3.9.91 \(\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2(\frac {x^2+\log ^2(x)}{x^2})} (((-1-4 x) \log (x)+(1+4 x) \log ^2(x)) \log (\frac {x^2+\log ^2(x)}{x^2})+(-x^3-x \log ^2(x)) \log ^2(\frac {x^2+\log ^2(x)}{x^2}))}{x^3+x \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ -5+e^{\left (-\frac {1}{4}-x\right ) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \]

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Rubi [F]  time = 13.57, 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 {e^{\frac {1}{4} (-1-4 x) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )} \left (\left ((-1-4 x) \log (x)+(1+4 x) \log ^2(x)\right ) \log \left (\frac {x^2+\log ^2(x)}{x^2}\right )+\left (-x^3-x \log ^2(x)\right ) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )\right )}{x^3+x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )} \left (\left ((-1-4 x) \log (x)+(1+4 x) \log ^2(x)\right ) \log \left (\frac {x^2+\log ^2(x)}{x^2}\right )+\left (-x^3-x \log ^2(x)\right ) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )\right )}{x \left (x^2+\log ^2(x)\right )} \, dx\\ &=\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \left (\left ((-1-4 x) \log (x)+(1+4 x) \log ^2(x)\right ) \log \left (\frac {x^2+\log ^2(x)}{x^2}\right )+\left (-x^3-x \log ^2(x)\right ) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )\right )}{x \left (x^2+\log ^2(x)\right )} \, dx\\ &=\int \left (\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (1+4 x) (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}-e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )\right ) \, dx\\ &=\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (1+4 x) (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx\\ &=-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx+\int \left (\frac {4 e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)}+\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}\right ) \, dx\\ &=4 \int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)} \, dx+\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx\\ &=4 \int \left (-\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)}+\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)}\right ) \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx+\int \left (-\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}+\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)} \, dx\right )+4 \int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)} \, dx-\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx+\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 24, normalized size = 0.96 \begin {gather*} e^{-\frac {1}{4} (1+4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, size = 23, normalized size = 0.92 \begin {gather*} e^{\left (-\frac {1}{4} \, {\left (4 \, x + 1\right )} \log \left (\frac {x^{2} + \log \relax (x)^{2}}{x^{2}}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 8.84, size = 33, normalized size = 1.32 \begin {gather*} e^{\left (-x \log \left (\frac {\log \relax (x)^{2}}{x^{2}} + 1\right )^{2} - \frac {1}{4} \, \log \left (\frac {\log \relax (x)^{2}}{x^{2}} + 1\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.37, size = 196, normalized size = 7.84

method	result	size
risch	\({\mathrm e}^{-\frac {\left (4 x +1\right ) \left (-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )^{2}+x^{2}\right )}{x^{2}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )^{2}+x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )^{2}+x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )^{2}+x^{2}\right )\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )^{2}+x^{2}\right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )^{2}+x^{2}\right )\right )+4 \ln \relax (x )-2 \ln \left (\ln \relax (x )^{2}+x^{2}\right )\right )^{2}}{16}}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-1/16*(4*x+1)*(-I*Pi*csgn(I*x^2)^3+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*x^2)*csgn(I*x)^2+I*Pi*csgn(I
/x^2*(ln(x)^2+x^2))^3-I*Pi*csgn(I/x^2*(ln(x)^2+x^2))^2*csgn(I/x^2)-I*Pi*csgn(I/x^2*(ln(x)^2+x^2))^2*csgn(I*(ln
(x)^2+x^2))+I*Pi*csgn(I/x^2*(ln(x)^2+x^2))*csgn(I/x^2)*csgn(I*(ln(x)^2+x^2))+4*ln(x)-2*ln(ln(x)^2+x^2))^2)

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maxima [B]  time = 0.90, size = 68, normalized size = 2.72 \begin {gather*} e^{\left (-x \log \left (x^{2} + \log \relax (x)^{2}\right )^{2} + 4 \, x \log \left (x^{2} + \log \relax (x)^{2}\right ) \log \relax (x) - 4 \, x \log \relax (x)^{2} - \frac {1}{4} \, \log \left (x^{2} + \log \relax (x)^{2}\right )^{2} + \log \left (x^{2} + \log \relax (x)^{2}\right ) \log \relax (x) - \log \relax (x)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.98, size = 83, normalized size = 3.32 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {{\ln \left (x^2+{\ln \relax (x)}^2\right )}^2}{4}}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x^2}\right )}^2}{4}}\,{\mathrm {e}}^{-x\,{\ln \left (x^2+{\ln \relax (x)}^2\right )}^2}\,{\mathrm {e}}^{-x\,{\ln \left (\frac {1}{x^2}\right )}^2}\,{\mathrm {e}}^{-\frac {\ln \left (\frac {1}{x^2}\right )\,\ln \left (x^2+{\ln \relax (x)}^2\right )}{2}}}{{\left (x^2+{\ln \relax (x)}^2\right )}^{2\,x\,\ln \left (\frac {1}{x^2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(-log(log(x)^2 + x^2)^2/4)*exp(-log(1/x^2)^2/4)*exp(-x*log(log(x)^2 + x^2)^2)*exp(-x*log(1/x^2)^2)*exp(-(l
og(1/x^2)*log(log(x)^2 + x^2))/2))/(log(x)^2 + x^2)^(2*x*log(1/x^2))

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sympy [A]  time = 1.39, size = 22, normalized size = 0.88 \begin {gather*} e^{\left (- x - \frac {1}{4}\right ) \log {\left (\frac {x^{2} + \log {\relax (x )}^{2}}{x^{2}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp((-x - 1/4)*log((x**2 + log(x)**2)/x**2)**2)

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