3.49.79 \(\int \frac {e^{1+e^{x^2} x+e^{x^2} \log (\frac {x^2}{256+x^2})} (e^{x^2} (512+256 x+513 x^3+2 x^5)+e^{x^2} (512 x^2+2 x^4) \log (\frac {x^2}{256+x^2}))}{256 x+x^3} \, dx\)

Optimal. Leaf size=24 \[ e^{1+e^{x^2} \left (x+\log \left (\frac {x^2}{256+x^2}\right )\right )} \]

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Rubi [F]  time = 10.57, 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^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(1 + E^x^2*x + E^x^2*Log[x^2/(256 + x^2)])*(E^x^2*(512 + 256*x + 513*x^3 + 2*x^5) + E^x^2*(512*x^2 + 2*
x^4)*Log[x^2/(256 + x^2)]))/(256*x + x^3),x]

[Out]

Defer[Int][E^(1 + E^x^2*x + x^2 + E^x^2*Log[x^2/(256 + x^2)]), x] + Defer[Int][E^(1 + E^x^2*x + x^2 + E^x^2*Lo
g[x^2/(256 + x^2)])/(16*I - x), x] + 2*Defer[Int][E^(1 + E^x^2*x + x^2 + E^x^2*Log[x^2/(256 + x^2)])/x, x] + 2
*Defer[Int][E^(1 + E^x^2*x + x^2 + E^x^2*Log[x^2/(256 + x^2)])*x^2, x] - Defer[Int][E^(1 + E^x^2*x + x^2 + E^x
^2*Log[x^2/(256 + x^2)])/(16*I + x), x] + 2*Defer[Int][E^(1 + E^x^2*x + x^2 + E^x^2*Log[x^2/(256 + x^2)])*x*Lo
g[x^2/(256 + x^2)], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{x \left (256+x^2\right )} \, dx\\ &=\int \left (\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x}-\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 \left (256+x^2\right )}\right ) \, dx\\ &=\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{x} \, dx-\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{256+x^2} \, dx\\ &=-\left (\frac {1}{256} \int \left (\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5\right )}{256+x^2}+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right )\right ) \, dx\right )+\frac {1}{256} \int \left (\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5\right )}{x}+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (256+x^2\right ) \log \left (\frac {x^2}{256+x^2}\right )\right ) \, dx\\ &=\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5\right )}{x} \, dx-\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5\right )}{256+x^2} \, dx-\frac {1}{128} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {1}{128} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (256+x^2\right ) \log \left (\frac {x^2}{256+x^2}\right ) \, dx\\ &=\frac {1}{256} \int \left (256 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}+\frac {512 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x}+513 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^4\right ) \, dx-\frac {1}{256} \int \left (e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^4+\frac {512 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x}{256+x^2}\right ) \, dx-\frac {1}{128} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {1}{128} \int \left (256 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right )+e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right )\right ) \, dx\\ &=-\left (\frac {1}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx\right )+2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x} \, dx-2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x}{256+x^2} \, dx+2 \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {513}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx+\int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \, dx\\ &=-\left (\frac {1}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx\right )+2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x} \, dx-2 \int \left (-\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{2 (16 i-x)}+\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{2 (16 i+x)}\right ) \, dx+2 \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {513}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx+\int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \, dx\\ &=-\left (\frac {1}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx\right )+2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x} \, dx+2 \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {513}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx+\int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \, dx+\int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{16 i-x} \, dx-\int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{16 i+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.39, size = 29, normalized size = 1.21 \begin {gather*} e^{1+e^{x^2} x} \left (\frac {x^2}{256+x^2}\right )^{e^{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 + E^x^2*x + E^x^2*Log[x^2/(256 + x^2)])*(E^x^2*(512 + 256*x + 513*x^3 + 2*x^5) + E^x^2*(512*x^
2 + 2*x^4)*Log[x^2/(256 + x^2)]))/(256*x + x^3),x]

[Out]

E^(1 + E^x^2*x)*(x^2/(256 + x^2))^E^x^2

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fricas [A]  time = 1.08, size = 26, normalized size = 1.08 \begin {gather*} e^{\left (x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )} \log \left (\frac {x^{2}}{x^{2} + 256}\right ) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+512*x^2)*exp(x^2)*log(x^2/(x^2+256))+(2*x^5+513*x^3+256*x+512)*exp(x^2))*exp(exp(x^2)*log(x^
2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x),x, algorithm="fricas")

[Out]

e^(x*e^(x^2) + e^(x^2)*log(x^2/(x^2 + 256)) + 1)

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giac [A]  time = 0.41, size = 26, normalized size = 1.08 \begin {gather*} e^{\left (x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )} \log \left (\frac {x^{2}}{x^{2} + 256}\right ) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+512*x^2)*exp(x^2)*log(x^2/(x^2+256))+(2*x^5+513*x^3+256*x+512)*exp(x^2))*exp(exp(x^2)*log(x^
2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x),x, algorithm="giac")

[Out]

e^(x*e^(x^2) + e^(x^2)*log(x^2/(x^2 + 256)) + 1)

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maple [C]  time = 0.21, size = 222, normalized size = 9.25

method	result	size
risch	\(x^{2 \,{\mathrm e}^{x^{2}}} \left (x^{2}+256\right )^{-{\mathrm e}^{x^{2}}} {\mathrm e}^{1-\frac {i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}+i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-\frac {i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}}{2}+\frac {i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{2}}{2}-\frac {i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}+256}\right )}{2}-\frac {i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{3}}{2}+\frac {i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}+256}\right )}{2}+{\mathrm e}^{x^{2}} x}\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4+512*x^2)*exp(x^2)*ln(x^2/(x^2+256))+(2*x^5+513*x^3+256*x+512)*exp(x^2))*exp(exp(x^2)*ln(x^2/(x^2+2
56))+exp(x^2)*x+1)/(x^3+256*x),x,method=_RETURNVERBOSE)

[Out]

x^(2*exp(x^2))*(x^2+256)^(-exp(x^2))*exp(1-1/2*I*exp(x^2)*Pi*csgn(I*x^2)^3+I*exp(x^2)*Pi*csgn(I*x^2)^2*csgn(I*
x)-1/2*I*exp(x^2)*Pi*csgn(I*x^2)*csgn(I*x)^2+1/2*I*exp(x^2)*Pi*csgn(I*x^2)*csgn(I*x^2/(x^2+256))^2-1/2*I*exp(x
^2)*Pi*csgn(I*x^2)*csgn(I*x^2/(x^2+256))*csgn(I/(x^2+256))-1/2*I*exp(x^2)*Pi*csgn(I*x^2/(x^2+256))^3+1/2*I*exp
(x^2)*Pi*csgn(I*x^2/(x^2+256))^2*csgn(I/(x^2+256))+exp(x^2)*x)

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maxima [A]  time = 0.47, size = 29, normalized size = 1.21 \begin {gather*} e^{\left (x e^{\left (x^{2}\right )} - e^{\left (x^{2}\right )} \log \left (x^{2} + 256\right ) + 2 \, e^{\left (x^{2}\right )} \log \relax (x) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+512*x^2)*exp(x^2)*log(x^2/(x^2+256))+(2*x^5+513*x^3+256*x+512)*exp(x^2))*exp(exp(x^2)*log(x^
2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x),x, algorithm="maxima")

[Out]

e^(x*e^(x^2) - e^(x^2)*log(x^2 + 256) + 2*e^(x^2)*log(x) + 1)

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mupad [B]  time = 3.47, size = 26, normalized size = 1.08 \begin {gather*} \mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{x^2}}\,{\left (\frac {x^2}{x^2+256}\right )}^{{\mathrm {e}}^{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*exp(x^2) + log(x^2/(x^2 + 256))*exp(x^2) + 1)*(exp(x^2)*(256*x + 513*x^3 + 2*x^5 + 512) + log(x^2/(
x^2 + 256))*exp(x^2)*(512*x^2 + 2*x^4)))/(256*x + x^3),x)

[Out]

exp(1)*exp(x*exp(x^2))*(x^2/(x^2 + 256))^exp(x^2)

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sympy [A]  time = 1.24, size = 24, normalized size = 1.00 \begin {gather*} e^{x e^{x^{2}} + e^{x^{2}} \log {\left (\frac {x^{2}}{x^{2} + 256} \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4+512*x**2)*exp(x**2)*ln(x**2/(x**2+256))+(2*x**5+513*x**3+256*x+512)*exp(x**2))*exp(exp(x**2
)*ln(x**2/(x**2+256))+exp(x**2)*x+1)/(x**3+256*x),x)

[Out]

exp(x*exp(x**2) + exp(x**2)*log(x**2/(x**2 + 256)) + 1)

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