\(\int \frac {3+x^2+e^4 x^2+(2 x+2 e^4 x) \log (3)+(1+e^4) \log ^2(3)+e^{2 x} (x^4+e^4 x^4) \log ^2(3)+e^x (3 x^2+3 x^3+(-2 x^3-2 e^4 x^3) \log (3)+(-2 x^2-2 e^4 x^2) \log ^2(3))}{-3 x-x^2+x^3+e^4 x^3+(-3-2 x+2 x^2+2 e^4 x^2) \log (3)+(-1+x+e^4 x) \log ^2(3)+e^x (3 x^3+(6 x^2+2 x^3-2 x^4-2 e^4 x^4) \log (3)+(2 x^2-2 x^3-2 e^4 x^3) \log ^2(3))+e^{2 x} (-3 x^4 \log (3)+(-x^4+x^5+e^4 x^5) \log ^2(3))} \, dx\) [702]

Optimal result

Integrand size = 259, antiderivative size = 35 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=1+\log \left (1-x-e^4 x+\frac {3}{\frac {x}{1-e^x x^2}+\log (3)}\right ) \]

[Out]

ln(3/(x/(1-exp(x)*x^2)+ln(3))-x-x*exp(4)+1)+1

Rubi [F]

\[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \]

[In]

Int[(3 + x^2 + E^4*x^2 + (2*x + 2*E^4*x)*Log[3] + (1 + E^4)*Log[3]^2 + E^(2*x)*(x^4 + E^4*x^4)*Log[3]^2 + E^x*
(3*x^2 + 3*x^3 + (-2*x^3 - 2*E^4*x^3)*Log[3] + (-2*x^2 - 2*E^4*x^2)*Log[3]^2))/(-3*x - x^2 + x^3 + E^4*x^3 + (
-3 - 2*x + 2*x^2 + 2*E^4*x^2)*Log[3] + (-1 + x + E^4*x)*Log[3]^2 + E^x*(3*x^3 + (6*x^2 + 2*x^3 - 2*x^4 - 2*E^4
*x^4)*Log[3] + (2*x^2 - 2*x^3 - 2*E^4*x^3)*Log[3]^2) + E^(2*x)*(-3*x^4*Log[3] + (-x^4 + x^5 + E^4*x^5)*Log[3]^
2)),x]

[Out]

Log[3 + Log[3] - (1 + E^4)*x*Log[3]] - (1 + Log[3])*Defer[Int][(-x - Log[3] + E^x*x^2*Log[3])^(-1), x] - Log[9
]*Defer[Int][1/(x*(-x - Log[3] + E^x*x^2*Log[3])), x] - Defer[Int][x/(-x - Log[3] + E^x*x^2*Log[3]), x] - ((3*
Log[3]^4 + 6*E^16*Log[3]^4 - 9*Log[3]*(3 + Log[9]) - 2*Log[3]^3*(3 + Log[9]) + Log[3]^2*(18 - 9*Log[9] + Log[9
]^2) + Log[27]*(9 + Log[9]^2 + Log[27]) + E^12*(19*Log[3]^4 - Log[3]^3*(21 + Log[9]) + Log[3]^2*(Log[9]^2 - Lo
g[27]) + Log[27]*(Log[9]^2 + Log[27])) + E^8*(27*Log[3]^4 - 6*Log[3]^3*(4 + Log[9]) - 9*Log[3]*(3 + Log[27]) +
Log[27]*(9 + Log[9]^2 + Log[9]*Log[81]) + Log[3]^2*(54 + Log[9]^2 - Log[9]*(27 - Log[81]) + Log[729])) + E^4*
(9*Log[3]^4 + Log[9]^2*Log[27] - Log[27]^2 + Log[9]*Log[27]*Log[81] + Log[729]^2 - Log[3]^3*(57 + Log[729]) +
Log[3]^2*(Log[9]^2 + Log[9]*Log[81] - Log[14348907])))*Defer[Int][((1 + E^4)*x^2 - 3*(1 + Log[3]/3) + 3*E^x*x^
2*(1 + Log[3]/3) - E^x*(1 + E^4)*x^3*Log[3] - x*(1 - (1 + E^4)*Log[3]))^(-1), x])/(3*(1 + E^4)^3*Log[3]^3) + 2
*(3 + Log[3])*Defer[Int][1/(x*((1 + E^4)*x^2 - 3*(1 + Log[3]/3) + 3*E^x*x^2*(1 + Log[3]/3) - E^x*(1 + E^4)*x^3
*Log[3] - x*(1 - (1 + E^4)*Log[3]))), x] - ((Log[3]^3 + Log[3]^2*(6 - Log[9]) + Log[3]*Log[9]^2 - E^12*Log[3]*
(Log[3]^2 - Log[9]^2 - Log[27]) - Log[9]*Log[27] - E^4*Log[3]*(3*Log[3]^2 - Log[9]^2 - Log[27] - Log[9]*Log[81
]) + E^8*(3*Log[3]^3 + Log[3]^2*(18 - Log[729]) - Log[9]*Log[729] + Log[3]*Log[9]*Log[729]))*Defer[Int][x/((1
+ E^4)*x^2 - 3*(1 + Log[3]/3) + 3*E^x*x^2*(1 + Log[3]/3) - E^x*(1 + E^4)*x^3*Log[3] - x*(1 - (1 + E^4)*Log[3])
), x])/(3*(1 + E^4)^2*Log[3]^2) - ((2*Log[3]^2 - Log[3]*Log[9] + Log[27] + E^8*(6*Log[3]^2 - 3*Log[3]*Log[9] +
Log[27]) + E^4*Log[729])*Defer[Int][x^2/((1 + E^4)*x^2 - 3*(1 + Log[3]/3) + 3*E^x*x^2*(1 + Log[3]/3) - E^x*(1
+ E^4)*x^3*Log[3] - x*(1 - (1 + E^4)*Log[3])), x])/(3*(1 + E^4)*Log[3]) + ((3 + Log[3])*(9*Log[3]^4 + Log[9]^
2*Log[27] + Log[27]^2 + E^12*(43*Log[3]^4 + Log[3]^2*(Log[9]^2 - Log[27]) + Log[27]*(Log[9]^2 + Log[27]) - Log
[3]^3*(9 + Log[9] + 3*Log[27] + 9*Log[81])) + E^8*(63*Log[3]^4 - 9*Log[3]*(3 + Log[27]) + 3*Log[3]^3*(4 - 3*Lo
g[27] - 4*Log[81]) + Log[27]*(9 + Log[9]^2 + Log[9]*Log[81]) + Log[3]^2*(54 + Log[9]^2 - Log[9]*(27 - Log[81])
+ Log[729])) - Log[9]*Log[19683] + Log[3]^3*(6 - Log[1594323]) - E^4*(54*Log[3] - 33*Log[3]^4 - Log[9]^2*Log[
27] + Log[27]^2 - Log[9]*Log[27]*Log[81] - 9*Log[729] - Log[729]^2 - Log[3]^2*(Log[9]^2 + Log[9]*Log[81] - Log
[14348907]) + 3*Log[3]^3*(7 + Log[14348907])) + Log[3]^2*(18 + Log[9]^2 - Log[387420489]))*Defer[Int][1/((3 +
Log[3] - (1 + E^4)*x*Log[3])*((1 + E^4)*x^2 - 3*(1 + Log[3]/3) + 3*E^x*x^2*(1 + Log[3]/3) - E^x*(1 + E^4)*x^3*
Log[3] - x*(1 - (1 + E^4)*Log[3]))), x])/(3*(1 + E^4)^3*Log[3]^3)

Rubi steps \begin{align*} \text {integral}& = \int \frac {3+\left (1+e^4\right ) x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \\ & = \int \frac {3+\left (1+e^4\right ) x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+\left (1+e^4\right ) x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \\ & = \int \frac {-e^{2 x} \left (1+e^4\right ) x^4 \log ^2(3)-3 \left (1+\frac {1}{3} \left (1+e^4\right ) \log ^2(3)\right )-x^2 \left (1+e^4-2 e^{4+x} \log ^2(3)+e^x \left (3-2 \log ^2(3)\right )\right )-\left (1+e^4\right ) x \log (9)+e^x x^3 \left (-3+\log (9)+e^4 \log (9)\right )}{\left (x+\log (3)-e^x x^2 \log (3)\right ) \left (3 \left (1+\frac {\log (3)}{3}\right )+e^x \left (1+e^4\right ) x^3 \log (3)-x \left (-1+\log (3)+e^4 \log (3)\right )-x^2 \left (1+e^4+e^x (3+\log (3))\right )\right )} \, dx \\ & = \int \left (\frac {\left (-1-e^4\right ) \log (3)}{3+\log (3)-\left (1+e^4\right ) x \log (3)}+\frac {x^2+x (1+\log (3))+\log (9)}{x \left (x+\log (3)-e^x x^2 \log (3)\right )}+\frac {6 (3+\log (3))^2+x^2 \left (9+2 \log ^2(3)+6 e^8 \log ^2(3)-3 \log (9)-\log (3) (15+\log (9))+e^4 \left (8 \log ^2(3)-3 \log (9)-\log (3) (18+\log (9))\right )\right )+3 x (3+\log (3)) \left (4-\log (27)-e^4 \log (81)\right )-x^3 \left (9+\log ^2(3)-\log ^2(9)+e^8 \left (\log ^2(3)-\log ^2(9)-\log (27)\right )+\log (27)+e^4 \left (9+2 \log ^2(3)-\log (9) \log (81)\right )\right )+x^4 \left (2 \log ^2(3)-\log (3) \log (9)+\log (27)+e^8 \left (6 \log ^2(3)-3 \log (3) \log (9)+\log (27)\right )+e^4 \log (729)\right )}{3 x \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right ) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )}\right ) \, dx \\ & = \log \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right )+\frac {1}{3} \int \frac {6 (3+\log (3))^2+x^2 \left (9+2 \log ^2(3)+6 e^8 \log ^2(3)-3 \log (9)-\log (3) (15+\log (9))+e^4 \left (8 \log ^2(3)-3 \log (9)-\log (3) (18+\log (9))\right )\right )+3 x (3+\log (3)) \left (4-\log (27)-e^4 \log (81)\right )-x^3 \left (9+\log ^2(3)-\log ^2(9)+e^8 \left (\log ^2(3)-\log ^2(9)-\log (27)\right )+\log (27)+e^4 \left (9+2 \log ^2(3)-\log (9) \log (81)\right )\right )+x^4 \left (2 \log ^2(3)-\log (3) \log (9)+\log (27)+e^8 \left (6 \log ^2(3)-3 \log (3) \log (9)+\log (27)\right )+e^4 \log (729)\right )}{x \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right ) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )} \, dx+\int \frac {x^2+x (1+\log (3))+\log (9)}{x \left (x+\log (3)-e^x x^2 \log (3)\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

\[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \]

[In]

Integrate[(3 + x^2 + E^4*x^2 + (2*x + 2*E^4*x)*Log[3] + (1 + E^4)*Log[3]^2 + E^(2*x)*(x^4 + E^4*x^4)*Log[3]^2
+ E^x*(3*x^2 + 3*x^3 + (-2*x^3 - 2*E^4*x^3)*Log[3] + (-2*x^2 - 2*E^4*x^2)*Log[3]^2))/(-3*x - x^2 + x^3 + E^4*x
^3 + (-3 - 2*x + 2*x^2 + 2*E^4*x^2)*Log[3] + (-1 + x + E^4*x)*Log[3]^2 + E^x*(3*x^3 + (6*x^2 + 2*x^3 - 2*x^4 -
2*E^4*x^4)*Log[3] + (2*x^2 - 2*x^3 - 2*E^4*x^3)*Log[3]^2) + E^(2*x)*(-3*x^4*Log[3] + (-x^4 + x^5 + E^4*x^5)*L
og[3]^2)),x]

[Out]

Integrate[(3 + x^2 + E^4*x^2 + (2*x + 2*E^4*x)*Log[3] + (1 + E^4)*Log[3]^2 + E^(2*x)*(x^4 + E^4*x^4)*Log[3]^2
+ E^x*(3*x^2 + 3*x^3 + (-2*x^3 - 2*E^4*x^3)*Log[3] + (-2*x^2 - 2*E^4*x^2)*Log[3]^2))/(-3*x - x^2 + x^3 + E^4*x
^3 + (-3 - 2*x + 2*x^2 + 2*E^4*x^2)*Log[3] + (-1 + x + E^4*x)*Log[3]^2 + E^x*(3*x^3 + (6*x^2 + 2*x^3 - 2*x^4 -
2*E^4*x^4)*Log[3] + (2*x^2 - 2*x^3 - 2*E^4*x^3)*Log[3]^2) + E^(2*x)*(-3*x^4*Log[3] + (-x^4 + x^5 + E^4*x^5)*L
og[3]^2)), x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(33)=66\).

Time = 0.75 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.43

method	result	size
norman	\(-\ln \left (x^{2} \ln \left (3\right ) {\mathrm e}^{x}-\ln \left (3\right )-x \right )+\ln \left ({\mathrm e}^{4} {\mathrm e}^{x} \ln \left (3\right ) x^{3}+{\mathrm e}^{x} \ln \left (3\right ) x^{3}-x^{2} \ln \left (3\right ) {\mathrm e}^{x}-{\mathrm e}^{4} x \ln \left (3\right )-x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{x} x^{2}-x \ln \left (3\right )-x^{2}+\ln \left (3\right )+x +3\right )\)	\(85\)
risch	\(\ln \left (\left ({\mathrm e}^{4} \ln \left (3\right )+\ln \left (3\right )\right ) x -\ln \left (3\right )-3\right )+\ln \left ({\mathrm e}^{x}-\frac {{\mathrm e}^{4} x \ln \left (3\right )+x^{2} {\mathrm e}^{4}+x \ln \left (3\right )+x^{2}-\ln \left (3\right )-x -3}{x^{2} \left ({\mathrm e}^{4} x \ln \left (3\right )+x \ln \left (3\right )-\ln \left (3\right )-3\right )}\right )-\ln \left ({\mathrm e}^{x}-\frac {\ln \left (3\right )+x}{\ln \left (3\right ) x^{2}}\right )\)	\(93\)
parallelrisch	\(-\ln \left (\frac {x^{2} \ln \left (3\right ) {\mathrm e}^{x}-\ln \left (3\right )-x}{\ln \left (3\right )}\right )+\ln \left (\frac {{\mathrm e}^{4} {\mathrm e}^{x} \ln \left (3\right ) x^{3}+{\mathrm e}^{x} \ln \left (3\right ) x^{3}-x^{2} \ln \left (3\right ) {\mathrm e}^{x}-{\mathrm e}^{4} x \ln \left (3\right )-x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{x} x^{2}-x \ln \left (3\right )-x^{2}+\ln \left (3\right )+x +3}{\left ({\mathrm e}^{4}+1\right ) \ln \left (3\right )}\right )\)	\(101\)

[In]

int(((x^4*exp(4)+x^4)*ln(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*ln(3)^2+(-2*x^3*exp(4)-2*x^3)*ln(3)+3*x^3+3*x^2)
*exp(x)+(exp(4)+1)*ln(3)^2+(2*x*exp(4)+2*x)*ln(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*ln(3)^2-3*x^4*ln(3)
)*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*ln(3)^2+(-2*x^4*exp(4)-2*x^4+2*x^3+6*x^2)*ln(3)+3*x^3)*exp(x)+(x*exp(4
)+x-1)*ln(3)^2+(2*x^2*exp(4)+2*x^2-2*x-3)*ln(3)+x^3*exp(4)+x^3-x^2-3*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (34) = 68\).

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.37 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\log \left ({\left (x e^{4} + x - 1\right )} \log \left (3\right ) - 3\right ) + \log \left (\frac {x^{2} e^{4} + x^{2} + {\left (3 \, x^{2} - {\left (x^{3} e^{4} + x^{3} - x^{2}\right )} \log \left (3\right )\right )} e^{x} + {\left (x e^{4} + x - 1\right )} \log \left (3\right ) - x - 3}{3 \, x^{2} - {\left (x^{3} e^{4} + x^{3} - x^{2}\right )} \log \left (3\right )}\right ) - \log \left (\frac {x^{2} e^{x} \log \left (3\right ) - x - \log \left (3\right )}{x^{2}}\right ) \]

[In]

integrate(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3)^2+(-2*x^3*exp(4)-2*x^3)*log(3)+3*x
^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+(2*x*exp(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2
-3*x^4*log(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4)-2*x^4+2*x^3+6*x^2)*log(3)+3*x^3)*
exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*exp(4)+2*x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x, algorithm="fricas"
)

[Out]

log((x*e^4 + x - 1)*log(3) - 3) + log((x^2*e^4 + x^2 + (3*x^2 - (x^3*e^4 + x^3 - x^2)*log(3))*e^x + (x*e^4 + x
- 1)*log(3) - x - 3)/(3*x^2 - (x^3*e^4 + x^3 - x^2)*log(3))) - log((x^2*e^x*log(3) - x - log(3))/x^2)

Sympy [F(-2)]

Exception generated. \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate(((x**4*exp(4)+x**4)*ln(3)**2*exp(x)**2+((-2*x**2*exp(4)-2*x**2)*ln(3)**2+(-2*x**3*exp(4)-2*x**3)*ln(
3)+3*x**3+3*x**2)*exp(x)+(exp(4)+1)*ln(3)**2+(2*x*exp(4)+2*x)*ln(3)+x**2*exp(4)+x**2+3)/(((x**5*exp(4)+x**5-x*
*4)*ln(3)**2-3*x**4*ln(3))*exp(x)**2+((-2*x**3*exp(4)-2*x**3+2*x**2)*ln(3)**2+(-2*x**4*exp(4)-2*x**4+2*x**3+6*
x**2)*ln(3)+3*x**3)*exp(x)+(x*exp(4)+x-1)*ln(3)**2+(2*x**2*exp(4)+2*x**2-2*x-3)*ln(3)+x**3*exp(4)+x**3-x**2-3*
x),x)

[Out]

Exception raised: PolynomialError >> 1/(x**5*log(3)**3 + 2*x**5*exp(4)*log(3)**3 + x**5*exp(8)*log(3)**3 - 6*x
**4*exp(4)*log(3)**2 - 2*x**4*exp(4)*log(3)**3 - 6*x**4*log(3)**2 - 2*x**4*log(3)**3 + x**3*log(3)**3 + 6*x**3
*log(3)**2 + 9*x*

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (34) = 68\).

Time = 0.43 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.51 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\log \left ({\left (e^{4} \log \left (3\right ) + \log \left (3\right )\right )} x - \log \left (3\right ) - 3\right ) + \log \left (-\frac {x^{2} {\left (e^{4} + 1\right )} + {\left (e^{4} \log \left (3\right ) + \log \left (3\right ) - 1\right )} x - {\left ({\left (e^{4} \log \left (3\right ) + \log \left (3\right )\right )} x^{3} - x^{2} {\left (\log \left (3\right ) + 3\right )}\right )} e^{x} - \log \left (3\right ) - 3}{{\left (e^{4} \log \left (3\right ) + \log \left (3\right )\right )} x^{3} - x^{2} {\left (\log \left (3\right ) + 3\right )}}\right ) - \log \left (\frac {x^{2} e^{x} \log \left (3\right ) - x - \log \left (3\right )}{x^{2} \log \left (3\right )}\right ) \]

[In]

integrate(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3)^2+(-2*x^3*exp(4)-2*x^3)*log(3)+3*x
^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+(2*x*exp(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2
-3*x^4*log(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4)-2*x^4+2*x^3+6*x^2)*log(3)+3*x^3)*
exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*exp(4)+2*x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x, algorithm="maxima"
)

[Out]

log((e^4*log(3) + log(3))*x - log(3) - 3) + log(-(x^2*(e^4 + 1) + (e^4*log(3) + log(3) - 1)*x - ((e^4*log(3) +
log(3))*x^3 - x^2*(log(3) + 3))*e^x - log(3) - 3)/((e^4*log(3) + log(3))*x^3 - x^2*(log(3) + 3))) - log((x^2*
e^x*log(3) - x - log(3))/(x^2*log(3)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (34) = 68\).

Time = 1.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.40 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\log \left (x^{3} e^{\left (x + 4\right )} \log \left (3\right ) + x^{3} e^{x} \log \left (3\right ) - x^{2} e^{x} \log \left (3\right ) - x^{2} e^{4} - 3 \, x^{2} e^{x} - x e^{4} \log \left (3\right ) - x^{2} - x \log \left (3\right ) + x + \log \left (3\right ) + 3\right ) - \log \left (x^{2} e^{x} \log \left (3\right ) - x - \log \left (3\right )\right ) \]

[In]

integrate(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3)^2+(-2*x^3*exp(4)-2*x^3)*log(3)+3*x
^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+(2*x*exp(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2
-3*x^4*log(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4)-2*x^4+2*x^3+6*x^2)*log(3)+3*x^3)*
exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*exp(4)+2*x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=-\int \frac {\ln \left (3\right )\,\left (2\,x+2\,x\,{\mathrm {e}}^4\right )-{\mathrm {e}}^x\,\left ({\ln \left (3\right )}^2\,\left (2\,x^2\,{\mathrm {e}}^4+2\,x^2\right )-3\,x^2-3\,x^3+\ln \left (3\right )\,\left (2\,x^3\,{\mathrm {e}}^4+2\,x^3\right )\right )+x^2\,{\mathrm {e}}^4+x^2+{\ln \left (3\right )}^2\,\left ({\mathrm {e}}^4+1\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (x^4\,{\mathrm {e}}^4+x^4\right )+3}{3\,x-{\ln \left (3\right )}^2\,\left (x+x\,{\mathrm {e}}^4-1\right )-x^3\,{\mathrm {e}}^4+{\mathrm {e}}^x\,\left (\ln \left (3\right )\,\left (2\,x^4\,{\mathrm {e}}^4-6\,x^2-2\,x^3+2\,x^4\right )+{\ln \left (3\right )}^2\,\left (2\,x^3\,{\mathrm {e}}^4-2\,x^2+2\,x^3\right )-3\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left ({\ln \left (3\right )}^2\,\left (x^5\,{\mathrm {e}}^4-x^4+x^5\right )-3\,x^4\,\ln \left (3\right )\right )+\ln \left (3\right )\,\left (2\,x-2\,x^2\,{\mathrm {e}}^4-2\,x^2+3\right )+x^2-x^3} \,d x \]

[In]

int(-(log(3)*(2*x + 2*x*exp(4)) - exp(x)*(log(3)^2*(2*x^2*exp(4) + 2*x^2) - 3*x^2 - 3*x^3 + log(3)*(2*x^3*exp(
4) + 2*x^3)) + x^2*exp(4) + x^2 + log(3)^2*(exp(4) + 1) + exp(2*x)*log(3)^2*(x^4*exp(4) + x^4) + 3)/(3*x - log
(3)^2*(x + x*exp(4) - 1) - x^3*exp(4) + exp(x)*(log(3)*(2*x^4*exp(4) - 6*x^2 - 2*x^3 + 2*x^4) + log(3)^2*(2*x^
3*exp(4) - 2*x^2 + 2*x^3) - 3*x^3) - exp(2*x)*(log(3)^2*(x^5*exp(4) - x^4 + x^5) - 3*x^4*log(3)) + log(3)*(2*x
- 2*x^2*exp(4) - 2*x^2 + 3) + x^2 - x^3),x)

[Out]

-int((log(3)*(2*x + 2*x*exp(4)) - exp(x)*(log(3)^2*(2*x^2*exp(4) + 2*x^2) - 3*x^2 - 3*x^3 + log(3)*(2*x^3*exp(
4) + 2*x^3)) + x^2*exp(4) + x^2 + log(3)^2*(exp(4) + 1) + exp(2*x)*log(3)^2*(x^4*exp(4) + x^4) + 3)/(3*x - log
(3)^2*(x + x*exp(4) - 1) - x^3*exp(4) + exp(x)*(log(3)*(2*x^4*exp(4) - 6*x^2 - 2*x^3 + 2*x^4) + log(3)^2*(2*x^
3*exp(4) - 2*x^2 + 2*x^3) - 3*x^3) - exp(2*x)*(log(3)^2*(x^5*exp(4) - x^4 + x^5) - 3*x^4*log(3)) + log(3)*(2*x
- 2*x^2*exp(4) - 2*x^2 + 3) + x^2 - x^3), x)