3.72.67 \(\int \frac {-25+100 x-19 x^2+e^{e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} (25-x^2+e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)} (50 x^2-20 x^3+2 x^4+(-100 x^2+40 x^3-4 x^4) \log (2)+(50 x^2-20 x^3+2 x^4) \log ^2(2)))}{x^2+e^{2 e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} x^2-4 x^3+4 x^4+e^{e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} (-2 x^2+4 x^3)} \, dx\)

Optimal. Leaf size=36 \[ 5+\frac {\left (-1+\frac {5}{x}\right )^2 x}{1-e^{e^{(-x+x \log (2))^2}}-2 x} \]

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Rubi [F]  time = 22.71, 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 {-25+100 x-19 x^2+e^{e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} \left (25-x^2+e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)} \left (50 x^2-20 x^3+2 x^4+\left (-100 x^2+40 x^3-4 x^4\right ) \log (2)+\left (50 x^2-20 x^3+2 x^4\right ) \log ^2(2)\right )\right )}{x^2+e^{2 e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} x^2-4 x^3+4 x^4+e^{e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} \left (-2 x^2+4 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-25 + 100*x - 19*x^2 + E^E^(x^2 - 2*x^2*Log[2] + x^2*Log[2]^2)*(25 - x^2 + E^(x^2 - 2*x^2*Log[2] + x^2*Lo
g[2]^2)*(50*x^2 - 20*x^3 + 2*x^4 + (-100*x^2 + 40*x^3 - 4*x^4)*Log[2] + (50*x^2 - 20*x^3 + 2*x^4)*Log[2]^2)))/
(x^2 + E^(2*E^(x^2 - 2*x^2*Log[2] + x^2*Log[2]^2))*x^2 - 4*x^3 + 4*x^4 + E^E^(x^2 - 2*x^2*Log[2] + x^2*Log[2]^
2)*(-2*x^2 + 4*x^3)),x]

[Out]

-20*Defer[Int][(-1 + E^(E^(x^2*(1 + Log[2]^2))/4^x^2) + 2*x)^(-2), x] + 25*(1 - Log[2])^2*Defer[Int][(2^(1 - 2
*x^2)*E^(E^(x^2*(1 + Log[2]^2))/4^x^2 + x^2*(1 + Log[2]^2)))/(-1 + E^(E^(x^2*(1 + Log[2]^2))/4^x^2) + 2*x)^2,
x] + 50*Defer[Int][1/(x*(-1 + E^(E^(x^2*(1 + Log[2]^2))/4^x^2) + 2*x)^2), x] + 2*Defer[Int][x/(-1 + E^(E^(x^2*
(1 + Log[2]^2))/4^x^2) + 2*x)^2, x] - 5*(1 - Log[2])^2*Defer[Int][(2^(2 - 2*x^2)*E^(E^(x^2*(1 + Log[2]^2))/4^x
^2 + x^2*(1 + Log[2]^2))*x)/(-1 + E^(E^(x^2*(1 + Log[2]^2))/4^x^2) + 2*x)^2, x] + (1 - Log[2])^2*Defer[Int][(2
^(1 - 2*x^2)*E^(E^(x^2*(1 + Log[2]^2))/4^x^2 + x^2*(1 + Log[2]^2))*x^2)/(-1 + E^(E^(x^2*(1 + Log[2]^2))/4^x^2)
+ 2*x)^2, x] - Defer[Int][(-1 + E^(E^(x^2*(1 + Log[2]^2))/4^x^2) + 2*x)^(-1), x] + 25*Defer[Int][1/(x^2*(-1 +
E^(E^(x^2*(1 + Log[2]^2))/4^x^2) + 2*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4^{-x^2} (5-x) \left (4^{x^2} e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}} (5+x)+4^{x^2} (-5+19 x)-2 \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) (-5+x) x^2 (-1+\log (2))^2\right )}{\left (1-e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}-2 x\right )^2 x^2} \, dx\\ &=\int \left (-\frac {(-5+x) \left (-5+5 e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+19 x+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}} x\right )}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) (-5+x)^2 (-1+\log (2))^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}\right ) \, dx\\ &=(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) (-5+x)^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \frac {(-5+x) \left (-5+5 e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+19 x+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}} x\right )}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx\\ &=(-1+\log (2))^2 \int \left (\frac {25\ 2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}-\frac {5\ 2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}\right ) \, dx-\int \left (-\frac {2 (-5+x)^2}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {(-5+x) (5+x)}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )}\right ) \, dx\\ &=2 \int \frac {(-5+x)^2}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\left (5 (1-\log (2))^2\right ) \int \frac {2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+\left (25 (1-\log (2))^2\right ) \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \frac {(-5+x) (5+x)}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )} \, dx\\ &=2 \int \left (-\frac {10}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {25}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}\right ) \, dx-\left (5 (1-\log (2))^2\right ) \int \frac {2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+\left (25 (1-\log (2))^2\right ) \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \left (\frac {1}{-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x}-\frac {25}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )}\right ) \, dx\\ &=2 \int \frac {x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-20 \int \frac {1}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+25 \int \frac {1}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )} \, dx+50 \int \frac {1}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\left (5 (1-\log (2))^2\right ) \int \frac {2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+\left (25 (1-\log (2))^2\right ) \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \frac {1}{-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.73, size = 39, normalized size = 1.08 \begin {gather*} -\frac {(-5+x)^2}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 + 100*x - 19*x^2 + E^E^(x^2 - 2*x^2*Log[2] + x^2*Log[2]^2)*(25 - x^2 + E^(x^2 - 2*x^2*Log[2] +
x^2*Log[2]^2)*(50*x^2 - 20*x^3 + 2*x^4 + (-100*x^2 + 40*x^3 - 4*x^4)*Log[2] + (50*x^2 - 20*x^3 + 2*x^4)*Log[2]
^2)))/(x^2 + E^(2*E^(x^2 - 2*x^2*Log[2] + x^2*Log[2]^2))*x^2 - 4*x^3 + 4*x^4 + E^E^(x^2 - 2*x^2*Log[2] + x^2*L
og[2]^2)*(-2*x^2 + 4*x^3)),x]

[Out]

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

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fricas [A]  time = 0.59, size = 44, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 10 \, x + 25}{2 \, x^{2} + x e^{\left (e^{\left (x^{2} \log \relax (2)^{2} - 2 \, x^{2} \log \relax (2) + x^{2}\right )}\right )} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x^4-20*x^3+50*x^2)*log(2)^2+(-4*x^4+40*x^3-100*x^2)*log(2)+2*x^4-20*x^3+50*x^2)*exp(x^2*log(2)
^2-2*x^2*log(2)+x^2)-x^2+25)*exp(exp(x^2*log(2)^2-2*x^2*log(2)+x^2))-19*x^2+100*x-25)/(x^2*exp(exp(x^2*log(2)^
2-2*x^2*log(2)+x^2))^2+(4*x^3-2*x^2)*exp(exp(x^2*log(2)^2-2*x^2*log(2)+x^2))+4*x^4-4*x^3+x^2),x, algorithm="fr
icas")

[Out]

-(x^2 - 10*x + 25)/(2*x^2 + x*e^(e^(x^2*log(2)^2 - 2*x^2*log(2) + x^2)) - x)

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giac [A]  time = 1.26, size = 44, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 10 \, x + 25}{2 \, x^{2} + x e^{\left (e^{\left (x^{2} \log \relax (2)^{2} - 2 \, x^{2} \log \relax (2) + x^{2}\right )}\right )} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x^4-20*x^3+50*x^2)*log(2)^2+(-4*x^4+40*x^3-100*x^2)*log(2)+2*x^4-20*x^3+50*x^2)*exp(x^2*log(2)
^2-2*x^2*log(2)+x^2)-x^2+25)*exp(exp(x^2*log(2)^2-2*x^2*log(2)+x^2))-19*x^2+100*x-25)/(x^2*exp(exp(x^2*log(2)^
2-2*x^2*log(2)+x^2))^2+(4*x^3-2*x^2)*exp(exp(x^2*log(2)^2-2*x^2*log(2)+x^2))+4*x^4-4*x^3+x^2),x, algorithm="gi
ac")

[Out]

-(x^2 - 10*x + 25)/(2*x^2 + x*e^(e^(x^2*log(2)^2 - 2*x^2*log(2) + x^2)) - x)

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maple [A]  time = 0.32, size = 39, normalized size = 1.08

method	result	size
risch	\(-\frac {x^{2}-10 x +25}{x \left (-1+{\mathrm e}^{\left (\frac {1}{4}\right )^{x^{2}} {\mathrm e}^{x^{2} \left (\ln \relax (2)^{2}+1\right )}}+2 x \right )}\)	\(39\)
norman	\(\frac {-25+\frac {19 x}{2}+\frac {{\mathrm e}^{{\mathrm e}^{x^{2} \ln \relax (2)^{2}-2 x^{2} \ln \relax (2)+x^{2}}} x}{2}}{x \left (-1+{\mathrm e}^{{\mathrm e}^{x^{2} \ln \relax (2)^{2}-2 x^{2} \ln \relax (2)+x^{2}}}+2 x \right )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((2*x^4-20*x^3+50*x^2)*ln(2)^2+(-4*x^4+40*x^3-100*x^2)*ln(2)+2*x^4-20*x^3+50*x^2)*exp(x^2*ln(2)^2-2*x^2*
ln(2)+x^2)-x^2+25)*exp(exp(x^2*ln(2)^2-2*x^2*ln(2)+x^2))-19*x^2+100*x-25)/(x^2*exp(exp(x^2*ln(2)^2-2*x^2*ln(2)
+x^2))^2+(4*x^3-2*x^2)*exp(exp(x^2*ln(2)^2-2*x^2*ln(2)+x^2))+4*x^4-4*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

-(x^2-10*x+25)/x/(-1+exp((1/4)^(x^2)*exp(x^2*(ln(2)^2+1)))+2*x)

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maxima [A]  time = 0.52, size = 44, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 10 \, x + 25}{2 \, x^{2} + x e^{\left (e^{\left (x^{2} \log \relax (2)^{2} - 2 \, x^{2} \log \relax (2) + x^{2}\right )}\right )} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x^4-20*x^3+50*x^2)*log(2)^2+(-4*x^4+40*x^3-100*x^2)*log(2)+2*x^4-20*x^3+50*x^2)*exp(x^2*log(2)
^2-2*x^2*log(2)+x^2)-x^2+25)*exp(exp(x^2*log(2)^2-2*x^2*log(2)+x^2))-19*x^2+100*x-25)/(x^2*exp(exp(x^2*log(2)^
2-2*x^2*log(2)+x^2))^2+(4*x^3-2*x^2)*exp(exp(x^2*log(2)^2-2*x^2*log(2)+x^2))+4*x^4-4*x^3+x^2),x, algorithm="ma
xima")

[Out]

-(x^2 - 10*x + 25)/(2*x^2 + x*e^(e^(x^2*log(2)^2 - 2*x^2*log(2) + x^2)) - x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {100\,x+{\mathrm {e}}^{{\mathrm {e}}^{x^2\,{\ln \relax (2)}^2-2\,x^2\,\ln \relax (2)+x^2}}\,\left ({\mathrm {e}}^{x^2\,{\ln \relax (2)}^2-2\,x^2\,\ln \relax (2)+x^2}\,\left ({\ln \relax (2)}^2\,\left (2\,x^4-20\,x^3+50\,x^2\right )-\ln \relax (2)\,\left (4\,x^4-40\,x^3+100\,x^2\right )+50\,x^2-20\,x^3+2\,x^4\right )-x^2+25\right )-19\,x^2-25}{x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2\,{\ln \relax (2)}^2-2\,x^2\,\ln \relax (2)+x^2}}-{\mathrm {e}}^{{\mathrm {e}}^{x^2\,{\ln \relax (2)}^2-2\,x^2\,\ln \relax (2)+x^2}}\,\left (2\,x^2-4\,x^3\right )+x^2-4\,x^3+4\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x + exp(exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2))*(exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2)*(log(2)^2*(50
*x^2 - 20*x^3 + 2*x^4) - log(2)*(100*x^2 - 40*x^3 + 4*x^4) + 50*x^2 - 20*x^3 + 2*x^4) - x^2 + 25) - 19*x^2 - 2
5)/(x^2*exp(2*exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2)) - exp(exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2))*(2*x^2 - 4
*x^3) + x^2 - 4*x^3 + 4*x^4),x)

[Out]

int((100*x + exp(exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2))*(exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2)*(log(2)^2*(50
*x^2 - 20*x^3 + 2*x^4) - log(2)*(100*x^2 - 40*x^3 + 4*x^4) + 50*x^2 - 20*x^3 + 2*x^4) - x^2 + 25) - 19*x^2 - 2
5)/(x^2*exp(2*exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2)) - exp(exp(x^2*log(2)^2 - 2*x^2*log(2) + x^2))*(2*x^2 - 4
*x^3) + x^2 - 4*x^3 + 4*x^4), x)

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sympy [A]  time = 0.50, size = 39, normalized size = 1.08 \begin {gather*} \frac {- x^{2} + 10 x - 25}{2 x^{2} + x e^{e^{- 2 x^{2} \log {\relax (2 )} + x^{2} \log {\relax (2 )}^{2} + x^{2}}} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x**4-20*x**3+50*x**2)*ln(2)**2+(-4*x**4+40*x**3-100*x**2)*ln(2)+2*x**4-20*x**3+50*x**2)*exp(x*
*2*ln(2)**2-2*x**2*ln(2)+x**2)-x**2+25)*exp(exp(x**2*ln(2)**2-2*x**2*ln(2)+x**2))-19*x**2+100*x-25)/(x**2*exp(
exp(x**2*ln(2)**2-2*x**2*ln(2)+x**2))**2+(4*x**3-2*x**2)*exp(exp(x**2*ln(2)**2-2*x**2*ln(2)+x**2))+4*x**4-4*x*
*3+x**2),x)

[Out]

(-x**2 + 10*x - 25)/(2*x**2 + x*exp(exp(-2*x**2*log(2) + x**2*log(2)**2 + x**2)) - x)

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