\(\int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+(147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2) \log (x)+(e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2) \log (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2)}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx\) [5074]

Optimal result

Integrand size = 207, antiderivative size = 30 \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx=(-2+\log (x)) \log \left (x \left (e^x+(x-4 x (2-i \pi -\log (3)))^2\right )\right ) \]

[Out]

ln((exp(x)+(x-4*x*(2-ln(3)-I*Pi))^2)*x)*(ln(x)-2)

Rubi [F]

\[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx=\int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx \]

[In]

Int[(E^x*(-2 - 2*x) - 294*x^2 + 336*x^2*(I*Pi + Log[3]) - 96*x^2*(I*Pi + Log[3])^2 + (147*x^2 + E^x*(1 + x) -
168*x^2*(I*Pi + Log[3]) + 48*x^2*(I*Pi + Log[3])^2)*Log[x] + (E^x + 49*x^2 - 56*x^2*(I*Pi + Log[3]) + 16*x^2*(
I*Pi + Log[3])^2)*Log[E^x*x + 49*x^3 - 56*x^3*(I*Pi + Log[3]) + 16*x^3*(I*Pi + Log[3])^2])/(E^x*x + 49*x^3 - 5
6*x^3*(I*Pi + Log[3]) + 16*x^3*(I*Pi + Log[3])^2),x]

[Out]

-3*x + (2 - Log[x])^2/2 + x*Log[x] + 2*(7*I + 4*Pi - (4*I)*Log[3])^2*Log[x]*Defer[Int][x/(-E^x - 49*x^2*(1 + (
-16*Pi^2 + 8*Log[3]*(-7 + Log[9]) + (8*I)*Pi*(-7 + Log[81]))/49)), x] + 2*(7*I + 4*Pi - (4*I)*Log[3])^2*Defer[
Int][x^2/(-E^x - 49*x^2*(1 + (-16*Pi^2 + 8*Log[3]*(-7 + Log[9]) + (8*I)*Pi*(-7 + Log[81]))/49)), x] + 4*(7*I +
4*Pi - (4*I)*Log[3])^2*Defer[Int][x/(E^x + 49*x^2*(1 + (-16*Pi^2 + 8*Log[3]*(-7 + Log[9]) + (8*I)*Pi*(-7 + Lo
g[81]))/49)), x] + (7*I + 4*Pi - (4*I)*Log[3])^2*Log[x]*Defer[Int][x^2/(E^x + 49*x^2*(1 + (-16*Pi^2 + 8*Log[3]
*(-7 + Log[9]) + (8*I)*Pi*(-7 + Log[81]))/49)), x] + Defer[Int][Log[E^x*x - x^3*(7*I + 4*Pi - (4*I)*Log[3])^2]
/x, x] - 2*(7*I + 4*Pi - (4*I)*Log[3])^2*Defer[Int][Defer[Int][-(x/(E^x + x^2*(49 - 16*Pi^2 + 8*Log[3]*(-7 + L
og[9]) + (8*I)*Pi*(-7 + Log[81])))), x]/x, x] - (7*I + 4*Pi - (4*I)*Log[3])^2*Defer[Int][Defer[Int][x^2/(E^x +
x^2*(49 - 16*Pi^2 + 8*Log[3]*(-7 + Log[9]) + (8*I)*Pi*(-7 + Log[81]))), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+16 x^3 (i \pi +\log (3))^2+x^3 (49-56 (i \pi +\log (3)))} \, dx \\ & = \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+x^3 \left (49-56 (i \pi +\log (3))+16 (i \pi +\log (3))^2\right )} \, dx \\ & = \int \frac {e^x (-2-2 x)-96 x^2 (i \pi +\log (3))^2+x^2 (-294+336 (i \pi +\log (3)))+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+x^3 \left (49-56 (i \pi +\log (3))+16 (i \pi +\log (3))^2\right )} \, dx \\ & = \int \frac {e^x (-2-2 x)+x^2 \left (-294+336 (i \pi +\log (3))-96 (i \pi +\log (3))^2\right )+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+x^3 \left (49-56 (i \pi +\log (3))+16 (i \pi +\log (3))^2\right )} \, dx \\ & = \int \frac {-2 e^x (1+x)+6 x^2 (7 i+4 \pi -4 i \log (3))^2+\left (e^x (1+x)-3 x^2 (7 i+4 \pi -4 i \log (3))^2\right ) \log (x)+\left (e^x-x^2 (7 i+4 \pi -4 i \log (3))^2\right ) \log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2} \, dx \\ & = \int \left (\frac {(2-x) x (7 i+4 \pi -4 i \log (3))^2 (2-\log (x))}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}+\frac {-2-2 x+\log (x)+x \log (x)+\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x}\right ) \, dx \\ & = (7 i+4 \pi -4 i \log (3))^2 \int \frac {(2-x) x (2-\log (x))}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\int \frac {-2-2 x+\log (x)+x \log (x)+\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x} \, dx \\ & = (7 i+4 \pi -4 i \log (3))^2 \int \left (\frac {2 x (2-\log (x))}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}+\frac {x^2 (-2+\log (x))}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}\right ) \, dx+\int \left (\frac {(1+x) (-2+\log (x))}{x}+\frac {\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x}\right ) \, dx \\ & = (7 i+4 \pi -4 i \log (3))^2 \int \frac {x^2 (-2+\log (x))}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x (2-\log (x))}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\int \frac {(1+x) (-2+\log (x))}{x} \, dx+\int \frac {\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x} \, dx \\ & = (7 i+4 \pi -4 i \log (3))^2 \int \left (\frac {2 x^2}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}+\frac {x^2 \log (x)}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}\right ) \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \left (\frac {2 x}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}+\frac {x \log (x)}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )}\right ) \, dx+\int (-2+\log (x)) \, dx+\int \frac {-2+\log (x)}{x} \, dx+\int \frac {\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x} \, dx \\ & = -2 x+\frac {1}{2} (2-\log (x))^2+(7 i+4 \pi -4 i \log (3))^2 \int \frac {x^2 \log (x)}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x^2}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x \log (x)}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (4 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\int \log (x) \, dx+\int \frac {\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x} \, dx \\ & = -3 x+\frac {1}{2} (2-\log (x))^2+x \log (x)-(7 i+4 \pi -4 i \log (3))^2 \int \frac {\int \frac {x^2}{e^x+x^2 \left (49-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )} \, dx}{x} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x^2}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx-\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {\int -\frac {x}{e^x+x^2 \left (49-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )} \, dx}{x} \, dx+\left (4 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left ((7 i+4 \pi -4 i \log (3))^2 \log (x)\right ) \int \frac {x^2}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2 \log (x)\right ) \int \frac {x}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\int \frac {\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x} \, dx \\ & = -3 x+\frac {1}{2} (2-\log (x))^2+x \log (x)-(7 i+4 \pi -4 i \log (3))^2 \int \frac {\int \frac {x^2}{e^x+x^2 \left (49-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )} \, dx}{x} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x^2}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {\int \frac {x}{e^x+x^2 \left (49-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )} \, dx}{x} \, dx+\left (4 (7 i+4 \pi -4 i \log (3))^2\right ) \int \frac {x}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left ((7 i+4 \pi -4 i \log (3))^2 \log (x)\right ) \int \frac {x^2}{e^x+49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\left (2 (7 i+4 \pi -4 i \log (3))^2 \log (x)\right ) \int \frac {x}{-e^x-49 x^2 \left (1+\frac {1}{49} \left (-16 \pi ^2+8 \log (3) (-7+\log (9))+8 i \pi (-7+\log (81))\right )\right )} \, dx+\int \frac {\log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )}{x} \, dx \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(30)=60\).

Time = 0.22 (sec) , antiderivative size = 216, normalized size of antiderivative = 7.20 \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx=2 i \arctan \left (\frac {8 \pi x^2 (-7+4 \log (3))}{-e^x-49 x^2+16 \pi ^2 x^2+56 x^2 \log (3)-16 x^2 \log ^2(3)}\right )-2 \log (x)+\log (x) \log \left (e^x x-x^3 (7 i+4 \pi -4 i \log (3))^2\right )-\log \left (e^{2 x}+98 e^x x^2-32 e^x \pi ^2 x^2+2401 x^4+1568 \pi ^2 x^4+256 \pi ^4 x^4-112 e^x x^2 \log (3)-5488 x^4 \log (3)-1792 \pi ^2 x^4 \log (3)+32 e^x x^2 \log ^2(3)+4704 x^4 \log ^2(3)+512 \pi ^2 x^4 \log ^2(3)-1792 x^4 \log ^3(3)+256 x^4 \log ^4(3)\right ) \]

[In]

Integrate[(E^x*(-2 - 2*x) - 294*x^2 + 336*x^2*(I*Pi + Log[3]) - 96*x^2*(I*Pi + Log[3])^2 + (147*x^2 + E^x*(1 +
x) - 168*x^2*(I*Pi + Log[3]) + 48*x^2*(I*Pi + Log[3])^2)*Log[x] + (E^x + 49*x^2 - 56*x^2*(I*Pi + Log[3]) + 16
*x^2*(I*Pi + Log[3])^2)*Log[E^x*x + 49*x^3 - 56*x^3*(I*Pi + Log[3]) + 16*x^3*(I*Pi + Log[3])^2])/(E^x*x + 49*x
^3 - 56*x^3*(I*Pi + Log[3]) + 16*x^3*(I*Pi + Log[3])^2),x]

[Out]

(2*I)*ArcTan[(8*Pi*x^2*(-7 + 4*Log[3]))/(-E^x - 49*x^2 + 16*Pi^2*x^2 + 56*x^2*Log[3] - 16*x^2*Log[3]^2)] - 2*L
og[x] + Log[x]*Log[E^x*x - x^3*(7*I + 4*Pi - (4*I)*Log[3])^2] - Log[E^(2*x) + 98*E^x*x^2 - 32*E^x*Pi^2*x^2 + 2
401*x^4 + 1568*Pi^2*x^4 + 256*Pi^4*x^4 - 112*E^x*x^2*Log[3] - 5488*x^4*Log[3] - 1792*Pi^2*x^4*Log[3] + 32*E^x*
x^2*Log[3]^2 + 4704*x^4*Log[3]^2 + 512*Pi^2*x^4*Log[3]^2 - 1792*x^4*Log[3]^3 + 256*x^4*Log[3]^4]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.43 (sec) , antiderivative size = 476, normalized size of antiderivative = 15.87

method	result	size
risch	\(\ln \left (x \right ) \ln \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )+\ln \left (x \right )^{2}-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )\right ) \operatorname {csgn}\left (i x \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )\right )}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )\right )}^{2}}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )\right ) {\operatorname {csgn}\left (i x \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )\right )}^{2}}{2}-\frac {i \pi \ln \left (x \right ) {\operatorname {csgn}\left (i x \left (x^{2} \ln \left (3\right )^{2}+\left (2 i \pi \,x^{2}-\frac {7}{2} x^{2}\right ) \ln \left (3\right )-\pi ^{2} x^{2}-\frac {7 i \pi \,x^{2}}{2}+\frac {49 x^{2}}{16}+\frac {{\mathrm e}^{x}}{16}\right )\right )}^{3}}{2}-2 \ln \left (x \right )-2 \ln \left (-16 \pi ^{2} x^{2}+32 i \ln \left (3\right ) \pi \,x^{2}+16 x^{2} \ln \left (3\right )^{2}-56 i \pi \,x^{2}-56 x^{2} \ln \left (3\right )+49 x^{2}+{\mathrm e}^{x}\right )\)	\(476\)

[In]

int(((exp(x)+16*x^2*(ln(3)+I*Pi)^2-56*x^2*(ln(3)+I*Pi)+49*x^2)*ln(exp(x)*x+16*x^3*(ln(3)+I*Pi)^2-56*x^3*(ln(3)
+I*Pi)+49*x^3)+((1+x)*exp(x)+48*x^2*(ln(3)+I*Pi)^2-168*x^2*(ln(3)+I*Pi)+147*x^2)*ln(x)+(-2-2*x)*exp(x)-96*x^2*
(ln(3)+I*Pi)^2+336*x^2*(ln(3)+I*Pi)-294*x^2)/(exp(x)*x+16*x^3*(ln(3)+I*Pi)^2-56*x^3*(ln(3)+I*Pi)+49*x^3),x,met
hod=_RETURNVERBOSE)

[Out]

ln(x)*ln(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x))+ln(x)^2-1/2*I*Pi*
ln(x)*csgn(I*x)*csgn(I*(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x)))*c
sgn(I*x*(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x)))+1/2*I*Pi*ln(x)*c
sgn(I*x)*csgn(I*x*(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x)))^2+1/2*
I*Pi*ln(x)*csgn(I*(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x)))*csgn(I
*x*(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x)))^2-1/2*I*Pi*ln(x)*csgn
(I*x*(x^2*ln(3)^2+(2*I*Pi*x^2-7/2*x^2)*ln(3)-Pi^2*x^2-7/2*I*Pi*x^2+49/16*x^2+1/16*exp(x)))^3-2*ln(x)-2*ln(-16*
Pi^2*x^2+32*I*ln(3)*Pi*x^2+16*x^2*ln(3)^2-56*I*Pi*x^2-56*x^2*ln(3)+49*x^2+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx={\left (\log \left (x\right ) - 2\right )} \log \left (-8 \, {\left (-4 i \, \pi + 7\right )} x^{3} \log \left (3\right ) + 16 \, x^{3} \log \left (3\right )^{2} + {\left (-56 i \, \pi - 16 \, \pi ^{2} + 49\right )} x^{3} + x e^{x}\right ) \]

[In]

integrate(((exp(x)+16*x^2*(log(3)+I*pi)^2-56*x^2*(log(3)+I*pi)+49*x^2)*log(exp(x)*x+16*x^3*(log(3)+I*pi)^2-56*
x^3*(log(3)+I*pi)+49*x^3)+((1+x)*exp(x)+48*x^2*(log(3)+I*pi)^2-168*x^2*(log(3)+I*pi)+147*x^2)*log(x)+(-2-2*x)*
exp(x)-96*x^2*(log(3)+I*pi)^2+336*x^2*(log(3)+I*pi)-294*x^2)/(exp(x)*x+16*x^3*(log(3)+I*pi)^2-56*x^3*(log(3)+I
*pi)+49*x^3),x, algorithm="fricas")

[Out]

(log(x) - 2)*log(-8*(-4*I*pi + 7)*x^3*log(3) + 16*x^3*log(3)^2 + (-56*I*pi - 16*pi^2 + 49)*x^3 + x*e^x)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx=\text {Timed out} \]

[In]

integrate(((exp(x)+16*x**2*(ln(3)+I*pi)**2-56*x**2*(ln(3)+I*pi)+49*x**2)*ln(exp(x)*x+16*x**3*(ln(3)+I*pi)**2-5
6*x**3*(ln(3)+I*pi)+49*x**3)+((1+x)*exp(x)+48*x**2*(ln(3)+I*pi)**2-168*x**2*(ln(3)+I*pi)+147*x**2)*ln(x)+(-2-2
*x)*exp(x)-96*x**2*(ln(3)+I*pi)**2+336*x**2*(ln(3)+I*pi)-294*x**2)/(exp(x)*x+16*x**3*(ln(3)+I*pi)**2-56*x**3*(
ln(3)+I*pi)+49*x**3),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx={\left (\log \left (x\right ) - 2\right )} \log \left ({\left (-56 i \, \pi - 16 \, \pi ^{2} - 8 \, {\left (-4 i \, \pi + 7\right )} \log \left (3\right ) + 16 \, \log \left (3\right )^{2} + 49\right )} x^{2} + e^{x}\right ) + \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \]

[In]

integrate(((exp(x)+16*x^2*(log(3)+I*pi)^2-56*x^2*(log(3)+I*pi)+49*x^2)*log(exp(x)*x+16*x^3*(log(3)+I*pi)^2-56*
x^3*(log(3)+I*pi)+49*x^3)+((1+x)*exp(x)+48*x^2*(log(3)+I*pi)^2-168*x^2*(log(3)+I*pi)+147*x^2)*log(x)+(-2-2*x)*
exp(x)-96*x^2*(log(3)+I*pi)^2+336*x^2*(log(3)+I*pi)-294*x^2)/(exp(x)*x+16*x^3*(log(3)+I*pi)^2-56*x^3*(log(3)+I
*pi)+49*x^3),x, algorithm="maxima")

[Out]

(log(x) - 2)*log((-56*I*pi - 16*pi^2 - 8*(-4*I*pi + 7)*log(3) + 16*log(3)^2 + 49)*x^2 + e^x) + log(x)^2 - 2*lo
g(x)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (29) = 58\).

Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.60 \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx=\log \left (-16 \, \pi ^{2} x^{2} + 32 i \, \pi x^{2} \log \left (3\right ) + 16 \, x^{2} \log \left (3\right )^{2} - 56 i \, \pi x^{2} - 56 \, x^{2} \log \left (3\right ) + 49 \, x^{2} + e^{x}\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \log \left (-16 \, \pi ^{2} x^{2} + 32 i \, \pi x^{2} \log \left (3\right ) + 16 \, x^{2} \log \left (3\right )^{2} - 56 i \, \pi x^{2} - 56 \, x^{2} \log \left (3\right ) + 49 \, x^{2} + e^{x}\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate(((exp(x)+16*x^2*(log(3)+I*pi)^2-56*x^2*(log(3)+I*pi)+49*x^2)*log(exp(x)*x+16*x^3*(log(3)+I*pi)^2-56*
x^3*(log(3)+I*pi)+49*x^3)+((1+x)*exp(x)+48*x^2*(log(3)+I*pi)^2-168*x^2*(log(3)+I*pi)+147*x^2)*log(x)+(-2-2*x)*
exp(x)-96*x^2*(log(3)+I*pi)^2+336*x^2*(log(3)+I*pi)-294*x^2)/(exp(x)*x+16*x^3*(log(3)+I*pi)^2-56*x^3*(log(3)+I
*pi)+49*x^3),x, algorithm="giac")

[Out]

log(-16*pi^2*x^2 + 32*I*pi*x^2*log(3) + 16*x^2*log(3)^2 - 56*I*pi*x^2 - 56*x^2*log(3) + 49*x^2 + e^x)*log(x) +
log(x)^2 - 2*log(-16*pi^2*x^2 + 32*I*pi*x^2*log(3) + 16*x^2*log(3)^2 - 56*I*pi*x^2 - 56*x^2*log(3) + 49*x^2 +
e^x) - 2*log(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^x (-2-2 x)-294 x^2+336 x^2 (i \pi +\log (3))-96 x^2 (i \pi +\log (3))^2+\left (147 x^2+e^x (1+x)-168 x^2 (i \pi +\log (3))+48 x^2 (i \pi +\log (3))^2\right ) \log (x)+\left (e^x+49 x^2-56 x^2 (i \pi +\log (3))+16 x^2 (i \pi +\log (3))^2\right ) \log \left (e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2\right )}{e^x x+49 x^3-56 x^3 (i \pi +\log (3))+16 x^3 (i \pi +\log (3))^2} \, dx=\int -\frac {96\,x^2\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2-\ln \left (16\,x^3\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2-56\,x^3\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x\,{\mathrm {e}}^x+49\,x^3\right )\,\left ({\mathrm {e}}^x+16\,x^2\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2-56\,x^2\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+49\,x^2\right )+{\mathrm {e}}^x\,\left (2\,x+2\right )-336\,x^2\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )-\ln \left (x\right )\,\left (48\,x^2\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2+{\mathrm {e}}^x\,\left (x+1\right )-168\,x^2\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+147\,x^2\right )+294\,x^2}{16\,x^3\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2-56\,x^3\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x\,{\mathrm {e}}^x+49\,x^3} \,d x \]

[In]

int(-(96*x^2*(Pi*1i + log(3))^2 - log(16*x^3*(Pi*1i + log(3))^2 - 56*x^3*(Pi*1i + log(3)) + x*exp(x) + 49*x^3)
*(exp(x) + 16*x^2*(Pi*1i + log(3))^2 - 56*x^2*(Pi*1i + log(3)) + 49*x^2) + exp(x)*(2*x + 2) - 336*x^2*(Pi*1i +
log(3)) - log(x)*(48*x^2*(Pi*1i + log(3))^2 + exp(x)*(x + 1) - 168*x^2*(Pi*1i + log(3)) + 147*x^2) + 294*x^2)
/(16*x^3*(Pi*1i + log(3))^2 - 56*x^3*(Pi*1i + log(3)) + x*exp(x) + 49*x^3),x)

[Out]

int(-(96*x^2*(Pi*1i + log(3))^2 - log(16*x^3*(Pi*1i + log(3))^2 - 56*x^3*(Pi*1i + log(3)) + x*exp(x) + 49*x^3)
*(exp(x) + 16*x^2*(Pi*1i + log(3))^2 - 56*x^2*(Pi*1i + log(3)) + 49*x^2) + exp(x)*(2*x + 2) - 336*x^2*(Pi*1i +
log(3)) - log(x)*(48*x^2*(Pi*1i + log(3))^2 + exp(x)*(x + 1) - 168*x^2*(Pi*1i + log(3)) + 147*x^2) + 294*x^2)
/(16*x^3*(Pi*1i + log(3))^2 - 56*x^3*(Pi*1i + log(3)) + x*exp(x) + 49*x^3), x)