right )-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-4 e^x \log (2 x) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {8 \text {Ei}(2+x) \log (2 x) \log \left (x^2\right )}{e^2}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )+\frac {5}{3} \log ^3\left (x^2\right )-2 \left (-80 \log \left (1+\frac {x}{2}\right ) \log (2 x)+40 \log ^2(2 x)-40 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}+10 \log ^2\left (x^2\right )-160 \text {Li}_2\left (-\frac {x}{2}\right )\right )-320 \text {Li}_2\left (-\frac {x}{2}\right )+2 \int \frac {e^x \left (2+x+\left (4+3 x+x^2\right ) \log (2 x)\right ) \log ^2\left (x^2\right )}{(2+x)^2} \, dx-2 \left (8 \int \left (\frac {\text {Ei}(x)}{x}-\frac {2 \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2 x}\right ) \, dx\right )-\frac {(16 \log (2 x)) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\frac {\left (8 \log \left (x^2\right )\right ) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\int \frac {e^x \left (2+x+\left (4+3 x+x^2\right ) \log (2 x)\right ) \log ^2\left (x^2\right )}{2+x} \, dx\\ &=8 \text {Ei}(x) \log (2 x)-160 \log \left (1+\frac {x}{2}\right ) \log (2 x)+80 \log ^2(2 x)-\frac {40}{3} \log ^3(2 x)+4 \text {Ei}(x) \log \left (x^2\right )-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-4 e^x \log (2 x) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {8 \text {Ei}(2+x) \log (2 x) \log \left (x^2\right )}{e^2}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )+\frac {5}{3} \log ^3\left (x^2\right )-2 \left (-80 \log \left (1+\frac {x}{2}\right ) \log (2 x)+40 \log ^2(2 x)-40 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}+10 \log ^2\left (x^2\right )-160 \text {Li}_2\left (-\frac {x}{2}\right )\right )-320 \text {Li}_2\left (-\frac {x}{2}\right )+2 \int \left (\frac {2 e^x \log ^2\left (x^2\right )}{(2+x)^2}+\frac {e^x x \log ^2\left (x^2\right )}{(2+x)^2}+\frac {4 e^x \log (2 x) \log ^2\left (x^2\right )}{(2+x)^2}+\frac {3 e^x x \log (2 x) \log ^2\left (x^2\right )}{(2+x)^2}+\frac {e^x x^2 \log (2 x) \log ^2\left (x^2\right )}{(2+x)^2}\right ) \, dx-2 \left (8 \int \frac {\text {Ei}(x)}{x} \, dx-\frac {16 \int \frac {\int \frac {\text {Ei}(2+x)}{x} \, dx}{x} \, dx}{e^2}\right )-\frac {(16 \log (2 x)) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\frac {\left (8 \log \left (x^2\right )\right ) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\int \left (\frac {2 e^x \log ^2\left (x^2\right )}{2+x}+\frac {e^x x \log ^2\left (x^2\right )}{2+x}+\frac {4 e^x \log (2 x) \log ^2\left (x^2\right )}{2+x}+\frac {3 e^x x \log (2 x) \log ^2\left (x^2\right )}{2+x}+\frac {e^x x^2 \log (2 x) \log ^2\left (x^2\right )}{2+x}\right ) \, dx\\ &=8 \text {Ei}(x) \log (2 x)-160 \log \left (1+\frac {x}{2}\right ) \log (2 x)+80 \log ^2(2 x)-\frac {40}{3} \log ^3(2 x)+4 \text {Ei}(x) \log \left (x^2\right )-80 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-4 e^x \log (2 x) \log \left (x^2\right )-\frac {160 \log (2 x) \log \left (x^2\right )}{2+x}+\frac {8 \text {Ei}(2+x) \log (2 x) \log \left (x^2\right )}{e^2}+20 \log ^2(2 x) \log \left (x^2\right )+20 \log ^2\left (x^2\right )+\frac {5}{3} \log ^3\left (x^2\right )-2 \left (-80 \log \left (1+\frac {x}{2}\right ) \log (2 x)+40 \log ^2(2 x)-40 \log \left (1+\frac {x}{2}\right ) \log \left (x^2\right )-\frac {80 \log (2 x) \log \left (x^2\right )}{2+x}+10 \log ^2\left (x^2\right )-160 \text {Li}_2\left (-\frac {x}{2}\right )\right )-320 \text {Li}_2\left (-\frac {x}{2}\right )+2 \int \frac {e^x x \log ^2\left (x^2\right )}{(2+x)^2} \, dx-2 \int \frac {e^x \log ^2\left (x^2\right )}{2+x} \, dx+2 \int \frac {e^x x^2 \log (2 x) \log ^2\left (x^2\right )}{(2+x)^2} \, dx-3 \int \frac {e^x x \log (2 x) \log ^2\left (x^2\right )}{2+x} \, dx+4 \int \frac {e^x \log ^2\left (x^2\right )}{(2+x)^2} \, dx-4 \int \frac {e^x \log (2 x) \log ^2\left (x^2\right )}{2+x} \, dx+6 \int \frac {e^x x \log (2 x) \log ^2\left (x^2\right )}{(2+x)^2} \, dx+8 \int \frac {e^x \log (2 x) \log ^2\left (x^2\right )}{(2+x)^2} \, dx-2 \left (8 (E_1(-x)+\text {Ei}(x)) \log (x)-8 \int \frac {E_1(-x)}{x} \, dx-\frac {16 \int \frac {\int \frac {\text {Ei}(2+x)}{x} \, dx}{x} \, dx}{e^2}\right )-\frac {(16 \log (2 x)) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\frac {\left (8 \log \left (x^2\right )\right ) \int \frac {\text {Ei}(2+x)}{x} \, dx}{e^2}-\int \frac {e^x x \log ^2\left (x^2\right )}{2+x} \, dx-\int \frac {e^x x^2 \log (2 x) \log ^2\left (x^2\right )}{2+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.84, size = 29, normalized size = 1.12 \begin {gather*} -\frac {\left (-20-10 x+e^x x^2\right ) \log (2 x) \log ^2\left (x^2\right )}{2+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((160 + 160*x + 40*x^2 + E^x*(-8*x^2 - 4*x^3))*Log[2*x]*Log[x^2] + (40 + 40*x + 10*x^2 + E^x*(-2*x^2
- x^3) + E^x*(-4*x^2 - 3*x^3 - x^4)*Log[2*x])*Log[x^2]^2)/(4*x + 4*x^2 + x^3),x]

[Out]

-(((-20 - 10*x + E^x*x^2)*Log[2*x]*Log[x^2]^2)/(2 + x))

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fricas [B]  time = 2.23, size = 75, normalized size = 2.88 \begin {gather*} -\frac {4 \, {\left ({\left (x^{2} e^{x} - 10 \, x - 20\right )} \log \left (2 \, x\right )^{3} - 2 \, {\left (x^{2} e^{x} \log \relax (2) - 10 \, {\left (x + 2\right )} \log \relax (2)\right )} \log \left (2 \, x\right )^{2} + {\left (x^{2} e^{x} \log \relax (2)^{2} - 10 \, {\left (x + 2\right )} \log \relax (2)^{2}\right )} \log \left (2 \, x\right )\right )}}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-3*x^3-4*x^2)*exp(x)*log(2*x)+(-x^3-2*x^2)*exp(x)+10*x^2+40*x+40)*log(x^2)^2+((-4*x^3-8*x^2)*
exp(x)+40*x^2+160*x+160)*log(2*x)*log(x^2))/(x^3+4*x^2+4*x),x, algorithm="fricas")

[Out]

-4*((x^2*e^x - 10*x - 20)*log(2*x)^3 - 2*(x^2*e^x*log(2) - 10*(x + 2)*log(2))*log(2*x)^2 + (x^2*e^x*log(2)^2 -
10*(x + 2)*log(2)^2)*log(2*x))/(x + 2)

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giac [B]  time = 0.44, size = 60, normalized size = 2.31 \begin {gather*} -\frac {4 \, {\left (x^{2} e^{x} \log \relax (2) \log \relax (x)^{2} + x^{2} e^{x} \log \relax (x)^{3} - 10 \, x \log \relax (2) \log \relax (x)^{2} - 10 \, x \log \relax (x)^{3} - 20 \, \log \relax (2) \log \relax (x)^{2} - 20 \, \log \relax (x)^{3}\right )}}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-3*x^3-4*x^2)*exp(x)*log(2*x)+(-x^3-2*x^2)*exp(x)+10*x^2+40*x+40)*log(x^2)^2+((-4*x^3-8*x^2)*
exp(x)+40*x^2+160*x+160)*log(2*x)*log(x^2))/(x^3+4*x^2+4*x),x, algorithm="giac")

[Out]

-4*(x^2*e^x*log(2)*log(x)^2 + x^2*e^x*log(x)^3 - 10*x*log(2)*log(x)^2 - 10*x*log(x)^3 - 20*log(2)*log(x)^2 - 2
0*log(x)^3)/(x + 2)

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maple [C]  time = 0.60, size = 806, normalized size = 31.00

method	result	size
risch	\(-\frac {4 \left ({\mathrm e}^{x} x^{2}-10 x -20\right ) \ln \relax (x )^{3}}{2+x}-\frac {2 \left (-i \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}+10 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+10 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-40 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+20 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-20 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}+20 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}+2 x^{2} \ln \relax (2) {\mathrm e}^{x}-20 x \ln \relax (2)-40 \ln \relax (2)\right ) \ln \relax (x )^{2}}{2+x}+\frac {\pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (\mathrm {csgn}\left (i x^{2}\right ) \pi \mathrm {csgn}\left (i x \right )^{4}-4 \mathrm {csgn}\left (i x^{2}\right )^{2} \pi \mathrm {csgn}\left (i x \right )^{3}+6 \mathrm {csgn}\left (i x^{2}\right )^{3} \pi \mathrm {csgn}\left (i x \right )^{2}-4 \mathrm {csgn}\left (i x^{2}\right )^{4} \pi \,\mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right )^{5} \pi -16 i \ln \relax (2) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )+8 i \ln \relax (2) \mathrm {csgn}\left (i x \right )^{2}+8 i \ln \relax (2) \mathrm {csgn}\left (i x^{2}\right )^{2}\right ) x^{2} {\mathrm e}^{x} \ln \relax (x )}{4 x +8}+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-160 \ln \relax (x ) \ln \relax (2) \mathrm {csgn}\left (i x \right )^{2} x -160 \ln \relax (x ) \ln \relax (2) \mathrm {csgn}\left (i x^{2}\right )^{2} x +640 \ln \relax (x ) \ln \relax (2) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )+120 i \pi \ln \relax (x ) x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{3}+240 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \ln \relax (2) x^{2} \mathrm {csgn}\left (i x^{2}\right )^{5} {\mathrm e}^{x}+8 i \pi \ln \relax (2) x^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}+40 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{5}-320 \ln \relax (x ) \ln \relax (2) \mathrm {csgn}\left (i x \right )^{2}-320 \ln \relax (x ) \ln \relax (2) \mathrm {csgn}\left (i x^{2}\right )^{2}-2 i \pi \ln \relax (2) x^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}-160 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{4}+20 i \pi \ln \relax (x ) x \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )-12 i \pi \ln \relax (2) x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}+20 i \pi \ln \relax (x ) x \mathrm {csgn}\left (i x^{2}\right )^{5}-80 i \pi \ln \relax (x ) x \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{2}+320 \ln \relax (x ) \ln \relax (2) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) x +8 i \pi \ln \relax (2) x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{4} {\mathrm e}^{x}+40 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )-160 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{2}-80 i \pi \ln \relax (x ) x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{4}\right )}{8 x +16}\)	\(806\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^4-3*x^3-4*x^2)*exp(x)*ln(2*x)+(-x^3-2*x^2)*exp(x)+10*x^2+40*x+40)*ln(x^2)^2+((-4*x^3-8*x^2)*exp(x)+4
0*x^2+160*x+160)*ln(2*x)*ln(x^2))/(x^3+4*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

-4*(exp(x)*x^2-10*x-20)/(2+x)*ln(x)^3-2*(-I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)*exp(x)+10*I*Pi*x*csgn(I*x^2)^3+10*I
*Pi*x*csgn(I*x)^2*csgn(I*x^2)-40*I*Pi*csgn(I*x)*csgn(I*x^2)^2+20*I*Pi*csgn(I*x^2)^3-20*I*Pi*x*csgn(I*x)*csgn(I
*x^2)^2-I*Pi*x^2*csgn(I*x^2)^3*exp(x)+20*I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2*exp(x
)+2*x^2*ln(2)*exp(x)-20*x*ln(2)-40*ln(2))/(2+x)*ln(x)^2+1/4*Pi*csgn(I*x^2)*(csgn(I*x^2)*Pi*csgn(I*x)^4-4*csgn(
I*x^2)^2*Pi*csgn(I*x)^3+6*csgn(I*x^2)^3*Pi*csgn(I*x)^2-4*csgn(I*x^2)^4*Pi*csgn(I*x)+csgn(I*x^2)^5*Pi-16*I*ln(2
)*csgn(I*x)*csgn(I*x^2)+8*I*ln(2)*csgn(I*x)^2+8*I*ln(2)*csgn(I*x^2)^2)*x^2/(2+x)*exp(x)*ln(x)+1/8*I*Pi*csgn(I*
x^2)*(-160*ln(x)*ln(2)*csgn(I*x)^2*x-160*ln(x)*ln(2)*csgn(I*x^2)^2*x+640*ln(x)*ln(2)*csgn(I*x)*csgn(I*x^2)+120
*I*Pi*ln(x)*x*csgn(I*x)^2*csgn(I*x^2)^3+240*I*Pi*ln(x)*csgn(I*x)^2*csgn(I*x^2)^3-2*I*Pi*ln(2)*x^2*csgn(I*x^2)^
5*exp(x)+8*I*Pi*ln(2)*x^2*csgn(I*x)^3*csgn(I*x^2)^2*exp(x)+40*I*Pi*ln(x)*csgn(I*x^2)^5-320*ln(x)*ln(2)*csgn(I*
x)^2-320*ln(x)*ln(2)*csgn(I*x^2)^2-2*I*Pi*ln(2)*x^2*csgn(I*x)^4*csgn(I*x^2)*exp(x)-160*I*Pi*ln(x)*csgn(I*x)*cs
gn(I*x^2)^4+20*I*Pi*ln(x)*x*csgn(I*x)^4*csgn(I*x^2)-12*I*Pi*ln(2)*x^2*csgn(I*x)^2*csgn(I*x^2)^3*exp(x)+20*I*Pi
*ln(x)*x*csgn(I*x^2)^5-80*I*Pi*ln(x)*x*csgn(I*x)^3*csgn(I*x^2)^2+320*ln(x)*ln(2)*csgn(I*x)*csgn(I*x^2)*x+8*I*P
i*ln(2)*x^2*csgn(I*x)*csgn(I*x^2)^4*exp(x)+40*I*Pi*ln(x)*csgn(I*x)^4*csgn(I*x^2)-160*I*Pi*ln(x)*csgn(I*x)^3*cs
gn(I*x^2)^2-80*I*Pi*ln(x)*x*csgn(I*x)*csgn(I*x^2)^4)/(2+x)

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maxima [B]  time = 0.54, size = 55, normalized size = 2.12 \begin {gather*} \frac {4 \, {\left (10 \, {\left (x + 2\right )} \log \relax (x)^{3} + 10 \, {\left (x \log \relax (2) + 2 \, \log \relax (2)\right )} \log \relax (x)^{2} - {\left (x^{2} \log \relax (2) \log \relax (x)^{2} + x^{2} \log \relax (x)^{3}\right )} e^{x}\right )}}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^4-3*x^3-4*x^2)*exp(x)*log(2*x)+(-x^3-2*x^2)*exp(x)+10*x^2+40*x+40)*log(x^2)^2+((-4*x^3-8*x^2)*
exp(x)+40*x^2+160*x+160)*log(2*x)*log(x^2))/(x^3+4*x^2+4*x),x, algorithm="maxima")

[Out]

4*(10*(x + 2)*log(x)^3 + 10*(x*log(2) + 2*log(2))*log(x)^2 - (x^2*log(2)*log(x)^2 + x^2*log(x)^3)*e^x)/(x + 2)

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mupad [B]  time = 1.15, size = 87, normalized size = 3.35 \begin {gather*} 40\,\ln \left (x^2\right )\,{\ln \relax (2)}^2-80\,{\ln \relax (2)}^2\,\ln \relax (x)-40\,\ln \relax (2)\,{\ln \relax (x)}^2+10\,{\ln \left (x^2\right )}^2\,\ln \relax (x)+40\,\ln \left (x^2\right )\,\ln \relax (2)\,\ln \relax (x)-\frac {x^2\,{\ln \left (x^2\right )}^2\,{\mathrm {e}}^x\,\ln \relax (2)}{x+2}-\frac {x^2\,{\ln \left (x^2\right )}^2\,{\mathrm {e}}^x\,\ln \relax (x)}{x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2)^2*(40*x - exp(x)*(2*x^2 + x^3) + 10*x^2 - log(2*x)*exp(x)*(4*x^2 + 3*x^3 + x^4) + 40) + log(2*x)
*log(x^2)*(160*x - exp(x)*(8*x^2 + 4*x^3) + 40*x^2 + 160))/(4*x + 4*x^2 + x^3),x)

[Out]

40*log(x^2)*log(2)^2 - 80*log(2)^2*log(x) - 40*log(2)*log(x)^2 + 10*log(x^2)^2*log(x) + 40*log(x^2)*log(2)*log
(x) - (x^2*log(x^2)^2*exp(x)*log(2))/(x + 2) - (x^2*log(x^2)^2*exp(x)*log(x))/(x + 2)

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sympy [B]  time = 0.52, size = 78, normalized size = 3.00 \begin {gather*} 40 \log {\relax (2 )}^{2} \log {\relax (x )} + 40 \log {\left (2 x \right )}^{3} - 80 \log {\relax (2 )} \log {\left (2 x \right )}^{2} + \frac {\left (- 4 x^{2} \log {\left (2 x \right )}^{3} + 8 x^{2} \log {\relax (2 )} \log {\left (2 x \right )}^{2} - 4 x^{2} \log {\relax (2 )}^{2} \log {\left (2 x \right )}\right ) e^{x}}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**4-3*x**3-4*x**2)*exp(x)*ln(2*x)+(-x**3-2*x**2)*exp(x)+10*x**2+40*x+40)*ln(x**2)**2+((-4*x**3-
8*x**2)*exp(x)+40*x**2+160*x+160)*ln(2*x)*ln(x**2))/(x**3+4*x**2+4*x),x)

[Out]

40*log(2)**2*log(x) + 40*log(2*x)**3 - 80*log(2)*log(2*x)**2 + (-4*x**2*log(2*x)**3 + 8*x**2*log(2)*log(2*x)**
2 - 4*x**2*log(2)**2*log(2*x))*exp(x)/(x + 2)

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