\(\int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+(1+6 e^x+12 e^{2 x}+8 e^{3 x}) \log (x)}{(4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+(x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x) \log (x)) \log (\frac {4+16 e^x+16 e^{2 x}+x+(x+4 e^x x+4 e^{2 x} x) \log (x)}{1+4 e^x+4 e^{2 x}})} \, dx\) [9309]

Optimal result

Integrand size = 159, antiderivative size = 21 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=4+\log \left (\log \left (4+\frac {x}{\left (1+2 e^x\right )^2}+x \log (x)\right )\right ) \]

[Out]

ln(ln(x*ln(x)+4+x/(1+2*exp(x))^2))+4

Rubi [F]

\[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx \]

[In]

Int[(2 + 12*E^(2*x) + 8*E^(3*x) + E^x*(8 - 4*x) + (1 + 6*E^x + 12*E^(2*x) + 8*E^(3*x))*Log[x])/((4 + 48*E^(2*x
) + 32*E^(3*x) + x + E^x*(24 + 2*x) + (x + 6*E^x*x + 12*E^(2*x)*x + 8*E^(3*x)*x)*Log[x])*Log[(4 + 16*E^x + 16*
E^(2*x) + x + (x + 4*E^x*x + 4*E^(2*x)*x)*Log[x])/(1 + 4*E^x + 4*E^(2*x))]),x]

[Out]

2*Defer[Int][1/((1 + 2*E^x)*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] + D
efer[Int][1/((4 + x*Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] + D
efer[Int][Log[x]/((4 + x*Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x
] - 28*Defer[Int][1/((4 + x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*Log
[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] - 64*Defer[Int][E^x/((4 +
x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*Log[x])*Log[(4 + 16*E^x + 16*
E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] - 9*Defer[Int][x/((4 + x*Log[x])*(4 + 16*E^x + 16*E^
(2*x) + x + x*Log[x] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x
*Log[x])/(1 + 2*E^x)^2]), x] - 16*Defer[Int][(x*Log[x])/((4 + x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x
] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x
)^2]), x] - 32*Defer[Int][(E^x*x*Log[x])/((4 + x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x] + 4*E^x*x*Log
[x] + 4*E^(2*x)*x*Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] - 2*D
efer[Int][(x^2*Log[x])/((4 + x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*
Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] - 2*Defer[Int][(x^2*Log
[x]^2)/((4 + x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*Log[x])*Log[(4 +
16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x] - 4*Defer[Int][(E^x*x^2*Log[x]^2)/((4 +
x*Log[x])*(4 + 16*E^x + 16*E^(2*x) + x + x*Log[x] + 4*E^x*x*Log[x] + 4*E^(2*x)*x*Log[x])*Log[(4 + 16*E^x + 16
*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2+12 e^{2 x}+8 e^{3 x}-4 e^x (-2+x)+\left (1+2 e^x\right )^3 \log (x)}{\left (1+2 e^x\right ) \left (4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx \\ & = \int \left (\frac {2}{\left (1+2 e^x\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {1+\log (x)}{(4+x \log (x)) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}-\frac {28+64 e^x+9 x+16 x \log (x)+32 e^x x \log (x)+2 x^2 \log (x)+2 x^2 \log ^2(x)+4 e^x x^2 \log ^2(x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\left (1+2 e^x\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx+\int \frac {1+\log (x)}{(4+x \log (x)) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-\int \frac {28+64 e^x+9 x+16 x \log (x)+32 e^x x \log (x)+2 x^2 \log (x)+2 x^2 \log ^2(x)+4 e^x x^2 \log ^2(x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx \\ & = 2 \int \frac {1}{\left (1+2 e^x\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx+\int \left (\frac {1}{(4+x \log (x)) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {\log (x)}{(4+x \log (x)) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}\right ) \, dx-\int \left (\frac {28}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {64 e^x}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {9 x}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {16 x \log (x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {32 e^x x \log (x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {2 x^2 \log (x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {2 x^2 \log ^2(x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}+\frac {4 e^x x^2 \log ^2(x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\left (1+2 e^x\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-2 \int \frac {x^2 \log (x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-2 \int \frac {x^2 \log ^2(x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-4 \int \frac {e^x x^2 \log ^2(x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-9 \int \frac {x}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-16 \int \frac {x \log (x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-28 \int \frac {1}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-32 \int \frac {e^x x \log (x)}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx-64 \int \frac {e^x}{(4+x \log (x)) \left (4+16 e^x+16 e^{2 x}+x+x \log (x)+4 e^x x \log (x)+4 e^{2 x} x \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx+\int \frac {1}{(4+x \log (x)) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx+\int \frac {\log (x)}{(4+x \log (x)) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (\log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )\right ) \]

[In]

Integrate[(2 + 12*E^(2*x) + 8*E^(3*x) + E^x*(8 - 4*x) + (1 + 6*E^x + 12*E^(2*x) + 8*E^(3*x))*Log[x])/((4 + 48*
E^(2*x) + 32*E^(3*x) + x + E^x*(24 + 2*x) + (x + 6*E^x*x + 12*E^(2*x)*x + 8*E^(3*x)*x)*Log[x])*Log[(4 + 16*E^x
+ 16*E^(2*x) + x + (x + 4*E^x*x + 4*E^(2*x)*x)*Log[x])/(1 + 4*E^x + 4*E^(2*x))]),x]

[Out]

Log[Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^x)^2]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(20)=40\).

Time = 106.69 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29

method	result	size
parallelrisch	\(\ln \left (\ln \left (\frac {\left (4 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x +x \right ) \ln \left (x \right )+16 \,{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+4+x}{4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1}\right )\right )\)	\(48\)
risch	\(\ln \left (\ln \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}}\right ) \operatorname {csgn}\left (i \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}\right )}^{3}-\pi \,\operatorname {csgn}\left (i \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left (\frac {1}{2}+{\mathrm e}^{x}\right )^{2}}\right )}^{3}-4 i \ln \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )}{2}\right )\)	\(385\)

[In]

int(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*ln(x)+8*exp(x)^3+12*exp(x)^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3+12*x*e
xp(x)^2+6*exp(x)*x+x)*ln(x)+32*exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/ln(((4*x*exp(x)^2+4*exp(x)*x+x)*ln(x)
+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(((4*x*exp(x)^2+4*exp(x)*x+x)*ln(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (\log \left (\frac {{\left (4 \, x e^{\left (2 \, x\right )} + 4 \, x e^{x} + x\right )} \log \left (x\right ) + x + 16 \, e^{\left (2 \, x\right )} + 16 \, e^{x} + 4}{4 \, e^{\left (2 \, x\right )} + 4 \, e^{x} + 1}\right )\right ) \]

[In]

integrate(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x)^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3
+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32*exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x
+x)*log(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x, algorithm="fricas")

[Out]

log(log(((4*x*e^(2*x) + 4*x*e^x + x)*log(x) + x + 16*e^(2*x) + 16*e^x + 4)/(4*e^(2*x) + 4*e^x + 1)))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).

Time = 3.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log {\left (\log {\left (\frac {x + \left (4 x e^{2 x} + 4 x e^{x} + x\right ) \log {\left (x \right )} + 16 e^{2 x} + 16 e^{x} + 4}{4 e^{2 x} + 4 e^{x} + 1} \right )} \right )} \]

[In]

integrate(((8*exp(x)**3+12*exp(x)**2+6*exp(x)+1)*ln(x)+8*exp(x)**3+12*exp(x)**2+(-4*x+8)*exp(x)+2)/((8*x*exp(x
)**3+12*x*exp(x)**2+6*exp(x)*x+x)*ln(x)+32*exp(x)**3+48*exp(x)**2+(2*x+24)*exp(x)+4+x)/ln(((4*x*exp(x)**2+4*ex
p(x)*x+x)*ln(x)+16*exp(x)**2+16*exp(x)+4+x)/(4*exp(x)**2+4*exp(x)+1)),x)

[Out]

log(log((x + (4*x*exp(2*x) + 4*x*exp(x) + x)*log(x) + 16*exp(2*x) + 16*exp(x) + 4)/(4*exp(2*x) + 4*exp(x) + 1)
))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 7.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (\log \left ({\left (4 \, x e^{\left (2 \, x\right )} + 4 \, x e^{x} + x\right )} \log \left (x\right ) + x + 16 \, e^{\left (2 \, x\right )} + 16 \, e^{x} + 4\right ) - 2 \, \log \left (2 \, e^{x} + 1\right )\right ) \]

[In]

integrate(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x)^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3
+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32*exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x
+x)*log(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x, algorithm="maxima")

[Out]

log(log((4*x*e^(2*x) + 4*x*e^x + x)*log(x) + x + 16*e^(2*x) + 16*e^x + 4) - 2*log(2*e^x + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).

Time = 0.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (-\log \left (4 \, x e^{\left (2 \, x\right )} \log \left (x\right ) + 4 \, x e^{x} \log \left (x\right ) + x \log \left (x\right ) + x + 16 \, e^{\left (2 \, x\right )} + 16 \, e^{x} + 4\right ) + \log \left (4 \, e^{\left (2 \, x\right )} + 4 \, e^{x} + 1\right )\right ) \]

[In]

integrate(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x)^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3
+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32*exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x
+x)*log(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x, algorithm="giac")

[Out]

log(-log(4*x*e^(2*x)*log(x) + 4*x*e^x*log(x) + x*log(x) + x + 16*e^(2*x) + 16*e^x + 4) + log(4*e^(2*x) + 4*e^x
+ 1))

Mupad [B] (verification not implemented)

Time = 14.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\ln \left (\ln \left (x+16\,{\mathrm {e}}^{2\,x}+16\,{\mathrm {e}}^x+x\,\ln \left (x\right )+4\,x\,{\mathrm {e}}^x\,\ln \left (x\right )+4\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+4\right )-\ln \left (4\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^x+1\right )\right ) \]

[In]

int((12*exp(2*x) + 8*exp(3*x) + log(x)*(12*exp(2*x) + 8*exp(3*x) + 6*exp(x) + 1) - exp(x)*(4*x - 8) + 2)/(log(
(x + 16*exp(2*x) + 16*exp(x) + log(x)*(x + 4*x*exp(2*x) + 4*x*exp(x)) + 4)/(4*exp(2*x) + 4*exp(x) + 1))*(x + 4
8*exp(2*x) + 32*exp(3*x) + exp(x)*(2*x + 24) + log(x)*(x + 12*x*exp(2*x) + 8*x*exp(3*x) + 6*x*exp(x)) + 4)),x)

[Out]

log(log(x + 16*exp(2*x) + 16*exp(x) + x*log(x) + 4*x*exp(x)*log(x) + 4*x*exp(2*x)*log(x) + 4) - log(4*exp(2*x)
+ 4*exp(x) + 1))