\(\int \frac {e^{1-x} (-64 x-32 x^2+4 x^3+2 x^4+(64+32 x-4 x^2-2 x^3) \log (2+x)+((48 x+21 x^2-13 x^3-7 x^4-x^5) \log (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4})+(-32+2 x+21 x^2+8 x^3+x^4) \log (2+x) \log (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4})) \log (\log (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4})))}{(32+30 x+9 x^2+x^3) \log (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}) \log ^2(\log (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}))} \, dx\) [840]

Optimal result

Integrand size = 243, antiderivative size = 36 \[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\frac {e^{1-x} x (x-\log (2+x))}{\log \left (\log \left (\frac {x^2}{\left (-x+(4+x)^2\right )^2}\right )\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx \]

[In]

Int[(E^(1 - x)*(-64*x - 32*x^2 + 4*x^3 + 2*x^4 + (64 + 32*x - 4*x^2 - 2*x^3)*Log[2 + x] + ((48*x + 21*x^2 - 13
*x^3 - 7*x^4 - x^5)*Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)] + (-32 + 2*x + 21*x^2 + 8*x^3 + x^4)*Log[2
+ x]*Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)])*Log[Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)]]))/((3
2 + 30*x + 9*x^2 + x^3)*Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)]*Log[Log[x^2/(256 + 224*x + 81*x^2 + 14*
x^3 + x^4)]]^2),x]

[Out]

-14*Defer[Int][E^(1 - x)/(Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x] + ((448*I)*Defer
[Int][E^(1 - x)/((-7 + I*Sqrt[15] - 2*x)*Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x])/
Sqrt[15] + 2*Defer[Int][(E^(1 - x)*x)/(Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x] + (
34*(15 + (7*I)*Sqrt[15])*Defer[Int][E^(1 - x)/((7 - I*Sqrt[15] + 2*x)*Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/
(16 + 7*x + x^2)^2]]^2), x])/15 + ((448*I)*Defer[Int][E^(1 - x)/((7 + I*Sqrt[15] + 2*x)*Log[x^2/(16 + 7*x + x^
2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x])/Sqrt[15] + (34*(15 - (7*I)*Sqrt[15])*Defer[Int][E^(1 - x)/((7 +
I*Sqrt[15] + 2*x)*Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x])/15 - 2*Defer[Int][(E^(
1 - x)*Log[2 + x])/(Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x] + ((128*I)*Defer[Int][
(E^(1 - x)*Log[2 + x])/((-7 + I*Sqrt[15] - 2*x)*Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2
), x])/Sqrt[15] + (14*(15 + (7*I)*Sqrt[15])*Defer[Int][(E^(1 - x)*Log[2 + x])/((7 - I*Sqrt[15] + 2*x)*Log[x^2/
(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x])/15 + ((128*I)*Defer[Int][(E^(1 - x)*Log[2 + x])/(
(7 + I*Sqrt[15] + 2*x)*Log[x^2/(16 + 7*x + x^2)^2]*Log[Log[x^2/(16 + 7*x + x^2)^2]]^2), x])/Sqrt[15] + (14*(15
- (7*I)*Sqrt[15])*Defer[Int][(E^(1 - x)*Log[2 + x])/((7 + I*Sqrt[15] + 2*x)*Log[x^2/(16 + 7*x + x^2)^2]*Log[L
og[x^2/(16 + 7*x + x^2)^2]]^2), x])/15 - Defer[Int][E^(1 - x)/Log[Log[x^2/(16 + 7*x + x^2)^2]], x] + 2*Defer[I
nt][(E^(1 - x)*x)/Log[Log[x^2/(16 + 7*x + x^2)^2]], x] - Defer[Int][(E^(1 - x)*x^2)/Log[Log[x^2/(16 + 7*x + x^
2)^2]], x] + 2*Defer[Int][E^(1 - x)/((2 + x)*Log[Log[x^2/(16 + 7*x + x^2)^2]]), x] - Defer[Int][(E^(1 - x)*Log
[2 + x])/Log[Log[x^2/(16 + 7*x + x^2)^2]], x] + Defer[Int][(E^(1 - x)*x*Log[2 + x])/Log[Log[x^2/(16 + 7*x + x^
2)^2]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4-2 \left (-32-16 x+2 x^2+x^3\right ) \log (2+x)-\left (16+7 x+x^2\right ) \left (x \left (-3+x^2\right )-\left (-2+x+x^2\right ) \log (2+x)\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx \\ & = \int \left (\frac {2 e^{1-x} \left (-16+x^2\right ) (x-\log (2+x))}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {e^{1-x} \left (3 x-x^3-2 \log (2+x)+x \log (2+x)+x^2 \log (2+x)\right )}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx \\ & = 2 \int \frac {e^{1-x} \left (-16+x^2\right ) (x-\log (2+x))}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+\int \frac {e^{1-x} \left (3 x-x^3-2 \log (2+x)+x \log (2+x)+x^2 \log (2+x)\right )}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx \\ & = 2 \int \left (\frac {e^{1-x} (x-\log (2+x))}{\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {e^{1-x} (32+7 x) (x-\log (2+x))}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx+\int \frac {e^{1-x} \left (-x \left (-3+x^2\right )+\left (-2+x+x^2\right ) \log (2+x)\right )}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx \\ & = 2 \int \frac {e^{1-x} (x-\log (2+x))}{\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx-2 \int \frac {e^{1-x} (32+7 x) (x-\log (2+x))}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+\int \left (\frac {3 e^{1-x} x}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {e^{1-x} x^3}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {2 e^{1-x} \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {e^{1-x} x \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {e^{1-x} x^2 \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx \\ & = 2 \int \left (\frac {e^{1-x} x}{\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {e^{1-x} \log (2+x)}{\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx-2 \int \left (\frac {32 e^{1-x} x}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {7 e^{1-x} x^2}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {32 e^{1-x} \log (2+x)}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {7 e^{1-x} x \log (2+x)}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx-2 \int \frac {e^{1-x} \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+3 \int \frac {e^{1-x} x}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx-\int \frac {e^{1-x} x^3}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+\int \frac {e^{1-x} x \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+\int \frac {e^{1-x} x^2 \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx \\ & = 2 \int \frac {e^{1-x} x}{\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx-2 \int \frac {e^{1-x} \log (2+x)}{\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx-2 \int \frac {e^{1-x} \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+3 \int \left (\frac {e^{1-x}}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {2 e^{1-x}}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx-14 \int \frac {e^{1-x} x^2}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+14 \int \frac {e^{1-x} x \log (2+x)}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx-64 \int \frac {e^{1-x} x}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx+64 \int \frac {e^{1-x} \log (2+x)}{\left (16+7 x+x^2\right ) \log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right ) \log ^2\left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \, dx-\int \left (\frac {4 e^{1-x}}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {2 e^{1-x} x}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {e^{1-x} x^2}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {8 e^{1-x}}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx+\int \left (\frac {e^{1-x} \log (2+x)}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}-\frac {2 e^{1-x} \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx+\int \left (-\frac {2 e^{1-x} \log (2+x)}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {e^{1-x} x \log (2+x)}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}+\frac {4 e^{1-x} \log (2+x)}{(2+x) \log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\frac {e^{1-x} x (x-\log (2+x))}{\log \left (\log \left (\frac {x^2}{\left (16+7 x+x^2\right )^2}\right )\right )} \]

[In]

Integrate[(E^(1 - x)*(-64*x - 32*x^2 + 4*x^3 + 2*x^4 + (64 + 32*x - 4*x^2 - 2*x^3)*Log[2 + x] + ((48*x + 21*x^
2 - 13*x^3 - 7*x^4 - x^5)*Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)] + (-32 + 2*x + 21*x^2 + 8*x^3 + x^4)*
Log[2 + x]*Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)])*Log[Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)]]
))/((32 + 30*x + 9*x^2 + x^3)*Log[x^2/(256 + 224*x + 81*x^2 + 14*x^3 + x^4)]*Log[Log[x^2/(256 + 224*x + 81*x^2
+ 14*x^3 + x^4)]]^2),x]

[Out]

(E^(1 - x)*x*(x - Log[2 + x]))/Log[Log[x^2/(16 + 7*x + x^2)^2]]

Maple [C] (warning: unable to verify)

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

Time = 0.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 5.42

\[\frac {x \left (x -\ln \left (2+x \right )\right ) {\mathrm e}^{1-x}}{\ln \left (2 \ln \left (x \right )-2 \ln \left (x^{2}+7 x +16\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (x^{2}+7 x +16\right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (x^{2}+7 x +16\right )^{2}\right )+\operatorname {csgn}\left (i \left (x^{2}+7 x +16\right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x^{2}}{\left (x^{2}+7 x +16\right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2}}{\left (x^{2}+7 x +16\right )^{2}}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2}}{\left (x^{2}+7 x +16\right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (x^{2}+7 x +16\right )^{2}}\right )\right )}{2}\right )}\]

[In]

int((((x^4+8*x^3+21*x^2+2*x-32)*ln(x^2/(x^4+14*x^3+81*x^2+224*x+256))*ln(2+x)+(-x^5-7*x^4-13*x^3+21*x^2+48*x)*
ln(x^2/(x^4+14*x^3+81*x^2+224*x+256)))*ln(ln(x^2/(x^4+14*x^3+81*x^2+224*x+256)))+(-2*x^3-4*x^2+32*x+64)*ln(2+x
)+2*x^4+4*x^3-32*x^2-64*x)/(x^3+9*x^2+30*x+32)/exp(-1+x)/ln(x^2/(x^4+14*x^3+81*x^2+224*x+256))/ln(ln(x^2/(x^4+
14*x^3+81*x^2+224*x+256)))^2,x)

[Out]

x*(x-ln(2+x))*exp(1-x)/ln(2*ln(x)-2*ln(x^2+7*x+16)+1/2*I*Pi*csgn(I*(x^2+7*x+16)^2)*(-csgn(I*(x^2+7*x+16)^2)+cs
gn(I*(x^2+7*x+16)))^2-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^2/(x^2+7*x+16)^2)*(-cs
gn(I*x^2/(x^2+7*x+16)^2)+csgn(I*x^2))*(-csgn(I*x^2/(x^2+7*x+16)^2)+csgn(I/(x^2+7*x+16)^2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\frac {x^{2} e^{\left (-x + 1\right )} - x e^{\left (-x + 1\right )} \log \left (x + 2\right )}{\log \left (\log \left (\frac {x^{2}}{x^{4} + 14 \, x^{3} + 81 \, x^{2} + 224 \, x + 256}\right )\right )} \]

[In]

integrate((((x^4+8*x^3+21*x^2+2*x-32)*log(x^2/(x^4+14*x^3+81*x^2+224*x+256))*log(2+x)+(-x^5-7*x^4-13*x^3+21*x^
2+48*x)*log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))*log(log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))+(-2*x^3-4*x^2+32*x
+64)*log(2+x)+2*x^4+4*x^3-32*x^2-64*x)/(x^3+9*x^2+30*x+32)/exp(-1+x)/log(x^2/(x^4+14*x^3+81*x^2+224*x+256))/lo
g(log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))^2,x, algorithm="fricas")

[Out]

(x^2*e^(-x + 1) - x*e^(-x + 1)*log(x + 2))/log(log(x^2/(x^4 + 14*x^3 + 81*x^2 + 224*x + 256)))

Sympy [A] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\frac {\left (x^{2} - x \log {\left (x + 2 \right )}\right ) e^{1 - x}}{\log {\left (\log {\left (\frac {x^{2}}{x^{4} + 14 x^{3} + 81 x^{2} + 224 x + 256} \right )} \right )}} \]

[In]

integrate((((x**4+8*x**3+21*x**2+2*x-32)*ln(x**2/(x**4+14*x**3+81*x**2+224*x+256))*ln(2+x)+(-x**5-7*x**4-13*x*
*3+21*x**2+48*x)*ln(x**2/(x**4+14*x**3+81*x**2+224*x+256)))*ln(ln(x**2/(x**4+14*x**3+81*x**2+224*x+256)))+(-2*
x**3-4*x**2+32*x+64)*ln(2+x)+2*x**4+4*x**3-32*x**2-64*x)/(x**3+9*x**2+30*x+32)/exp(-1+x)/ln(x**2/(x**4+14*x**3
+81*x**2+224*x+256))/ln(ln(x**2/(x**4+14*x**3+81*x**2+224*x+256)))**2,x)

[Out]

(x**2 - x*log(x + 2))*exp(1 - x)/log(log(x**2/(x**4 + 14*x**3 + 81*x**2 + 224*x + 256)))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\frac {x^{2} e - x e \log \left (x + 2\right )}{{\left (i \, \pi + \log \left (2\right )\right )} e^{x} + e^{x} \log \left (\log \left (x^{2} + 7 \, x + 16\right ) - \log \left (x\right )\right )} \]

[In]

integrate((((x^4+8*x^3+21*x^2+2*x-32)*log(x^2/(x^4+14*x^3+81*x^2+224*x+256))*log(2+x)+(-x^5-7*x^4-13*x^3+21*x^
2+48*x)*log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))*log(log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))+(-2*x^3-4*x^2+32*x
+64)*log(2+x)+2*x^4+4*x^3-32*x^2-64*x)/(x^3+9*x^2+30*x+32)/exp(-1+x)/log(x^2/(x^4+14*x^3+81*x^2+224*x+256))/lo
g(log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))^2,x, algorithm="maxima")

[Out]

(x^2*e - x*e*log(x + 2))/((I*pi + log(2))*e^x + e^x*log(log(x^2 + 7*x + 16) - log(x)))

Giac [F]

\[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + 4 \, x^{3} - 32 \, x^{2} - 2 \, {\left (x^{3} + 2 \, x^{2} - 16 \, x - 32\right )} \log \left (x + 2\right ) + {\left ({\left (x^{4} + 8 \, x^{3} + 21 \, x^{2} + 2 \, x - 32\right )} \log \left (x + 2\right ) \log \left (\frac {x^{2}}{x^{4} + 14 \, x^{3} + 81 \, x^{2} + 224 \, x + 256}\right ) - {\left (x^{5} + 7 \, x^{4} + 13 \, x^{3} - 21 \, x^{2} - 48 \, x\right )} \log \left (\frac {x^{2}}{x^{4} + 14 \, x^{3} + 81 \, x^{2} + 224 \, x + 256}\right )\right )} \log \left (\log \left (\frac {x^{2}}{x^{4} + 14 \, x^{3} + 81 \, x^{2} + 224 \, x + 256}\right )\right ) - 64 \, x\right )} e^{\left (-x + 1\right )}}{{\left (x^{3} + 9 \, x^{2} + 30 \, x + 32\right )} \log \left (\frac {x^{2}}{x^{4} + 14 \, x^{3} + 81 \, x^{2} + 224 \, x + 256}\right ) \log \left (\log \left (\frac {x^{2}}{x^{4} + 14 \, x^{3} + 81 \, x^{2} + 224 \, x + 256}\right )\right )^{2}} \,d x } \]

[In]

integrate((((x^4+8*x^3+21*x^2+2*x-32)*log(x^2/(x^4+14*x^3+81*x^2+224*x+256))*log(2+x)+(-x^5-7*x^4-13*x^3+21*x^
2+48*x)*log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))*log(log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))+(-2*x^3-4*x^2+32*x
+64)*log(2+x)+2*x^4+4*x^3-32*x^2-64*x)/(x^3+9*x^2+30*x+32)/exp(-1+x)/log(x^2/(x^4+14*x^3+81*x^2+224*x+256))/lo
g(log(x^2/(x^4+14*x^3+81*x^2+224*x+256)))^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 10.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.28 \[ \int \frac {e^{1-x} \left (-64 x-32 x^2+4 x^3+2 x^4+\left (64+32 x-4 x^2-2 x^3\right ) \log (2+x)+\left (\left (48 x+21 x^2-13 x^3-7 x^4-x^5\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )+\left (-32+2 x+21 x^2+8 x^3+x^4\right ) \log (2+x) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right ) \log \left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )\right )}{\left (32+30 x+9 x^2+x^3\right ) \log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right ) \log ^2\left (\log \left (\frac {x^2}{256+224 x+81 x^2+14 x^3+x^4}\right )\right )} \, dx=\frac {x^2\,{\mathrm {e}}^{1-x}}{\ln \left (\ln \left (x^2\right )-\ln \left (x^4+14\,x^3+81\,x^2+224\,x+256\right )\right )}-\frac {x\,\ln \left (x+2\right )\,{\mathrm {e}}^{1-x}}{\ln \left (\ln \left (x^2\right )-\ln \left (x^4+14\,x^3+81\,x^2+224\,x+256\right )\right )} \]

[In]

int(-(exp(1 - x)*(64*x - log(x + 2)*(32*x - 4*x^2 - 2*x^3 + 64) + 32*x^2 - 4*x^3 - 2*x^4 + log(log(x^2/(224*x
+ 81*x^2 + 14*x^3 + x^4 + 256)))*(log(x^2/(224*x + 81*x^2 + 14*x^3 + x^4 + 256))*(13*x^3 - 21*x^2 - 48*x + 7*x
^4 + x^5) - log(x + 2)*log(x^2/(224*x + 81*x^2 + 14*x^3 + x^4 + 256))*(2*x + 21*x^2 + 8*x^3 + x^4 - 32))))/(lo
g(x^2/(224*x + 81*x^2 + 14*x^3 + x^4 + 256))*log(log(x^2/(224*x + 81*x^2 + 14*x^3 + x^4 + 256)))^2*(30*x + 9*x
^2 + x^3 + 32)),x)

[Out]

(x^2*exp(1 - x))/log(log(x^2) - log(224*x + 81*x^2 + 14*x^3 + x^4 + 256)) - (x*log(x + 2)*exp(1 - x))/log(log(
x^2) - log(224*x + 81*x^2 + 14*x^3 + x^4 + 256))