3.79.43 \(\int \frac {e^{2+\frac {1}{4+e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)}-4 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)}}-16 x^4+32 x^2 \log (4)-16 \log ^2(4)} (-128 x^4+128 x^2 \log (4))}{-8 x+e^{6-48 x^4+96 x^2 \log (4)-48 \log ^2(4)} x-6 e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)} x+12 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)} x} \, dx\)

Optimal. Leaf size=28 \[ 1-e^{\frac {1}{\left (2-e^{2-16 \left (x^2-\log (4)\right )^2}\right )^2}} \]

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Rubi [F]  time = 27.06, 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 {\exp \left (2+\frac {1}{4+e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)}-4 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)}}-16 x^4+32 x^2 \log (4)-16 \log ^2(4)\right ) \left (-128 x^4+128 x^2 \log (4)\right )}{-8 x+e^{6-48 x^4+96 x^2 \log (4)-48 \log ^2(4)} x-6 e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)} x+12 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2 + (4 + E^(4 - 32*x^4 + 64*x^2*Log[4] - 32*Log[4]^2) - 4*E^(2 - 16*x^4 + 32*x^2*Log[4] - 16*Log[4]^2)
)^(-1) - 16*x^4 + 32*x^2*Log[4] - 16*Log[4]^2)*(-128*x^4 + 128*x^2*Log[4]))/(-8*x + E^(6 - 48*x^4 + 96*x^2*Log
[4] - 48*Log[4]^2)*x - 6*E^(4 - 32*x^4 + 64*x^2*Log[4] - 32*Log[4]^2)*x + 12*E^(2 - 16*x^4 + 32*x^2*Log[4] - 1
6*Log[4]^2)*x),x]

[Out]

-1/2*(Log[4]*Defer[Subst][Defer[Int][(2^(7 + 64*x)*E^(E^(32*(x^2 + Log[4]^2))/(2^(64*x)*E^2 - 2*E^(16*(x^2 + L
og[4]^2)))^2 + 32*x^2 + 2*(1 + 16*Log[4]^2)))/(-(2^(64*x)*E^2) + 2*E^(16*x^2 + 16*Log[4]^2))^3, x], x, x^2]) +
Defer[Subst][Defer[Int][(2^(7 + 64*x)*E^(E^(32*(x^2 + Log[4]^2))/(2^(64*x)*E^2 - 2*E^(16*(x^2 + Log[4]^2)))^2
+ 32*x^2 + 2*(1 + 16*Log[4]^2))*x)/(-(2^(64*x)*E^2) + 2*E^(16*x^2 + 16*Log[4]^2))^3, x], x, x^2]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (2+\frac {1}{4+e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)}-4 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)}}-16 x^4+32 x^2 \log (4)-16 \log ^2(4)\right ) x^2 \left (-128 x^2+128 \log (4)\right )}{-8 x+e^{6-48 x^4+96 x^2 \log (4)-48 \log ^2(4)} x-6 e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)} x+12 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)} x} \, dx\\ &=\int \frac {2^{7+64 x^2} \exp \left (\frac {e^{32 \left (x^4+\log ^2(4)\right )}}{\left (2^{64 x^2} e^2-2 e^{16 \left (x^4+\log ^2(4)\right )}\right )^2}+32 x^4+2 \left (1+16 \log ^2(4)\right )\right ) x \left (-x^2+\log (4)\right )}{\left (2^{64 x^2} e^2-2 e^{16 \left (x^4+\log ^2(4)\right )}\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2^{7+64 x} \exp \left (\frac {e^{32 \left (x^2+\log ^2(4)\right )}}{\left (2^{64 x} e^2-2 e^{16 \left (x^2+\log ^2(4)\right )}\right )^2}+32 x^2+2 \left (1+16 \log ^2(4)\right )\right ) (-x+\log (4))}{\left (2^{64 x} e^2-2 e^{16 \left (x^2+\log ^2(4)\right )}\right )^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2^{7+64 x} \exp \left (\frac {e^{32 \left (x^2+\log ^2(4)\right )}}{\left (2^{64 x} e^2-2 e^{16 \left (x^2+\log ^2(4)\right )}\right )^2}+32 x^2+2 \left (1+16 \log ^2(4)\right )\right ) x}{\left (-2^{64 x} e^2+2 e^{16 x^2+16 \log ^2(4)}\right )^3}-\frac {2^{7+64 x} \exp \left (\frac {e^{32 \left (x^2+\log ^2(4)\right )}}{\left (2^{64 x} e^2-2 e^{16 \left (x^2+\log ^2(4)\right )}\right )^2}+32 x^2+2 \left (1+16 \log ^2(4)\right )\right ) \log (4)}{\left (-2^{64 x} e^2+2 e^{16 x^2+16 \log ^2(4)}\right )^3}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2^{7+64 x} \exp \left (\frac {e^{32 \left (x^2+\log ^2(4)\right )}}{\left (2^{64 x} e^2-2 e^{16 \left (x^2+\log ^2(4)\right )}\right )^2}+32 x^2+2 \left (1+16 \log ^2(4)\right )\right ) x}{\left (-2^{64 x} e^2+2 e^{16 x^2+16 \log ^2(4)}\right )^3} \, dx,x,x^2\right )-\frac {1}{2} \log (4) \operatorname {Subst}\left (\int \frac {2^{7+64 x} \exp \left (\frac {e^{32 \left (x^2+\log ^2(4)\right )}}{\left (2^{64 x} e^2-2 e^{16 \left (x^2+\log ^2(4)\right )}\right )^2}+32 x^2+2 \left (1+16 \log ^2(4)\right )\right )}{\left (-2^{64 x} e^2+2 e^{16 x^2+16 \log ^2(4)}\right )^3} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 9.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{2+\frac {1}{4+e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)}-4 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)}}-16 x^4+32 x^2 \log (4)-16 \log ^2(4)} \left (-128 x^4+128 x^2 \log (4)\right )}{-8 x+e^{6-48 x^4+96 x^2 \log (4)-48 \log ^2(4)} x-6 e^{4-32 x^4+64 x^2 \log (4)-32 \log ^2(4)} x+12 e^{2-16 x^4+32 x^2 \log (4)-16 \log ^2(4)} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^(2 + (4 + E^(4 - 32*x^4 + 64*x^2*Log[4] - 32*Log[4]^2) - 4*E^(2 - 16*x^4 + 32*x^2*Log[4] - 16*Log
[4]^2))^(-1) - 16*x^4 + 32*x^2*Log[4] - 16*Log[4]^2)*(-128*x^4 + 128*x^2*Log[4]))/(-8*x + E^(6 - 48*x^4 + 96*x
^2*Log[4] - 48*Log[4]^2)*x - 6*E^(4 - 32*x^4 + 64*x^2*Log[4] - 32*Log[4]^2)*x + 12*E^(2 - 16*x^4 + 32*x^2*Log[
4] - 16*Log[4]^2)*x),x]

[Out]

Integrate[(E^(2 + (4 + E^(4 - 32*x^4 + 64*x^2*Log[4] - 32*Log[4]^2) - 4*E^(2 - 16*x^4 + 32*x^2*Log[4] - 16*Log
[4]^2))^(-1) - 16*x^4 + 32*x^2*Log[4] - 16*Log[4]^2)*(-128*x^4 + 128*x^2*Log[4]))/(-8*x + E^(6 - 48*x^4 + 96*x
^2*Log[4] - 48*Log[4]^2)*x - 6*E^(4 - 32*x^4 + 64*x^2*Log[4] - 32*Log[4]^2)*x + 12*E^(2 - 16*x^4 + 32*x^2*Log[
4] - 16*Log[4]^2)*x), x]

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fricas [B]  time = 0.61, size = 180, normalized size = 6.43 \begin {gather*} -e^{\left (16 \, x^{4} - 64 \, x^{2} \log \relax (2) + 64 \, \log \relax (2)^{2} + \frac {64 \, x^{4} - 256 \, x^{2} \log \relax (2) - 8 \, {\left (8 \, x^{4} - 32 \, x^{2} \log \relax (2) + 32 \, \log \relax (2)^{2} - 1\right )} e^{\left (-16 \, x^{4} + 64 \, x^{2} \log \relax (2) - 64 \, \log \relax (2)^{2} + 2\right )} + 2 \, {\left (8 \, x^{4} - 32 \, x^{2} \log \relax (2) + 32 \, \log \relax (2)^{2} - 1\right )} e^{\left (-32 \, x^{4} + 128 \, x^{2} \log \relax (2) - 128 \, \log \relax (2)^{2} + 4\right )} + 256 \, \log \relax (2)^{2} - 9}{4 \, e^{\left (-16 \, x^{4} + 64 \, x^{2} \log \relax (2) - 64 \, \log \relax (2)^{2} + 2\right )} - e^{\left (-32 \, x^{4} + 128 \, x^{2} \log \relax (2) - 128 \, \log \relax (2)^{2} + 4\right )} - 4} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*x^4+256*x^2*log(2))*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)*exp(1/(exp(-16*x^4+64*x^2*log(2)-
64*log(2)^2+2)^2-4*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)+4))/(x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)^3-
6*x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)^2+12*x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)-8*x),x, algorithm
="fricas")

[Out]

-e^(16*x^4 - 64*x^2*log(2) + 64*log(2)^2 + (64*x^4 - 256*x^2*log(2) - 8*(8*x^4 - 32*x^2*log(2) + 32*log(2)^2 -
1)*e^(-16*x^4 + 64*x^2*log(2) - 64*log(2)^2 + 2) + 2*(8*x^4 - 32*x^2*log(2) + 32*log(2)^2 - 1)*e^(-32*x^4 + 1
28*x^2*log(2) - 128*log(2)^2 + 4) + 256*log(2)^2 - 9)/(4*e^(-16*x^4 + 64*x^2*log(2) - 64*log(2)^2 + 2) - e^(-3
2*x^4 + 128*x^2*log(2) - 128*log(2)^2 + 4) - 4) - 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*x^4+256*x^2*log(2))*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)*exp(1/(exp(-16*x^4+64*x^2*log(2)-
64*log(2)^2+2)^2-4*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)+4))/(x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)^3-
6*x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)^2+12*x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)-8*x),x, algorithm
="giac")

[Out]

undef

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maple [A]  time = 0.11, size = 56, normalized size = 2.00

method	result	size
risch	\(-{\mathrm e}^{-\frac {1}{-{\mathrm e}^{-32 x^{4}+128 x^{2} \ln \relax (2)-128 \ln \relax (2)^{2}+4}+4 \,{\mathrm e}^{-16 x^{4}+64 x^{2} \ln \relax (2)-64 \ln \relax (2)^{2}+2}-4}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-128*x^4+256*x^2*ln(2))*exp(-16*x^4+64*x^2*ln(2)-64*ln(2)^2+2)*exp(1/(exp(-16*x^4+64*x^2*ln(2)-64*ln(2)^2
+2)^2-4*exp(-16*x^4+64*x^2*ln(2)-64*ln(2)^2+2)+4))/(x*exp(-16*x^4+64*x^2*ln(2)-64*ln(2)^2+2)^3-6*x*exp(-16*x^4
+64*x^2*ln(2)-64*ln(2)^2+2)^2+12*x*exp(-16*x^4+64*x^2*ln(2)-64*ln(2)^2+2)-8*x),x,method=_RETURNVERBOSE)

[Out]

-exp(-1/(-exp(-32*x^4+128*x^2*ln(2)-128*ln(2)^2+4)+4*exp(-16*x^4+64*x^2*ln(2)-64*ln(2)^2+2)-4))

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maxima [B]  time = 1.56, size = 68, normalized size = 2.43 \begin {gather*} -e^{\left (\frac {e^{\left (32 \, x^{4} + 128 \, \log \relax (2)^{2}\right )}}{4 \, e^{\left (32 \, x^{4} + 128 \, \log \relax (2)^{2}\right )} - 4 \, e^{\left (16 \, x^{4} + 64 \, x^{2} \log \relax (2) + 64 \, \log \relax (2)^{2} + 2\right )} + e^{\left (128 \, x^{2} \log \relax (2) + 4\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*x^4+256*x^2*log(2))*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)*exp(1/(exp(-16*x^4+64*x^2*log(2)-
64*log(2)^2+2)^2-4*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)+4))/(x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)^3-
6*x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)^2+12*x*exp(-16*x^4+64*x^2*log(2)-64*log(2)^2+2)-8*x),x, algorithm
="maxima")

[Out]

-e^(e^(32*x^4 + 128*log(2)^2)/(4*e^(32*x^4 + 128*log(2)^2) - 4*e^(16*x^4 + 64*x^2*log(2) + 64*log(2)^2 + 2) +
e^(128*x^2*log(2) + 4)))

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mupad [B]  time = 7.00, size = 54, normalized size = 1.93 \begin {gather*} -{\mathrm {e}}^{\frac {1}{2^{128\,x^2}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-128\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-32\,x^4}-4\,2^{64\,x^2}\,{\mathrm {e}}^2\,{\mathrm {e}}^{-64\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-16\,x^4}+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1/(exp(128*x^2*log(2) - 128*log(2)^2 - 32*x^4 + 4) - 4*exp(64*x^2*log(2) - 64*log(2)^2 - 16*x^4 + 2)
+ 4))*exp(64*x^2*log(2) - 64*log(2)^2 - 16*x^4 + 2)*(256*x^2*log(2) - 128*x^4))/(8*x - 12*x*exp(64*x^2*log(2)
- 64*log(2)^2 - 16*x^4 + 2) + 6*x*exp(128*x^2*log(2) - 128*log(2)^2 - 32*x^4 + 4) - x*exp(192*x^2*log(2) - 19
2*log(2)^2 - 48*x^4 + 6)),x)

[Out]

-exp(1/(2^(128*x^2)*exp(4)*exp(-128*log(2)^2)*exp(-32*x^4) - 4*2^(64*x^2)*exp(2)*exp(-64*log(2)^2)*exp(-16*x^4
) + 4))

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sympy [B]  time = 0.58, size = 54, normalized size = 1.93 \begin {gather*} - e^{\frac {1}{e^{- 32 x^{4} + 128 x^{2} \log {\relax (2 )} - 128 \log {\relax (2 )}^{2} + 4} - 4 e^{- 16 x^{4} + 64 x^{2} \log {\relax (2 )} - 64 \log {\relax (2 )}^{2} + 2} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*x**4+256*x**2*ln(2))*exp(-16*x**4+64*x**2*ln(2)-64*ln(2)**2+2)*exp(1/(exp(-16*x**4+64*x**2*ln(
2)-64*ln(2)**2+2)**2-4*exp(-16*x**4+64*x**2*ln(2)-64*ln(2)**2+2)+4))/(x*exp(-16*x**4+64*x**2*ln(2)-64*ln(2)**2
+2)**3-6*x*exp(-16*x**4+64*x**2*ln(2)-64*ln(2)**2+2)**2+12*x*exp(-16*x**4+64*x**2*ln(2)-64*ln(2)**2+2)-8*x),x)

[Out]

-exp(1/(exp(-32*x**4 + 128*x**2*log(2) - 128*log(2)**2 + 4) - 4*exp(-16*x**4 + 64*x**2*log(2) - 64*log(2)**2 +
2) + 4))

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