3.4.79 \(\int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} (-4 x^2+2 x^2 \log (2))+e^{3 x^2} (-16 x^3+8 x^3 \log (2))+(-4 x^3-3 x^2 \log (2)) \log (4)+e^{2 x^2} (-24 x^4+12 x^4 \log (2)+(-2 x-4 x^3+(-1-4 x^2) \log (2)) \log (4))+e^{x^2} (-16 x^5+8 x^5 \log (2)+(-6 x^2-4 x^4+(-4 x-4 x^3) \log (2)) \log (4))}{16 x^6+16 x^7+4 x^8+e^{4 x^2} (16 x^2+16 x^3+4 x^4)+e^{3 x^2} (64 x^3+64 x^4+16 x^5)+(8 x^3+4 x^4) \log (4)+\log ^2(4)+e^{2 x^2} (96 x^4+96 x^5+24 x^6+(8 x+4 x^2) \log (4))+e^{x^2} (64 x^5+64 x^6+16 x^7+(16 x^2+8 x^3) \log (4))} \, dx\) [379]
Optimal. Leaf size=31 \[ \frac {-x-\log (2)}{4+2 x+\frac {\log (4)}{x \left (e^{x^2}+x\right )^2}} \]
[Out]
(-x-ln(2))/(2*x+2*ln(2)/(exp(x^2)+x)^2/x+4)
________________________________________________________________________________________
Rubi [F]
time = 142.50, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^(3*x^2)*(-16*x^3 + 8*x^3*Log[2]) + (-4*x^3
- 3*x^2*Log[2])*Log[4] + E^(2*x^2)*(-24*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4]) + E
^x^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2])*Log[4]))/(16*x^6 + 16*x^7 + 4*x^8 + E^
(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[4]^2 +
E^(2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64*x^5 + 64*x^6 + 16*x^7 + (16*x^2 + 8*x
^3)*Log[4])),x]
[Out]
(4 - Log[4])/(4*(2 + x)) - 768*Log[4]*Defer[Int][(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x)
+ Log[4])^(-2), x] + 16*(2 + Log[2])*Log[4]*Defer[Int][(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2
+ x) + Log[4])^(-2), x] + 320*(4 + Log[2])*Log[4]*Defer[Int][(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2
*x^2*(2 + x) + Log[4])^(-2), x] + 2*Log[2]*Log[4]^2*Defer[Int][(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^
2*x^2*(2 + x) + Log[4])^(-2), x] - 24*Log[4]*(4 - Log[128])*Defer[Int][(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x)
+ 4*E^x^2*x^2*(2 + x) + Log[4])^(-2), x] - 64*Log[4]*(7 + Log[256])*Defer[Int][(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*
(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^(-2), x] + 320*Log[4]*Defer[Int][E^x^2/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*
(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 8*(1 + Log[2])*Log[4]*Defer[Int][E^x^2/(4*x^3 + 2*x^4 + 2*E^(2
*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 128*(4 + Log[2])*Log[4]*Defer[Int][E^x^2/(4*x^3 + 2*x^
4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 2*Log[4]*(16 - 7*Log[16])*Defer[Int][E^x^2/(
4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 24*Log[4]*(7 + Log[256])*Defer[I
nt][E^x^2/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + (Log[4]^2*Log[16]*Def
er[Int][1/(x*(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2), x])/16 + 320*Log[4]*De
fer[Int][x/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 4*(2 + Log[2])*Log[4
]*Defer[Int][x/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 128*(4 + Log[2])
*Log[4]*Defer[Int][x/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 8*Log[4]*(
4 - Log[128])*Defer[Int][x/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 24*L
og[4]*(7 + Log[256])*Defer[Int][x/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x]
- 128*Log[4]*Defer[Int][(E^x^2*x)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x
] + 48*(4 + Log[2])*Log[4]*Defer[Int][(E^x^2*x)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) +
Log[4])^2, x] - (Log[4]*(16 - 7*Log[16])*Defer[Int][(E^x^2*x)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^
2*x^2*(2 + x) + Log[4])^2, x])/2 - 8*Log[4]*(7 + Log[256])*Defer[Int][(E^x^2*x)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x
*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 128*Log[4]*Defer[Int][x^2/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 +
x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 48*(4 + Log[2])*Log[4]*Defer[Int][x^2/(4*x^3 + 2*x^4 + 2*E^(2*x^2)
*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 2*Log[4]*(4 - Log[128])*Defer[Int][x^2/(4*x^3 + 2*x^4 + 2*E
^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 8*Log[4]*(7 + Log[256])*Defer[Int][x^2/(4*x^3 + 2*x
^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 48*Log[4]*Defer[Int][(E^x^2*x^2)/(4*x^3 + 2
*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 16*(4 + Log[2])*Log[4]*Defer[Int][(E^x^2*
x^2)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 2*Log[4]*(7 + Log[256])*De
fer[Int][(E^x^2*x^2)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 48*Log[4]*
Defer[Int][x^3/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 16*(4 + Log[2])*
Log[4]*Defer[Int][x^3/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] + 2*Log[4]*
(7 + Log[256])*Defer[Int][x^3/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] - 1
6*Log[4]*Defer[Int][(E^x^2*x^3)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2, x] +
4*(4 + Log[2])*Log[4]*Defer[Int][(E^x^2*x^3)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + L
og[4])^2, x] - 16*Log[4]*Defer[Int][x^4/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])
^2, x] + 4*(4 + Log[2])*Log[4]*Defer[Int][x^4/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + L
og[4])^2, x] + 4*Log[4]*Defer[Int][(E^x^2*x^4)/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) +
Log[4])^2, x] + 4*Log[4]*Defer[Int][x^5/(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])
^2, x] - 512*Log[4]*Defer[Int][1/((2 + x)^2*(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log
[4])^2), x] + 32*(2 + Log[2])*Log[4]*Defer[Int][1/((2 + x)^2*(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*
x^2*(2 + x) + Log[4])^2), x] + 256*(4 + Log[2])*Log[4]*Defer[Int][1/((2 + x)^2*(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*
(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2), x] + 8*Log[2]*Log[4]^2*Defer[Int][1/((2 + x)^2*(4*x^3 + 2*x^4 + 2*
E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2), x] + (1 + Log[2])*Log[4]^2*Defer[Int][1/((2 + x)^2*(4*
x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x) + 4*E^x^2*x^2*(2 + x) + Log[4])^2), x] - (Log[4]^2*Log[16]*Defer[Int][1/((
2 + x)^2*(4*x^3 + 2*x^4 + 2*E^(2*x^2)*x*(2 + x)...
Rubi steps
Aborted
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Mathematica [F]
time = 8.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^(3*x^2)*(-16*x^3 + 8*x^3*Log[2]) + (-
4*x^3 - 3*x^2*Log[2])*Log[4] + E^(2*x^2)*(-24*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4
]) + E^x^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2])*Log[4]))/(16*x^6 + 16*x^7 + 4*x^
8 + E^(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[
4]^2 + E^(2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64*x^5 + 64*x^6 + 16*x^7 + (16*x^2
+ 8*x^3)*Log[4])),x]
[Out]
Integrate[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^(3*x^2)*(-16*x^3 + 8*x^3*Log[2]) + (-
4*x^3 - 3*x^2*Log[2])*Log[4] + E^(2*x^2)*(-24*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4
]) + E^x^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2])*Log[4]))/(16*x^6 + 16*x^7 + 4*x^
8 + E^(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[
4]^2 + E^(2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64*x^5 + 64*x^6 + 16*x^7 + (16*x^2
+ 8*x^3)*Log[4])), x]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs.
\(2(31)=62\).
time = 0.14, size = 79, normalized size = 2.55
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {1}{2+x}-\frac {\ln \left (2\right )}{2 \left (2+x \right )}+\frac {\ln \left (2\right ) \left (\ln \left (2\right )+x \right )}{2 \left (2+x \right ) \left (x^{2} {\mathrm e}^{2 x^{2}}+2 x^{3} {\mathrm e}^{x^{2}}+x^{4}+2 x \,{\mathrm e}^{2 x^{2}}+4 x^{2} {\mathrm e}^{x^{2}}+2 x^{3}+\ln \left (2\right )\right )}\) |
\(79\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2*x^2*ln(2)-4*x^2)*exp(x^2)^4+(8*x^3*ln(2)-16*x^3)*exp(x^2)^3+(2*((-4*x^2-1)*ln(2)-4*x^3-2*x)*ln(2)+12*x
^4*ln(2)-24*x^4)*exp(x^2)^2+(2*((-4*x^3-4*x)*ln(2)-4*x^4-6*x^2)*ln(2)+8*x^5*ln(2)-16*x^5)*exp(x^2)+2*(-3*x^2*l
n(2)-4*x^3)*ln(2)+2*x^6*ln(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x^2)^4+(16*x^5+64*x^4+64*x^3)*exp(x^2)^3+(2*(4
*x^2+8*x)*ln(2)+24*x^6+96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*ln(2)+16*x^7+64*x^6+64*x^5)*exp(x^2)+4*ln(2
)^2+2*(4*x^4+8*x^3)*ln(2)+4*x^8+16*x^7+16*x^6),x,method=_RETURNVERBOSE)
[Out]
1/(2+x)-1/2*ln(2)/(2+x)+1/2*ln(2)*(ln(2)+x)/(2+x)/(x^2*exp(2*x^2)+2*x^3*exp(x^2)+x^4+2*x*exp(2*x^2)+4*x^2*exp(
x^2)+2*x^3+ln(2))
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (25) = 50\).
time = 0.61, size = 82, normalized size = 2.65 \begin {gather*} -\frac {x^{3} {\left (\log \left (2\right ) - 2\right )} + 2 \, x^{2} {\left (\log \left (2\right ) - 2\right )} e^{\left (x^{2}\right )} + x {\left (\log \left (2\right ) - 2\right )} e^{\left (2 \, x^{2}\right )} - \log \left (2\right )}{2 \, {\left (x^{4} + 2 \, x^{3} + {\left (x^{2} + 2 \, x\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (x^{2}\right )} + \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^3+(2*((-4*x^2-1)*log(2)-4*x^3-2*x)*l
og(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2+(2*((-4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^
2)+2*(-3*x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x^2)^4+(16*x^5+64*x^4+64*x^3)
*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6+96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x
^5)*exp(x^2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x, algorithm="maxima")
[Out]
-1/2*(x^3*(log(2) - 2) + 2*x^2*(log(2) - 2)*e^(x^2) + x*(log(2) - 2)*e^(2*x^2) - log(2))/(x^4 + 2*x^3 + (x^2 +
2*x)*e^(2*x^2) + 2*(x^3 + 2*x^2)*e^(x^2) + log(2))
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (25) = 50\).
time = 0.31, size = 93, normalized size = 3.00 \begin {gather*} \frac {2 \, x^{3} - {\left (x \log \left (2\right ) - 2 \, x\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} \log \left (2\right ) - 2 \, x^{2}\right )} e^{\left (x^{2}\right )} - {\left (x^{3} - 1\right )} \log \left (2\right )}{2 \, {\left (x^{4} + 2 \, x^{3} + {\left (x^{2} + 2 \, x\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (x^{2}\right )} + \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^3+(2*((-4*x^2-1)*log(2)-4*x^3-2*x)*l
og(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2+(2*((-4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^
2)+2*(-3*x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x^2)^4+(16*x^5+64*x^4+64*x^3)
*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6+96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x
^5)*exp(x^2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x, algorithm="fricas")
[Out]
1/2*(2*x^3 - (x*log(2) - 2*x)*e^(2*x^2) - 2*(x^2*log(2) - 2*x^2)*e^(x^2) - (x^3 - 1)*log(2))/(x^4 + 2*x^3 + (x
^2 + 2*x)*e^(2*x^2) + 2*(x^3 + 2*x^2)*e^(x^2) + log(2))
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (26) = 52\).
time = 0.33, size = 87, normalized size = 2.81 \begin {gather*} \frac {x \log {\left (2 \right )} + \log {\left (2 \right )}^{2}}{2 x^{5} + 8 x^{4} + 8 x^{3} + 2 x \log {\left (2 \right )} + \left (2 x^{3} + 8 x^{2} + 8 x\right ) e^{2 x^{2}} + \left (4 x^{4} + 16 x^{3} + 16 x^{2}\right ) e^{x^{2}} + 4 \log {\left (2 \right )}} - \frac {-2 + \log {\left (2 \right )}}{2 x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x**2*ln(2)-4*x**2)*exp(x**2)**4+(8*x**3*ln(2)-16*x**3)*exp(x**2)**3+(2*((-4*x**2-1)*ln(2)-4*x**3
-2*x)*ln(2)+12*x**4*ln(2)-24*x**4)*exp(x**2)**2+(2*((-4*x**3-4*x)*ln(2)-4*x**4-6*x**2)*ln(2)+8*x**5*ln(2)-16*x
**5)*exp(x**2)+2*(-3*x**2*ln(2)-4*x**3)*ln(2)+2*x**6*ln(2)-4*x**6)/((4*x**4+16*x**3+16*x**2)*exp(x**2)**4+(16*
x**5+64*x**4+64*x**3)*exp(x**2)**3+(2*(4*x**2+8*x)*ln(2)+24*x**6+96*x**5+96*x**4)*exp(x**2)**2+(2*(8*x**3+16*x
**2)*ln(2)+16*x**7+64*x**6+64*x**5)*exp(x**2)+4*ln(2)**2+2*(4*x**4+8*x**3)*ln(2)+4*x**8+16*x**7+16*x**6),x)
[Out]
(x*log(2) + log(2)**2)/(2*x**5 + 8*x**4 + 8*x**3 + 2*x*log(2) + (2*x**3 + 8*x**2 + 8*x)*exp(2*x**2) + (4*x**4
+ 16*x**3 + 16*x**2)*exp(x**2) + 4*log(2)) - (-2 + log(2))/(2*x + 4)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs.
\(2 (25) = 50\).
time = 2.87, size = 205, normalized size = 6.61 \begin {gather*} -\frac {x^{4} \log \left (2\right ) + 2 \, x^{3} e^{\left (x^{2}\right )} \log \left (2\right ) - 2 \, x^{4} - 4 \, x^{3} e^{\left (x^{2}\right )} + 2 \, x^{3} \log \left (2\right ) + x^{2} e^{\left (2 \, x^{2}\right )} \log \left (2\right ) + 4 \, x^{2} e^{\left (x^{2}\right )} \log \left (2\right ) - 4 \, x^{3} - 2 \, x^{2} e^{\left (2 \, x^{2}\right )} - 8 \, x^{2} e^{\left (x^{2}\right )} + 2 \, x e^{\left (2 \, x^{2}\right )} \log \left (2\right ) - 4 \, x e^{\left (2 \, x^{2}\right )} - 2 \, x \log \left (2\right ) - \log \left (2\right )^{2} - 2 \, \log \left (2\right )}{2 \, {\left (x^{5} + 2 \, x^{4} e^{\left (x^{2}\right )} + 4 \, x^{4} + x^{3} e^{\left (2 \, x^{2}\right )} + 8 \, x^{3} e^{\left (x^{2}\right )} + 4 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x^{2}\right )} + 8 \, x^{2} e^{\left (x^{2}\right )} + 4 \, x e^{\left (2 \, x^{2}\right )} + x \log \left (2\right ) + 2 \, \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^2*log(2)-4*x^2)*exp(x^2)^4+(8*x^3*log(2)-16*x^3)*exp(x^2)^3+(2*((-4*x^2-1)*log(2)-4*x^3-2*x)*l
og(2)+12*x^4*log(2)-24*x^4)*exp(x^2)^2+(2*((-4*x^3-4*x)*log(2)-4*x^4-6*x^2)*log(2)+8*x^5*log(2)-16*x^5)*exp(x^
2)+2*(-3*x^2*log(2)-4*x^3)*log(2)+2*x^6*log(2)-4*x^6)/((4*x^4+16*x^3+16*x^2)*exp(x^2)^4+(16*x^5+64*x^4+64*x^3)
*exp(x^2)^3+(2*(4*x^2+8*x)*log(2)+24*x^6+96*x^5+96*x^4)*exp(x^2)^2+(2*(8*x^3+16*x^2)*log(2)+16*x^7+64*x^6+64*x
^5)*exp(x^2)+4*log(2)^2+2*(4*x^4+8*x^3)*log(2)+4*x^8+16*x^7+16*x^6),x, algorithm="giac")
[Out]
-1/2*(x^4*log(2) + 2*x^3*e^(x^2)*log(2) - 2*x^4 - 4*x^3*e^(x^2) + 2*x^3*log(2) + x^2*e^(2*x^2)*log(2) + 4*x^2*
e^(x^2)*log(2) - 4*x^3 - 2*x^2*e^(2*x^2) - 8*x^2*e^(x^2) + 2*x*e^(2*x^2)*log(2) - 4*x*e^(2*x^2) - 2*x*log(2) -
log(2)^2 - 2*log(2))/(x^5 + 2*x^4*e^(x^2) + 4*x^4 + x^3*e^(2*x^2) + 8*x^3*e^(x^2) + 4*x^3 + 4*x^2*e^(2*x^2) +
8*x^2*e^(x^2) + 4*x*e^(2*x^2) + x*log(2) + 2*log(2))
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,\ln \left (2\right )\,\left (4\,x^3+3\,\ln \left (2\right )\,x^2\right )-{\mathrm {e}}^{4\,x^2}\,\left (2\,x^2\,\ln \left (2\right )-4\,x^2\right )-{\mathrm {e}}^{3\,x^2}\,\left (8\,x^3\,\ln \left (2\right )-16\,x^3\right )-2\,x^6\,\ln \left (2\right )+{\mathrm {e}}^{x^2}\,\left (2\,\ln \left (2\right )\,\left (\ln \left (2\right )\,\left (4\,x^3+4\,x\right )+6\,x^2+4\,x^4\right )-8\,x^5\,\ln \left (2\right )+16\,x^5\right )+{\mathrm {e}}^{2\,x^2}\,\left (2\,\ln \left (2\right )\,\left (2\,x+\ln \left (2\right )\,\left (4\,x^2+1\right )+4\,x^3\right )-12\,x^4\,\ln \left (2\right )+24\,x^4\right )+4\,x^6}{{\mathrm {e}}^{2\,x^2}\,\left (2\,\ln \left (2\right )\,\left (4\,x^2+8\,x\right )+96\,x^4+96\,x^5+24\,x^6\right )+{\mathrm {e}}^{x^2}\,\left (2\,\ln \left (2\right )\,\left (8\,x^3+16\,x^2\right )+64\,x^5+64\,x^6+16\,x^7\right )+2\,\ln \left (2\right )\,\left (4\,x^4+8\,x^3\right )+4\,{\ln \left (2\right )}^2+16\,x^6+16\,x^7+4\,x^8+{\mathrm {e}}^{4\,x^2}\,\left (4\,x^4+16\,x^3+16\,x^2\right )+{\mathrm {e}}^{3\,x^2}\,\left (16\,x^5+64\,x^4+64\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(2*log(2)*(3*x^2*log(2) + 4*x^3) - exp(4*x^2)*(2*x^2*log(2) - 4*x^2) - exp(3*x^2)*(8*x^3*log(2) - 16*x^3)
- 2*x^6*log(2) + exp(x^2)*(2*log(2)*(log(2)*(4*x + 4*x^3) + 6*x^2 + 4*x^4) - 8*x^5*log(2) + 16*x^5) + exp(2*x
^2)*(2*log(2)*(2*x + log(2)*(4*x^2 + 1) + 4*x^3) - 12*x^4*log(2) + 24*x^4) + 4*x^6)/(exp(2*x^2)*(2*log(2)*(8*x
+ 4*x^2) + 96*x^4 + 96*x^5 + 24*x^6) + exp(x^2)*(2*log(2)*(16*x^2 + 8*x^3) + 64*x^5 + 64*x^6 + 16*x^7) + 2*lo
g(2)*(8*x^3 + 4*x^4) + 4*log(2)^2 + 16*x^6 + 16*x^7 + 4*x^8 + exp(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + exp(3*x^2
)*(64*x^3 + 64*x^4 + 16*x^5)),x)
[Out]
int(-(2*log(2)*(3*x^2*log(2) + 4*x^3) - exp(4*x^2)*(2*x^2*log(2) - 4*x^2) - exp(3*x^2)*(8*x^3*log(2) - 16*x^3)
- 2*x^6*log(2) + exp(x^2)*(2*log(2)*(log(2)*(4*x + 4*x^3) + 6*x^2 + 4*x^4) - 8*x^5*log(2) + 16*x^5) + exp(2*x
^2)*(2*log(2)*(2*x + log(2)*(4*x^2 + 1) + 4*x^3) - 12*x^4*log(2) + 24*x^4) + 4*x^6)/(exp(2*x^2)*(2*log(2)*(8*x
+ 4*x^2) + 96*x^4 + 96*x^5 + 24*x^6) + exp(x^2)*(2*log(2)*(16*x^2 + 8*x^3) + 64*x^5 + 64*x^6 + 16*x^7) + 2*lo
g(2)*(8*x^3 + 4*x^4) + 4*log(2)^2 + 16*x^6 + 16*x^7 + 4*x^8 + exp(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + exp(3*x^2
)*(64*x^3 + 64*x^4 + 16*x^5)), x)
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