\(\int \frac {x+3 x^2+3 x^3+x^4+e^5 (-32-96 x-96 x^2-32 x^3)+e^{12+4 x^2+2 x^4} (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6)+(e^{10} (-256-256 x)+e^5 (-8-8 x)) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} (e^{10} (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6)+e^5 (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7)+e^{10} (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5) \log (3))}{32+96 x+96 x^2+32 x^3} \, dx\) [7934]

Optimal result

Integrand size = 244, antiderivative size = 32 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\left (\frac {x}{8}+e^5 \left (-4+e^{\left (1+x^2\right )^2}-\frac {\log (3)}{1+x}\right )\right )^2 \]

[Out]

(exp(5)*(exp((x^2+1)^2)-ln(3)/(1+x)-4)+1/8*x)^2

Rubi [F]

\[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx \]

[In]

Int[(x + 3*x^2 + 3*x^3 + x^4 + E^5*(-32 - 96*x - 96*x^2 - 32*x^3) + E^(12 + 4*x^2 + 2*x^4)*(256*x + 768*x^2 +
1024*x^3 + 1024*x^4 + 768*x^5 + 256*x^6) + (E^10*(-256 - 256*x) + E^5*(-8 - 8*x))*Log[3] - 64*E^10*Log[3]^2 +
E^(1 + 2*x^2 + x^4)*(E^10*(-1024*x - 3072*x^2 - 4096*x^3 - 4096*x^4 - 3072*x^5 - 1024*x^6) + E^5*(8 + 24*x + 5
6*x^2 + 104*x^3 + 128*x^4 + 128*x^5 + 96*x^6 + 32*x^7) + E^10*(64 - 192*x - 512*x^2 - 512*x^3 - 512*x^4 - 256*
x^5)*Log[3]))/(32 + 96*x + 96*x^2 + 32*x^3),x]

[Out]

E^(12 + 4*x^2 + 2*x^4) - E^5*x + x^2/64 + (E^5*Log[3])/(4*(1 + x)) + (E^10*(4 + 4*x + Log[3])^2)/(1 + x)^2 - (
2*E^(11 + 2*x^2 + x^4)*(4*x^5 + x*(4 + Log[3]) + x^2*(8 + Log[3]) + x^3*(8 + Log[3]) + x^4*(8 + Log[3])))/((1
+ x)^2*(x + x^3)) + Defer[Int][E^(6 + 2*x^2 + x^4), x]/4 + Defer[Int][E^(6 + 2*x^2 + x^4)*x^2, x] + Defer[Int]
[E^(6 + 2*x^2 + x^4)*x^4, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{32}+8 e^{12+4 x^2+2 x^4} x \left (1+x^2\right )+\frac {1}{4} e^{6+2 x^2+x^4} \left (1+2 x^2\right )^2-\frac {2 e^{10} \log (3) (4+4 x+\log (3))}{(1+x)^3}-\frac {e^5 \left (4+8 x+4 x^2+\log (3)\right )}{4 (1+x)^2}-\frac {2 e^{11+2 x^2+x^4} \left (16 x^5-\log (3)+4 x (4+\log (3))+4 x^2 (8+\log (3))+4 x^3 (8+\log (3))+4 x^4 (8+\log (3))\right )}{(1+x)^2}\right ) \, dx \\ & = \frac {x^2}{64}+\frac {1}{4} \int e^{6+2 x^2+x^4} \left (1+2 x^2\right )^2 \, dx-2 \int \frac {e^{11+2 x^2+x^4} \left (16 x^5-\log (3)+4 x (4+\log (3))+4 x^2 (8+\log (3))+4 x^3 (8+\log (3))+4 x^4 (8+\log (3))\right )}{(1+x)^2} \, dx+8 \int e^{12+4 x^2+2 x^4} x \left (1+x^2\right ) \, dx-\frac {1}{4} e^5 \int \frac {4+8 x+4 x^2+\log (3)}{(1+x)^2} \, dx-\left (2 e^{10} \log (3)\right ) \int \frac {4+4 x+\log (3)}{(1+x)^3} \, dx \\ & = e^{12+4 x^2+2 x^4}+\frac {x^2}{64}+\frac {e^{10} (4+4 x+\log (3))^2}{(1+x)^2}-\frac {2 e^{11+2 x^2+x^4} \left (4 x^5+x (4+\log (3))+x^2 (8+\log (3))+x^3 (8+\log (3))+x^4 (8+\log (3))\right )}{(1+x)^2 \left (x+x^3\right )}+\frac {1}{4} \int \left (e^{6+2 x^2+x^4}+4 e^{6+2 x^2+x^4} x^2+4 e^{6+2 x^2+x^4} x^4\right ) \, dx-\frac {1}{4} e^5 \int \left (4+\frac {\log (3)}{(1+x)^2}\right ) \, dx \\ & = e^{12+4 x^2+2 x^4}-e^5 x+\frac {x^2}{64}+\frac {e^5 \log (3)}{4 (1+x)}+\frac {e^{10} (4+4 x+\log (3))^2}{(1+x)^2}-\frac {2 e^{11+2 x^2+x^4} \left (4 x^5+x (4+\log (3))+x^2 (8+\log (3))+x^3 (8+\log (3))+x^4 (8+\log (3))\right )}{(1+x)^2 \left (x+x^3\right )}+\frac {1}{4} \int e^{6+2 x^2+x^4} \, dx+\int e^{6+2 x^2+x^4} x^2 \, dx+\int e^{6+2 x^2+x^4} x^4 \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(32)=64\).

Time = 0.99 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.41 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=e^{2 \left (6+2 x^2+x^4\right )}+\frac {1}{4} e^{6+2 x^2+x^4} x+\frac {x^2}{64}-\frac {e^5 \left (4+8 x+4 x^2-\log (3)\right )}{4 (1+x)}-\frac {2 e^{11+2 x^2+x^4} (4+4 x+\log (3))}{1+x}+\frac {e^{10} \log (3) (8+8 x+\log (3))}{(1+x)^2} \]

[In]

Integrate[(x + 3*x^2 + 3*x^3 + x^4 + E^5*(-32 - 96*x - 96*x^2 - 32*x^3) + E^(12 + 4*x^2 + 2*x^4)*(256*x + 768*
x^2 + 1024*x^3 + 1024*x^4 + 768*x^5 + 256*x^6) + (E^10*(-256 - 256*x) + E^5*(-8 - 8*x))*Log[3] - 64*E^10*Log[3
]^2 + E^(1 + 2*x^2 + x^4)*(E^10*(-1024*x - 3072*x^2 - 4096*x^3 - 4096*x^4 - 3072*x^5 - 1024*x^6) + E^5*(8 + 24
*x + 56*x^2 + 104*x^3 + 128*x^4 + 128*x^5 + 96*x^6 + 32*x^7) + E^10*(64 - 192*x - 512*x^2 - 512*x^3 - 512*x^4
- 256*x^5)*Log[3]))/(32 + 96*x + 96*x^2 + 32*x^3),x]

[Out]

E^(2*(6 + 2*x^2 + x^4)) + (E^(6 + 2*x^2 + x^4)*x)/4 + x^2/64 - (E^5*(4 + 8*x + 4*x^2 - Log[3]))/(4*(1 + x)) -
(2*E^(11 + 2*x^2 + x^4)*(4 + 4*x + Log[3]))/(1 + x) + (E^10*Log[3]*(8 + 8*x + Log[3]))/(1 + x)^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(28)=56\).

Time = 0.60 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.56

method	result	size
risch	\(\frac {x^{2}}{64}-x \,{\mathrm e}^{5}+\frac {\frac {\left (8 \,{\mathrm e}^{5} \ln \left (3\right )+256 \,{\mathrm e}^{10} \ln \left (3\right )\right ) x}{32}+{\mathrm e}^{10} \ln \left (3\right )^{2}+8 \,{\mathrm e}^{10} \ln \left (3\right )+\frac {{\mathrm e}^{5} \ln \left (3\right )}{4}}{x^{2}+2 x +1}+{\mathrm e}^{2 x^{4}+4 x^{2}+12}-\frac {\left (8 \,{\mathrm e}^{5} \ln \left (3\right )+32 x \,{\mathrm e}^{5}-x^{2}+32 \,{\mathrm e}^{5}-x \right ) {\mathrm e}^{x^{4}+2 x^{2}+6}}{4 \left (1+x \right )}\)	\(114\)
parts	\(\frac {x^{2}}{64}-x \,{\mathrm e}^{5}+\frac {{\mathrm e}^{10} \ln \left (3\right )^{2}}{\left (1+x \right )^{2}}+\frac {\ln \left (3\right ) \left ({\mathrm e}^{5}+32 \,{\mathrm e}^{10}\right )}{4+4 x}+\frac {\left (-2 \,{\mathrm e}^{10} \ln \left (3\right )-8 \,{\mathrm e}^{10}\right ) {\mathrm e}^{x^{4}+2 x^{2}+1}+\left (\frac {{\mathrm e}^{5}}{4}-8 \,{\mathrm e}^{10}\right ) x \,{\mathrm e}^{x^{4}+2 x^{2}+1}+\frac {{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{5} x^{2}}{4}}{1+x}+{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2}\)	\(134\)
norman	\(\frac {\left (\frac {1}{32}-{\mathrm e}^{5}\right ) x^{3}+\left (-2 \,{\mathrm e}^{10} \ln \left (3\right )-8 \,{\mathrm e}^{10}\right ) {\mathrm e}^{x^{4}+2 x^{2}+1}+\left (-\frac {1}{32}+3 \,{\mathrm e}^{5}+8 \,{\mathrm e}^{10} \ln \left (3\right )+\frac {{\mathrm e}^{5} \ln \left (3\right )}{4}\right ) x +{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2}+\left (\frac {{\mathrm e}^{5}}{2}-8 \,{\mathrm e}^{10}\right ) x^{2} {\mathrm e}^{x^{4}+2 x^{2}+1}+\left (-2 \,{\mathrm e}^{10} \ln \left (3\right )-16 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{5}}{4}\right ) x \,{\mathrm e}^{x^{4}+2 x^{2}+1}+{\mathrm e}^{2 x^{4}+4 x^{2}+2} {\mathrm e}^{10} x^{2}+\frac {x^{4}}{64}+\frac {{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{5} x^{3}}{4}+2 \,{\mathrm e}^{2 x^{4}+4 x^{2}+2} {\mathrm e}^{10} x -\frac {1}{64}+2 \,{\mathrm e}^{5}+8 \,{\mathrm e}^{10} \ln \left (3\right )+\frac {{\mathrm e}^{5} \ln \left (3\right )}{4}+{\mathrm e}^{10} \ln \left (3\right )^{2}}{\left (1+x \right )^{2}}\)	\(235\)
parallelrisch	\(\frac {-1-2 x +16 x \,{\mathrm e}^{5} \ln \left (3\right )-64 x^{3} {\mathrm e}^{5}-128 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{10} \ln \left (3\right ) x +16 \,{\mathrm e}^{5} \ln \left (3\right )+64 \,{\mathrm e}^{10} \ln \left (3\right )^{2}+192 x \,{\mathrm e}^{5}+128 \,{\mathrm e}^{5}+x^{4}+2 x^{3}+16 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{5} x^{3}-128 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{10} \ln \left (3\right )-1024 \,{\mathrm e}^{x^{4}+2 x^{2}+1} x \,{\mathrm e}^{10}+32 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{5} x^{2}+512 \,{\mathrm e}^{10} \ln \left (3\right ) x +16 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{5} x +64 \,{\mathrm e}^{2 x^{4}+4 x^{2}+2} {\mathrm e}^{10} x^{2}+128 \,{\mathrm e}^{2 x^{4}+4 x^{2}+2} {\mathrm e}^{10} x -512 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{10} x^{2}+64 \,{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2}-512 \,{\mathrm e}^{x^{4}+2 x^{2}+1} {\mathrm e}^{10}+512 \,{\mathrm e}^{10} \ln \left (3\right )}{64 x^{2}+128 x +64}\)	\(289\)

[In]

int(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp(x^4+2*x^2+1)^2+((-256*x^5-512*x^4-512*x^3-
512*x^2-192*x+64)*exp(5)^2*ln(3)+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^7+96*x^
6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64*exp(5)^2*ln(3)^2+((-256*x-256)*exp(5)^2+(
-8*x-8)*exp(5))*ln(3)+(-32*x^3-96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x,method=_RET
URNVERBOSE)

[Out]

1/64*x^2-x*exp(5)+(1/32*(8*exp(5)*ln(3)+256*exp(10)*ln(3))*x+exp(10)*ln(3)^2+8*exp(10)*ln(3)+1/4*exp(5)*ln(3))
/(x^2+2*x+1)+exp(2*x^4+4*x^2+12)-1/4/(1+x)*(8*exp(5)*ln(3)+32*x*exp(5)-x^2+32*exp(5)-x)*exp(x^4+2*x^2+6)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (32) = 64\).

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.25 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {x^{4} + 2 \, x^{3} + 64 \, e^{10} \log \left (3\right )^{2} + x^{2} - 64 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{5} + 64 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 16 \, {\left (8 \, {\left (x + 1\right )} e^{10} \log \left (3\right ) + 32 \, {\left (x^{2} + 2 \, x + 1\right )} e^{10} - {\left (x^{3} + 2 \, x^{2} + x\right )} e^{5}\right )} e^{\left (x^{4} + 2 \, x^{2} + 1\right )} + 16 \, {\left (32 \, {\left (x + 1\right )} e^{10} + {\left (x + 1\right )} e^{5}\right )} \log \left (3\right )}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[In]

integrate(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp(x^4+2*x^2+1)^2+((-256*x^5-512*x^4-51
2*x^3-512*x^2-192*x+64)*exp(5)^2*log(3)+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^
7+96*x^6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64*exp(5)^2*log(3)^2+((-256*x-256)*ex
p(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3-96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x, a
lgorithm="fricas")

[Out]

1/64*(x^4 + 2*x^3 + 64*e^10*log(3)^2 + x^2 - 64*(x^3 + 2*x^2 + x)*e^5 + 64*(x^2 + 2*x + 1)*e^(2*x^4 + 4*x^2 +
12) - 16*(8*(x + 1)*e^10*log(3) + 32*(x^2 + 2*x + 1)*e^10 - (x^3 + 2*x^2 + x)*e^5)*e^(x^4 + 2*x^2 + 1) + 16*(3
2*(x + 1)*e^10 + (x + 1)*e^5)*log(3))/(x^2 + 2*x + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (24) = 48\).

Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.25 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {x^{2}}{64} - x e^{5} + \frac {x \left (e^{5} \log {\left (3 \right )} + 32 e^{10} \log {\left (3 \right )}\right ) + e^{5} \log {\left (3 \right )} + 4 e^{10} \log {\left (3 \right )}^{2} + 32 e^{10} \log {\left (3 \right )}}{4 x^{2} + 8 x + 4} + \frac {\left (4 x e^{10} + 4 e^{10}\right ) e^{2 x^{4} + 4 x^{2} + 2} + \left (x^{2} e^{5} - 32 x e^{10} + x e^{5} - 32 e^{10} - 8 e^{10} \log {\left (3 \right )}\right ) e^{x^{4} + 2 x^{2} + 1}}{4 x + 4} \]

[In]

integrate(((256*x**6+768*x**5+1024*x**4+1024*x**3+768*x**2+256*x)*exp(5)**2*exp(x**4+2*x**2+1)**2+((-256*x**5-
512*x**4-512*x**3-512*x**2-192*x+64)*exp(5)**2*ln(3)+(-1024*x**6-3072*x**5-4096*x**4-4096*x**3-3072*x**2-1024*
x)*exp(5)**2+(32*x**7+96*x**6+128*x**5+128*x**4+104*x**3+56*x**2+24*x+8)*exp(5))*exp(x**4+2*x**2+1)-64*exp(5)*
*2*ln(3)**2+((-256*x-256)*exp(5)**2+(-8*x-8)*exp(5))*ln(3)+(-32*x**3-96*x**2-96*x-32)*exp(5)+x**4+3*x**3+3*x**
2+x)/(32*x**3+96*x**2+96*x+32),x)

[Out]

x**2/64 - x*exp(5) + (x*(exp(5)*log(3) + 32*exp(10)*log(3)) + exp(5)*log(3) + 4*exp(10)*log(3)**2 + 32*exp(10)
*log(3))/(4*x**2 + 8*x + 4) + ((4*x*exp(10) + 4*exp(10))*exp(2*x**4 + 4*x**2 + 2) + (x**2*exp(5) - 32*x*exp(10
) + x*exp(5) - 32*exp(10) - 8*exp(10)*log(3))*exp(x**4 + 2*x**2 + 1))/(4*x + 4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (32) = 64\).

Time = 0.36 (sec) , antiderivative size = 323, normalized size of antiderivative = 10.09 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {1}{64} \, x^{2} - \frac {1}{2} \, {\left (2 \, x - \frac {6 \, x + 5}{x^{2} + 2 \, x + 1} - 6 \, \log \left (x + 1\right )\right )} e^{5} - \frac {3}{2} \, {\left (\frac {4 \, x + 3}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right )\right )} e^{5} + \frac {4 \, {\left (2 \, x + 1\right )} e^{10} \log \left (3\right )}{x^{2} + 2 \, x + 1} + \frac {{\left (2 \, x + 1\right )} e^{5} \log \left (3\right )}{8 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {e^{10} \log \left (3\right )^{2}}{x^{2} + 2 \, x + 1} + \frac {3 \, {\left (2 \, x + 1\right )} e^{5}}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {4 \, e^{10} \log \left (3\right )}{x^{2} + 2 \, x + 1} + \frac {e^{5} \log \left (3\right )}{8 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {8 \, x + 7}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {3 \, {\left (6 \, x + 5\right )}}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {3 \, {\left (4 \, x + 3\right )}}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {2 \, x + 1}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {4 \, {\left (x e^{12} + e^{12}\right )} e^{\left (2 \, x^{4} + 4 \, x^{2}\right )} + {\left (x^{2} e^{6} - x {\left (32 \, e^{11} - e^{6}\right )} - 8 \, {\left (\log \left (3\right ) + 4\right )} e^{11}\right )} e^{\left (x^{4} + 2 \, x^{2}\right )}}{4 \, {\left (x + 1\right )}} + \frac {e^{5}}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[In]

integrate(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp(x^4+2*x^2+1)^2+((-256*x^5-512*x^4-51
2*x^3-512*x^2-192*x+64)*exp(5)^2*log(3)+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^
7+96*x^6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64*exp(5)^2*log(3)^2+((-256*x-256)*ex
p(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3-96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x, a
lgorithm="maxima")

[Out]

1/64*x^2 - 1/2*(2*x - (6*x + 5)/(x^2 + 2*x + 1) - 6*log(x + 1))*e^5 - 3/2*((4*x + 3)/(x^2 + 2*x + 1) + 2*log(x
+ 1))*e^5 + 4*(2*x + 1)*e^10*log(3)/(x^2 + 2*x + 1) + 1/8*(2*x + 1)*e^5*log(3)/(x^2 + 2*x + 1) + e^10*log(3)^
2/(x^2 + 2*x + 1) + 3/2*(2*x + 1)*e^5/(x^2 + 2*x + 1) + 4*e^10*log(3)/(x^2 + 2*x + 1) + 1/8*e^5*log(3)/(x^2 +
2*x + 1) + 1/64*(8*x + 7)/(x^2 + 2*x + 1) - 3/64*(6*x + 5)/(x^2 + 2*x + 1) + 3/64*(4*x + 3)/(x^2 + 2*x + 1) -
1/64*(2*x + 1)/(x^2 + 2*x + 1) + 1/4*(4*(x*e^12 + e^12)*e^(2*x^4 + 4*x^2) + (x^2*e^6 - x*(32*e^11 - e^6) - 8*(
log(3) + 4)*e^11)*e^(x^4 + 2*x^2))/(x + 1) + 1/2*e^5/(x^2 + 2*x + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (32) = 64\).

Time = 0.34 (sec) , antiderivative size = 246, normalized size of antiderivative = 7.69 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {x^{4} - 64 \, x^{3} e^{5} + 16 \, x^{3} e^{\left (x^{4} + 2 \, x^{2} + 6\right )} + 2 \, x^{3} - 128 \, x^{2} e^{5} + 64 \, x^{2} e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 512 \, x^{2} e^{\left (x^{4} + 2 \, x^{2} + 11\right )} + 32 \, x^{2} e^{\left (x^{4} + 2 \, x^{2} + 6\right )} + 512 \, x e^{10} \log \left (3\right ) + 16 \, x e^{5} \log \left (3\right ) - 128 \, x e^{\left (x^{4} + 2 \, x^{2} + 11\right )} \log \left (3\right ) + 64 \, e^{10} \log \left (3\right )^{2} + x^{2} - 64 \, x e^{5} + 128 \, x e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 1024 \, x e^{\left (x^{4} + 2 \, x^{2} + 11\right )} + 16 \, x e^{\left (x^{4} + 2 \, x^{2} + 6\right )} + 512 \, e^{10} \log \left (3\right ) + 16 \, e^{5} \log \left (3\right ) - 128 \, e^{\left (x^{4} + 2 \, x^{2} + 11\right )} \log \left (3\right ) + 64 \, e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 512 \, e^{\left (x^{4} + 2 \, x^{2} + 11\right )}}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[In]

integrate(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp(x^4+2*x^2+1)^2+((-256*x^5-512*x^4-51
2*x^3-512*x^2-192*x+64)*exp(5)^2*log(3)+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^
7+96*x^6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64*exp(5)^2*log(3)^2+((-256*x-256)*ex
p(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3-96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x, a
lgorithm="giac")

[Out]

1/64*(x^4 - 64*x^3*e^5 + 16*x^3*e^(x^4 + 2*x^2 + 6) + 2*x^3 - 128*x^2*e^5 + 64*x^2*e^(2*x^4 + 4*x^2 + 12) - 51
2*x^2*e^(x^4 + 2*x^2 + 11) + 32*x^2*e^(x^4 + 2*x^2 + 6) + 512*x*e^10*log(3) + 16*x*e^5*log(3) - 128*x*e^(x^4 +
2*x^2 + 11)*log(3) + 64*e^10*log(3)^2 + x^2 - 64*x*e^5 + 128*x*e^(2*x^4 + 4*x^2 + 12) - 1024*x*e^(x^4 + 2*x^2
+ 11) + 16*x*e^(x^4 + 2*x^2 + 6) + 512*e^10*log(3) + 16*e^5*log(3) - 128*e^(x^4 + 2*x^2 + 11)*log(3) + 64*e^(
2*x^4 + 4*x^2 + 12) - 512*e^(x^4 + 2*x^2 + 11))/(x^2 + 2*x + 1)

Mupad [B] (verification not implemented)

Time = 14.07 (sec) , antiderivative size = 218, normalized size of antiderivative = 6.81 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=-\frac {x^2\,\left (64\,{\mathrm {e}}^5+8\,{\mathrm {e}}^5\,\ln \left (3\right )+256\,{\mathrm {e}}^{10}\,\ln \left (3\right )+32\,{\mathrm {e}}^{10}\,{\ln \left (3\right )}^2-\frac {1}{2}\right )-64\,x\,{\mathrm {e}}^{2\,x^4+4\,x^2+12}-8\,x^3\,{\mathrm {e}}^{x^4+2\,x^2+6}-32\,{\mathrm {e}}^{2\,x^4+4\,x^2+12}+x^3\,\left (32\,{\mathrm {e}}^5-1\right )+{\mathrm {e}}^{x^4+2\,x^2+11}\,\left (64\,\ln \left (3\right )+256\right )-32\,x^2\,{\mathrm {e}}^{2\,x^4+4\,x^2+12}-\frac {x^4}{2}+x\,\left (32\,{\mathrm {e}}^5+8\,{\mathrm {e}}^5\,\ln \left (3\right )+256\,{\mathrm {e}}^{10}\,\ln \left (3\right )+64\,{\mathrm {e}}^{10}\,{\ln \left (3\right )}^2\right )+8\,x\,{\mathrm {e}}^{x^4+2\,x^2+6}\,\left (64\,{\mathrm {e}}^5+8\,{\mathrm {e}}^5\,\ln \left (3\right )-1\right )+16\,x^2\,{\mathrm {e}}^{x^4+2\,x^2+6}\,\left (16\,{\mathrm {e}}^5-1\right )}{32\,x^2+64\,x+32} \]

[In]

int((x - log(3)*(exp(5)*(8*x + 8) + exp(10)*(256*x + 256)) - exp(2*x^2 + x^4 + 1)*(exp(10)*(1024*x + 3072*x^2
+ 4096*x^3 + 4096*x^4 + 3072*x^5 + 1024*x^6) - exp(5)*(24*x + 56*x^2 + 104*x^3 + 128*x^4 + 128*x^5 + 96*x^6 +
32*x^7 + 8) + exp(10)*log(3)*(192*x + 512*x^2 + 512*x^3 + 512*x^4 + 256*x^5 - 64)) - exp(5)*(96*x + 96*x^2 + 3
2*x^3 + 32) - 64*exp(10)*log(3)^2 + 3*x^2 + 3*x^3 + x^4 + exp(4*x^2 + 2*x^4 + 2)*exp(10)*(256*x + 768*x^2 + 10
24*x^3 + 1024*x^4 + 768*x^5 + 256*x^6))/(96*x + 96*x^2 + 32*x^3 + 32),x)

[Out]

-(x^2*(64*exp(5) + 8*exp(5)*log(3) + 256*exp(10)*log(3) + 32*exp(10)*log(3)^2 - 1/2) - 64*x*exp(4*x^2 + 2*x^4
+ 12) - 8*x^3*exp(2*x^2 + x^4 + 6) - 32*exp(4*x^2 + 2*x^4 + 12) + x^3*(32*exp(5) - 1) + exp(2*x^2 + x^4 + 11)*
(64*log(3) + 256) - 32*x^2*exp(4*x^2 + 2*x^4 + 12) - x^4/2 + x*(32*exp(5) + 8*exp(5)*log(3) + 256*exp(10)*log(
3) + 64*exp(10)*log(3)^2) + 8*x*exp(2*x^2 + x^4 + 6)*(64*exp(5) + 8*exp(5)*log(3) - 1) + 16*x^2*exp(2*x^2 + x^
4 + 6)*(16*exp(5) - 1))/(64*x + 32*x^2 + 32)