3.19.96 \(\int \frac {-64-32 \log (x^2)+e^{\frac {e^{x^2}}{x}+x^2} (-x+2 x^3) \log ^3(x^2)}{x^3 \log ^3(x^2)} \, dx\)

Optimal. Leaf size=24 \[ 3+e^{\frac {e^{x^2}}{x}}+\frac {16}{x^2 \log ^2\left (x^2\right )} \]

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Rubi [F]  time = 1.20, 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 {-64-32 \log \left (x^2\right )+e^{\frac {e^{x^2}}{x}+x^2} \left (-x+2 x^3\right ) \log ^3\left (x^2\right )}{x^3 \log ^3\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-64 - 32*Log[x^2] + E^(E^x^2/x + x^2)*(-x + 2*x^3)*Log[x^2]^3)/(x^3*Log[x^2]^3),x]

[Out]

8/x^2 + 16*ExpIntegralEi[-Log[x^2]] + 8/(x^2*Log[x^2]) + 8*ExpIntegralEi[-Log[x^2]]*Log[x^2] - 8*ExpIntegralEi
[-Log[x^2]]*(2 + Log[x^2]) + (8*(2 + Log[x^2]))/(x^2*Log[x^2]^2) - (8*(2 + Log[x^2]))/(x^2*Log[x^2]) + 2*Defer
[Int][E^(E^x^2/x + x^2), x] - Defer[Int][E^(E^x^2/x + x^2)/x^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 23, normalized size = 0.96 \begin {gather*} e^{\frac {e^{x^2}}{x}}+\frac {16}{x^2 \log ^2\left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64 - 32*Log[x^2] + E^(E^x^2/x + x^2)*(-x + 2*x^3)*Log[x^2]^3)/(x^3*Log[x^2]^3),x]

[Out]

E^(E^x^2/x) + 16/(x^2*Log[x^2]^2)

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fricas [B]  time = 0.65, size = 46, normalized size = 1.92 \begin {gather*} \frac {{\left (x^{2} e^{\left (\frac {x^{3} + e^{\left (x^{2}\right )}}{x}\right )} \log \left (x^{2}\right )^{2} + 16 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{x^{2} \log \left (x^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3-x)*exp(x^2)*log(x^2)^3*exp(exp(x^2)/x)-32*log(x^2)-64)/x^3/log(x^2)^3,x, algorithm="fricas")

[Out]

(x^2*e^((x^3 + e^(x^2))/x)*log(x^2)^2 + 16*e^(x^2))*e^(-x^2)/(x^2*log(x^2)^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} - x\right )} e^{\left (x^{2} + \frac {e^{\left (x^{2}\right )}}{x}\right )} \log \left (x^{2}\right )^{3} - 32 \, \log \left (x^{2}\right ) - 64}{x^{3} \log \left (x^{2}\right )^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3-x)*exp(x^2)*log(x^2)^3*exp(exp(x^2)/x)-32*log(x^2)-64)/x^3/log(x^2)^3,x, algorithm="giac")

[Out]

integrate(((2*x^3 - x)*e^(x^2 + e^(x^2)/x)*log(x^2)^3 - 32*log(x^2) - 64)/(x^3*log(x^2)^3), x)

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maple [A]  time = 0.13, size = 22, normalized size = 0.92

method	result	size
default	\(\frac {16}{x^{2} \ln \left (x^{2}\right )^{2}}+{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{x}}\)	\(22\)
risch	\(-\frac {64}{x^{2} \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )\right )^{2}}+{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{x}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3-x)*exp(x^2)*ln(x^2)^3*exp(exp(x^2)/x)-32*ln(x^2)-64)/x^3/ln(x^2)^3,x,method=_RETURNVERBOSE)

[Out]

16/x^2/ln(x^2)^2+exp(exp(x^2)/x)

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maxima [A]  time = 0.72, size = 27, normalized size = 1.12 \begin {gather*} \frac {x^{2} e^{\left (\frac {e^{\left (x^{2}\right )}}{x}\right )} \log \relax (x)^{2} + 4}{x^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3-x)*exp(x^2)*log(x^2)^3*exp(exp(x^2)/x)-32*log(x^2)-64)/x^3/log(x^2)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.21, size = 21, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{x}}+\frac {16}{x^2\,{\ln \left (x^2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*log(x^2) + log(x^2)^3*exp(x^2)*exp(exp(x^2)/x)*(x - 2*x^3) + 64)/(x^3*log(x^2)^3),x)

[Out]

exp(exp(x^2)/x) + 16/(x^2*log(x^2)^2)

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sympy [A]  time = 0.42, size = 19, normalized size = 0.79 \begin {gather*} e^{\frac {e^{x^{2}}}{x}} + \frac {16}{x^{2} \log {\left (x^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3-x)*exp(x**2)*ln(x**2)**3*exp(exp(x**2)/x)-32*ln(x**2)-64)/x**3/ln(x**2)**3,x)

[Out]

exp(exp(x**2)/x) + 16/(x**2*log(x**2)**2)

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