3.97.49 \(\int \frac {e^{\frac {x^2+(2+x^2) \log ^2(x^2)+x \log (e^{4/5} x) \log ^2(x^2)}{x}} (x^2+(8+4 x^2) \log (x^2)+4 x \log (e^{4/5} x) \log (x^2)+(-2+x+x^2) \log ^2(x^2))}{x^2} \, dx\)

Optimal. Leaf size=26 \[ e^{x+\left (\frac {2}{x}+x+\log \left (e^{4/5} x\right )\right ) \log ^2\left (x^2\right )} \]

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Rubi [F]  time = 7.31, 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 (\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}\right ) \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((x^2 + (2 + x^2)*Log[x^2]^2 + x*Log[E^(4/5)*x]*Log[x^2]^2)/x)*(x^2 + (8 + 4*x^2)*Log[x^2] + 4*x*Log[E^
(4/5)*x]*Log[x^2] + (-2 + x + x^2)*Log[x^2]^2))/x^2,x]

[Out]

Defer[Int][E^(x + (4/5 + 2/x + x + Log[x])*Log[x^2]^2), x] + 4*Defer[Int][E^(x + (4/5 + 2/x + x + Log[x])*Log[
x^2]^2)*Log[x^2], x] + 8*Defer[Int][(E^(x + ((2 + x^2)*Log[x^2]^2)/x + Log[E^(4/5)*x]*Log[x^2]^2)*Log[x^2])/x^
2, x] + (16*Defer[Int][(E^(x + ((2 + x^2)*Log[x^2]^2)/x + Log[E^(4/5)*x]*Log[x^2]^2)*Log[x^2])/x, x])/5 + 4*De
fer[Int][(E^(x + ((2 + x^2)*Log[x^2]^2)/x + Log[E^(4/5)*x]*Log[x^2]^2)*Log[x]*Log[x^2])/x, x] + Defer[Int][E^(
x + (4/5 + 2/x + x + Log[x])*Log[x^2]^2)*Log[x^2]^2, x] - 2*Defer[Int][(E^(x + ((2 + x^2)*Log[x^2]^2)/x + Log[
E^(4/5)*x]*Log[x^2]^2)*Log[x^2]^2)/x^2, x] + Defer[Int][(E^(x + ((2 + x^2)*Log[x^2]^2)/x + Log[E^(4/5)*x]*Log[
x^2]^2)*Log[x^2]^2)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx\\ &=\int \left (\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right )+\frac {4 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \left (10+4 x+5 x^2+5 x \log (x)\right ) \log \left (x^2\right )}{5 x^2}+\frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) (-1+x) (2+x) \log ^2\left (x^2\right )}{x^2}\right ) \, dx\\ &=\frac {4}{5} \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \left (10+4 x+5 x^2+5 x \log (x)\right ) \log \left (x^2\right )}{x^2} \, dx+\int \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \, dx+\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) (-1+x) (2+x) \log ^2\left (x^2\right )}{x^2} \, dx\\ &=\frac {4}{5} \int \left (5 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )+\frac {10 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2}+\frac {4 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x}+\frac {5 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log (x) \log \left (x^2\right )}{x}\right ) \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \, dx+\int \left (\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )-\frac {2 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x^2}+\frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x^2} \, dx\right )+\frac {16}{5} \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x} \, dx+4 \int \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right ) \, dx+4 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log (x) \log \left (x^2\right )}{x} \, dx+8 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2} \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \, dx+\int \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right ) \, dx+\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x} \, dx\\ &=-\left (2 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x^2} \, dx\right )+\frac {16}{5} \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x} \, dx+4 \int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \log \left (x^2\right ) \, dx+4 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log (x) \log \left (x^2\right )}{x} \, dx+8 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2} \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \log ^2\left (x^2\right ) \, dx+\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.05, size = 30, normalized size = 1.15 \begin {gather*} e^{x+\left (\frac {4}{5}+\frac {2}{x}+x\right ) \log ^2\left (x^2\right )} x^{\log ^2\left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x^2 + (2 + x^2)*Log[x^2]^2 + x*Log[E^(4/5)*x]*Log[x^2]^2)/x)*(x^2 + (8 + 4*x^2)*Log[x^2] + 4*x*
Log[E^(4/5)*x]*Log[x^2] + (-2 + x + x^2)*Log[x^2]^2))/x^2,x]

[Out]

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

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fricas [A]  time = 0.54, size = 39, normalized size = 1.50 \begin {gather*} e^{\left (\frac {5 \, x \log \left (x^{2}\right )^{3} + 2 \, {\left (5 \, x^{2} + 4 \, x + 10\right )} \log \left (x^{2}\right )^{2} + 10 \, x^{2}}{10 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log(x^2)+x^2)*exp((x*log(x^2)^2*log(x*e
xp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^2,x, algorithm="fricas")

[Out]

e^(1/10*(5*x*log(x^2)^3 + 2*(5*x^2 + 4*x + 10)*log(x^2)^2 + 10*x^2)/x)

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giac [A]  time = 0.67, size = 34, normalized size = 1.31 \begin {gather*} e^{\left (x \log \left (x^{2}\right )^{2} + \log \left (x^{2}\right )^{2} \log \left (x e^{\frac {4}{5}}\right ) + x + \frac {2 \, \log \left (x^{2}\right )^{2}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log(x^2)+x^2)*exp((x*log(x^2)^2*log(x*e
xp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^2,x, algorithm="giac")

[Out]

e^(x*log(x^2)^2 + log(x^2)^2*log(x*e^(4/5)) + x + 2*log(x^2)^2/x)

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maple [C]  time = 0.26, size = 537, normalized size = 20.65

method	result	size
risch	\(x^{-\frac {3 \pi ^{2}}{2}} x^{2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )} x^{-\frac {\pi ^{2}}{2}} x^{-\frac {16 i \pi \,\mathrm {csgn}\left (i x^{2}\right )}{5}} x^{\frac {8 i \pi \,\mathrm {csgn}\left (i x \right )}{x}} x^{-\frac {8 i \pi \,\mathrm {csgn}\left (i x^{2}\right )}{x}} x^{4 i x \pi \,\mathrm {csgn}\left (i x \right )} x^{-4 i x \pi \,\mathrm {csgn}\left (i x^{2}\right )} x^{\frac {16 i \pi \,\mathrm {csgn}\left (i x \right )}{5}} {\mathrm e}^{\frac {-5 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6} x^{2}+20 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right ) x^{2}-30 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2} x^{2}+20 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3} x^{2}-5 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4} x^{2}-4 x \,\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+16 x \,\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right )-24 x \,\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2}+16 x \,\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3}-4 x \,\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4}-10 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+40 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{5} \mathrm {csgn}\left (i x \right )-60 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x \right )^{2}+40 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x \right )^{3}-10 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )^{4}-40 i x \ln \relax (x )^{2} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}-40 i x \ln \relax (x )^{2} \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+80 i x \ln \relax (x )^{2} \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+80 x \ln \relax (x )^{3}+80 x^{2} \ln \relax (x )^{2}+64 x \ln \relax (x )^{2}+160 \ln \relax (x )^{2}+20 x^{2}}{20 x}}\)	\(537\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*ln(x^2)*ln(x*exp(4/5))+(x^2+x-2)*ln(x^2)^2+(4*x^2+8)*ln(x^2)+x^2)*exp((x*ln(x^2)^2*ln(x*exp(4/5))+(x^
2+2)*ln(x^2)^2+x^2)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^(-3/2*Pi^2)*(x^(Pi^2*csgn(I*x^2)*csgn(I*x)))^2*(x^(-1/4*Pi^2))^2*(x^(-8/5*I*Pi*csgn(I*x^2)))^2*x^(8*I/x*Pi*c
sgn(I*x))*(x^(-4*I/x*Pi*csgn(I*x^2)))^2*x^(4*I*x*Pi*csgn(I*x))*(x^(-2*I*x*Pi*csgn(I*x^2)))^2*x^(16/5*I*Pi*csgn
(I*x))*exp(1/20*(-5*Pi^2*csgn(I*x^2)^6*x^2+20*Pi^2*csgn(I*x^2)^5*csgn(I*x)*x^2-30*Pi^2*csgn(I*x^2)^4*csgn(I*x)
^2*x^2+20*Pi^2*csgn(I*x^2)^3*csgn(I*x)^3*x^2-5*Pi^2*csgn(I*x^2)^2*csgn(I*x)^4*x^2-4*x*Pi^2*csgn(I*x^2)^6+16*x*
Pi^2*csgn(I*x^2)^5*csgn(I*x)-24*x*Pi^2*csgn(I*x^2)^4*csgn(I*x)^2+16*x*Pi^2*csgn(I*x^2)^3*csgn(I*x)^3-4*x*Pi^2*
csgn(I*x^2)^2*csgn(I*x)^4-10*Pi^2*csgn(I*x^2)^6+40*Pi^2*csgn(I*x^2)^5*csgn(I*x)-60*Pi^2*csgn(I*x^2)^4*csgn(I*x
)^2+40*Pi^2*csgn(I*x^2)^3*csgn(I*x)^3-10*Pi^2*csgn(I*x^2)^2*csgn(I*x)^4-40*I*x*ln(x)^2*Pi*csgn(I*x^2)*csgn(I*x
)^2-40*I*x*ln(x)^2*Pi*csgn(I*x^2)^3+80*I*x*ln(x)^2*Pi*csgn(I*x^2)^2*csgn(I*x)+80*x*ln(x)^3+80*x^2*ln(x)^2+64*x
*ln(x)^2+160*ln(x)^2+20*x^2)/x)

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maxima [A]  time = 0.56, size = 31, normalized size = 1.19 \begin {gather*} e^{\left (4 \, x \log \relax (x)^{2} + 4 \, \log \relax (x)^{3} + \frac {16}{5} \, \log \relax (x)^{2} + x + \frac {8 \, \log \relax (x)^{2}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log(x^2)+x^2)*exp((x*log(x^2)^2*log(x*e
xp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^2,x, algorithm="maxima")

[Out]

e^(4*x*log(x)^2 + 4*log(x)^3 + 16/5*log(x)^2 + x + 8*log(x)^2/x)

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mupad [B]  time = 5.72, size = 41, normalized size = 1.58 \begin {gather*} x^{{\ln \left (x^2\right )}^2}\,{\mathrm {e}}^{\frac {2\,{\ln \left (x^2\right )}^2}{x}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x^2\right )}^2}{5}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\ln \left (x^2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x^2)^2*(x^2 + 2) + x^2 + x*log(x^2)^2*log(x*exp(4/5)))/x)*(log(x^2)*(4*x^2 + 8) + log(x^2)^2*(x
+ x^2 - 2) + x^2 + 4*x*log(x^2)*log(x*exp(4/5))))/x^2,x)

[Out]

x^(log(x^2)^2)*exp((2*log(x^2)^2)/x)*exp((4*log(x^2)^2)/5)*exp(x)*exp(x*log(x^2)^2)

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sympy [A]  time = 0.44, size = 36, normalized size = 1.38 \begin {gather*} e^{\frac {x^{2} + x \left (\frac {\log {\left (x^{2} \right )}}{2} + \frac {4}{5}\right ) \log {\left (x^{2} \right )}^{2} + \left (x^{2} + 2\right ) \log {\left (x^{2} \right )}^{2}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*ln(x**2)*ln(x*exp(4/5))+(x**2+x-2)*ln(x**2)**2+(4*x**2+8)*ln(x**2)+x**2)*exp((x*ln(x**2)**2*ln(
x*exp(4/5))+(x**2+2)*ln(x**2)**2+x**2)/x)/x**2,x)

[Out]

exp((x**2 + x*(log(x**2)/2 + 4/5)*log(x**2)**2 + (x**2 + 2)*log(x**2)**2)/x)

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