\(\int \frac {(-4-6 x^2+3 x^4) \log ^2(x)+e^{\frac {8 x}{\log (x)}} (-8 x+8 x \log (x)-\log ^2(x))+e^{\frac {4 x}{\log (x)}} (24 x-8 x^3+(-24 x+8 x^3) \log (x)+(6+2 x^2) \log ^2(x))}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} (-6 x+2 x^3) \log ^2(x)+(4 x-6 x^3+x^5) \log ^2(x)} \, dx\) [6587]

Optimal result

Integrand size = 145, antiderivative size = 36 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=3+\log \left (\frac {5-x^2 \left (-\frac {3-e^{\frac {4 x}{\log (x)}}}{x}+x\right )^2}{x}\right ) \]

[Out]

ln((-(x-(3-exp(4*x/ln(x)))/x)^2*x^2+5)/x)+3

Rubi [F]

\[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=\int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx \]

[In]

Int[((-4 - 6*x^2 + 3*x^4)*Log[x]^2 + E^((8*x)/Log[x])*(-8*x + 8*x*Log[x] - Log[x]^2) + E^((4*x)/Log[x])*(24*x
- 8*x^3 + (-24*x + 8*x^3)*Log[x] + (6 + 2*x^2)*Log[x]^2))/(E^((8*x)/Log[x])*x*Log[x]^2 + E^((4*x)/Log[x])*(-6*
x + 2*x^3)*Log[x]^2 + (4*x - 6*x^3 + x^5)*Log[x]^2),x]

[Out]

(8*x)/Log[x] - Log[x] - 12*Defer[Int][x/(4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x]
)*x^2 + x^4), x] + 4*Defer[Int][(E^((4*x)/Log[x])*x)/(4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^
((4*x)/Log[x])*x^2 + x^4), x] + 4*Defer[Int][x^3/(4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*
x)/Log[x])*x^2 + x^4), x] + 32*Defer[Int][1/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/L
og[x])*x^2 + x^4)*Log[x]^2), x] - 24*Defer[Int][E^((4*x)/Log[x])/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) -
6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x]^2), x] - 48*Defer[Int][x^2/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/L
og[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x]^2), x] + 8*Defer[Int][(E^((4*x)/Log[x])*x^2)/((4 - 6*E^(
(4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x]^2), x] + 8*Defer[Int][x^4/((4
- 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x]^2), x] - 32*Defer[Int][
1/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x]), x] + 24*Defer[I
nt][E^((4*x)/Log[x])/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x
]), x] + 48*Defer[Int][x^2/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4)
*Log[x]), x] - 8*Defer[Int][(E^((4*x)/Log[x])*x^2)/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^(
(4*x)/Log[x])*x^2 + x^4)*Log[x]), x] - 8*Defer[Int][x^4/((4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 +
2*E^((4*x)/Log[x])*x^2 + x^4)*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{x \left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx \\ & = \int \left (\frac {-8 x+8 x \log (x)-\log ^2(x)}{x \log ^2(x)}-\frac {4 \left (-8+6 e^{\frac {4 x}{\log (x)}}+12 x^2-2 e^{\frac {4 x}{\log (x)}} x^2-2 x^4+8 \log (x)-6 e^{\frac {4 x}{\log (x)}} \log (x)-12 x^2 \log (x)+2 e^{\frac {4 x}{\log (x)}} x^2 \log (x)+2 x^4 \log (x)+3 x \log ^2(x)-e^{\frac {4 x}{\log (x)}} x \log ^2(x)-x^3 \log ^2(x)\right )}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)}\right ) \, dx \\ & = -\left (4 \int \frac {-8+6 e^{\frac {4 x}{\log (x)}}+12 x^2-2 e^{\frac {4 x}{\log (x)}} x^2-2 x^4+8 \log (x)-6 e^{\frac {4 x}{\log (x)}} \log (x)-12 x^2 \log (x)+2 e^{\frac {4 x}{\log (x)}} x^2 \log (x)+2 x^4 \log (x)+3 x \log ^2(x)-e^{\frac {4 x}{\log (x)}} x \log ^2(x)-x^3 \log ^2(x)}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx\right )+\int \frac {-8 x+8 x \log (x)-\log ^2(x)}{x \log ^2(x)} \, dx \\ & = -\left (4 \int \frac {-2 \left (4-6 x^2+x^4+e^{\frac {4 x}{\log (x)}} \left (-3+x^2\right )\right )+2 \left (4-6 x^2+x^4+e^{\frac {4 x}{\log (x)}} \left (-3+x^2\right )\right ) \log (x)-x \left (-3+e^{\frac {4 x}{\log (x)}}+x^2\right ) \log ^2(x)}{\left (4+e^{\frac {8 x}{\log (x)}}-6 x^2+x^4+2 e^{\frac {4 x}{\log (x)}} \left (-3+x^2\right )\right ) \log ^2(x)} \, dx\right )+\int \left (-\frac {1}{x}-\frac {8}{\log ^2(x)}+\frac {8}{\log (x)}\right ) \, dx \\ & = -\log (x)-4 \int \left (\frac {3 x}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4}-\frac {e^{\frac {4 x}{\log (x)}} x}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4}-\frac {x^3}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4}-\frac {8}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)}+\frac {6 e^{\frac {4 x}{\log (x)}}}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)}+\frac {12 x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)}-\frac {2 e^{\frac {4 x}{\log (x)}} x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)}-\frac {2 x^4}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)}+\frac {8}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)}-\frac {6 e^{\frac {4 x}{\log (x)}}}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)}-\frac {12 x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)}+\frac {2 e^{\frac {4 x}{\log (x)}} x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)}+\frac {2 x^4}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)}\right ) \, dx-8 \int \frac {1}{\log ^2(x)} \, dx+8 \int \frac {1}{\log (x)} \, dx \\ & = \frac {8 x}{\log (x)}-\log (x)+8 \text {li}(x)+4 \int \frac {e^{\frac {4 x}{\log (x)}} x}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4} \, dx+4 \int \frac {x^3}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4} \, dx+8 \int \frac {e^{\frac {4 x}{\log (x)}} x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx+8 \int \frac {x^4}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\log (x)} \, dx-8 \int \frac {e^{\frac {4 x}{\log (x)}} x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx-8 \int \frac {x^4}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx-12 \int \frac {x}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4} \, dx-24 \int \frac {e^{\frac {4 x}{\log (x)}}}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx+24 \int \frac {e^{\frac {4 x}{\log (x)}}}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx+32 \int \frac {1}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx-32 \int \frac {1}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx-48 \int \frac {x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx+48 \int \frac {x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx \\ & = \frac {8 x}{\log (x)}-\log (x)+4 \int \frac {e^{\frac {4 x}{\log (x)}} x}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4} \, dx+4 \int \frac {x^3}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4} \, dx+8 \int \frac {e^{\frac {4 x}{\log (x)}} x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx+8 \int \frac {x^4}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx-8 \int \frac {e^{\frac {4 x}{\log (x)}} x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx-8 \int \frac {x^4}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx-12 \int \frac {x}{4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4} \, dx-24 \int \frac {e^{\frac {4 x}{\log (x)}}}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx+24 \int \frac {e^{\frac {4 x}{\log (x)}}}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx+32 \int \frac {1}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx-32 \int \frac {1}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx-48 \int \frac {x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log ^2(x)} \, dx+48 \int \frac {x^2}{\left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=-\log (x)+\log \left (4-6 e^{\frac {4 x}{\log (x)}}+e^{\frac {8 x}{\log (x)}}-6 x^2+2 e^{\frac {4 x}{\log (x)}} x^2+x^4\right ) \]

[In]

Integrate[((-4 - 6*x^2 + 3*x^4)*Log[x]^2 + E^((8*x)/Log[x])*(-8*x + 8*x*Log[x] - Log[x]^2) + E^((4*x)/Log[x])*
(24*x - 8*x^3 + (-24*x + 8*x^3)*Log[x] + (6 + 2*x^2)*Log[x]^2))/(E^((8*x)/Log[x])*x*Log[x]^2 + E^((4*x)/Log[x]
)*(-6*x + 2*x^3)*Log[x]^2 + (4*x - 6*x^3 + x^5)*Log[x]^2),x]

[Out]

-Log[x] + Log[4 - 6*E^((4*x)/Log[x]) + E^((8*x)/Log[x]) - 6*x^2 + 2*E^((4*x)/Log[x])*x^2 + x^4]

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {8 x}{\ln \left (x \right )}}+\left (2 x^{2}-6\right ) {\mathrm e}^{\frac {4 x}{\ln \left (x \right )}}+x^{4}-6 x^{2}+4\right )\)	\(41\)
parallelrisch	\(\ln \left (x^{4}+2 \,{\mathrm e}^{\frac {4 x}{\ln \left (x \right )}} x^{2}-6 x^{2}-6 \,{\mathrm e}^{\frac {4 x}{\ln \left (x \right )}}+{\mathrm e}^{\frac {8 x}{\ln \left (x \right )}}+4\right )-\ln \left (x \right )\)	\(50\)

[In]

int(((-ln(x)^2+8*x*ln(x)-8*x)*exp(4*x/ln(x))^2+((2*x^2+6)*ln(x)^2+(8*x^3-24*x)*ln(x)-8*x^3+24*x)*exp(4*x/ln(x)
)+(3*x^4-6*x^2-4)*ln(x)^2)/(x*ln(x)^2*exp(4*x/ln(x))^2+(2*x^3-6*x)*ln(x)^2*exp(4*x/ln(x))+(x^5-6*x^3+4*x)*ln(x
)^2),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(exp(8*x/ln(x))+(2*x^2-6)*exp(4*x/ln(x))+x^4-6*x^2+4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=\log \left (x^{4} - 6 \, x^{2} + 2 \, {\left (x^{2} - 3\right )} e^{\left (\frac {4 \, x}{\log \left (x\right )}\right )} + e^{\left (\frac {8 \, x}{\log \left (x\right )}\right )} + 4\right ) - \log \left (x\right ) \]

[In]

integrate(((-log(x)^2+8*x*log(x)-8*x)*exp(4*x/log(x))^2+((2*x^2+6)*log(x)^2+(8*x^3-24*x)*log(x)-8*x^3+24*x)*ex
p(4*x/log(x))+(3*x^4-6*x^2-4)*log(x)^2)/(x*log(x)^2*exp(4*x/log(x))^2+(2*x^3-6*x)*log(x)^2*exp(4*x/log(x))+(x^
5-6*x^3+4*x)*log(x)^2),x, algorithm="fricas")

[Out]

log(x^4 - 6*x^2 + 2*(x^2 - 3)*e^(4*x/log(x)) + e^(8*x/log(x)) + 4) - log(x)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=- \log {\left (x \right )} + \log {\left (x^{4} - 6 x^{2} + \left (2 x^{2} - 6\right ) e^{\frac {4 x}{\log {\left (x \right )}}} + e^{\frac {8 x}{\log {\left (x \right )}}} + 4 \right )} \]

[In]

integrate(((-ln(x)**2+8*x*ln(x)-8*x)*exp(4*x/ln(x))**2+((2*x**2+6)*ln(x)**2+(8*x**3-24*x)*ln(x)-8*x**3+24*x)*e
xp(4*x/ln(x))+(3*x**4-6*x**2-4)*ln(x)**2)/(x*ln(x)**2*exp(4*x/ln(x))**2+(2*x**3-6*x)*ln(x)**2*exp(4*x/ln(x))+(
x**5-6*x**3+4*x)*ln(x)**2),x)

[Out]

-log(x) + log(x**4 - 6*x**2 + (2*x**2 - 6)*exp(4*x/log(x)) + exp(8*x/log(x)) + 4)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=\log \left (x^{4} - 6 \, x^{2} + 2 \, {\left (x^{2} - 3\right )} e^{\left (\frac {4 \, x}{\log \left (x\right )}\right )} + e^{\left (\frac {8 \, x}{\log \left (x\right )}\right )} + 4\right ) - \log \left (x\right ) \]

[In]

integrate(((-log(x)^2+8*x*log(x)-8*x)*exp(4*x/log(x))^2+((2*x^2+6)*log(x)^2+(8*x^3-24*x)*log(x)-8*x^3+24*x)*ex
p(4*x/log(x))+(3*x^4-6*x^2-4)*log(x)^2)/(x*log(x)^2*exp(4*x/log(x))^2+(2*x^3-6*x)*log(x)^2*exp(4*x/log(x))+(x^
5-6*x^3+4*x)*log(x)^2),x, algorithm="maxima")

[Out]

log(x^4 - 6*x^2 + 2*(x^2 - 3)*e^(4*x/log(x)) + e^(8*x/log(x)) + 4) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=\log \left (x^{4} + 2 \, x^{2} e^{\left (\frac {4 \, x}{\log \left (x\right )}\right )} - 6 \, x^{2} + e^{\left (\frac {8 \, x}{\log \left (x\right )}\right )} - 6 \, e^{\left (\frac {4 \, x}{\log \left (x\right )}\right )} + 4\right ) - \log \left (x\right ) \]

[In]

integrate(((-log(x)^2+8*x*log(x)-8*x)*exp(4*x/log(x))^2+((2*x^2+6)*log(x)^2+(8*x^3-24*x)*log(x)-8*x^3+24*x)*ex
p(4*x/log(x))+(3*x^4-6*x^2-4)*log(x)^2)/(x*log(x)^2*exp(4*x/log(x))^2+(2*x^3-6*x)*log(x)^2*exp(4*x/log(x))+(x^
5-6*x^3+4*x)*log(x)^2),x, algorithm="giac")

[Out]

log(x^4 + 2*x^2*e^(4*x/log(x)) - 6*x^2 + e^(8*x/log(x)) - 6*e^(4*x/log(x)) + 4) - log(x)

Mupad [B] (verification not implemented)

Time = 11.60 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-4-6 x^2+3 x^4\right ) \log ^2(x)+e^{\frac {8 x}{\log (x)}} \left (-8 x+8 x \log (x)-\log ^2(x)\right )+e^{\frac {4 x}{\log (x)}} \left (24 x-8 x^3+\left (-24 x+8 x^3\right ) \log (x)+\left (6+2 x^2\right ) \log ^2(x)\right )}{e^{\frac {8 x}{\log (x)}} x \log ^2(x)+e^{\frac {4 x}{\log (x)}} \left (-6 x+2 x^3\right ) \log ^2(x)+\left (4 x-6 x^3+x^5\right ) \log ^2(x)} \, dx=\ln \left ({\mathrm {e}}^{\frac {8\,x}{\ln \left (x\right )}}-6\,{\mathrm {e}}^{\frac {4\,x}{\ln \left (x\right )}}+2\,x^2\,{\mathrm {e}}^{\frac {4\,x}{\ln \left (x\right )}}-6\,x^2+x^4+4\right )-\ln \left (x\right ) \]

[In]

int(-(exp((8*x)/log(x))*(8*x + log(x)^2 - 8*x*log(x)) - exp((4*x)/log(x))*(24*x + log(x)^2*(2*x^2 + 6) - log(x
)*(24*x - 8*x^3) - 8*x^3) + log(x)^2*(6*x^2 - 3*x^4 + 4))/(log(x)^2*(4*x - 6*x^3 + x^5) - exp((4*x)/log(x))*lo
g(x)^2*(6*x - 2*x^3) + x*exp((8*x)/log(x))*log(x)^2),x)

[Out]

log(exp((8*x)/log(x)) - 6*exp((4*x)/log(x)) + 2*x^2*exp((4*x)/log(x)) - 6*x^2 + x^4 + 4) - log(x)