3.8.79 \(\int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} (-72+4 x^2)+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} (-18+x^2+x^3)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{16} e^{-3+\frac {9}{x^2}} \left (4+e^x\right ) x \]

________________________________________________________________________________________

Rubi [F]  time = 0.19, 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 {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((9 - 3*x^2 - x^2*Log[16])/x^2)*(-72 + 4*x^2) + E^(x + (9 - 3*x^2 - x^2*Log[16])/x^2)*(-18 + x^2 + x^3)
)/x^2,x]

[Out]

(E^(-3 + 9/x^2)*x)/4 + Defer[Int][E^(-3 + 9/x^2 + x), x]/16 - (9*Defer[Int][E^(-3 + 9/x^2 + x)/x^2, x])/8 + De
fer[Int][E^(-3 + 9/x^2 + x)*x, x]/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{4} e^{-3+\frac {9}{x^2}}+\frac {1}{16} e^{-3+\frac {9}{x^2}+x}-\frac {9 e^{-3+\frac {9}{x^2}}}{2 x^2}-\frac {9 e^{-3+\frac {9}{x^2}+x}}{8 x^2}+\frac {1}{16} e^{-3+\frac {9}{x^2}+x} x\right ) \, dx\\ &=\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx+\frac {1}{4} \int e^{-3+\frac {9}{x^2}} \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx-\frac {9}{2} \int \frac {e^{-3+\frac {9}{x^2}}}{x^2} \, dx\\ &=\frac {1}{4} e^{-3+\frac {9}{x^2}} x+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx+\frac {9}{2} \int \frac {e^{-3+\frac {9}{x^2}}}{x^2} \, dx+\frac {9}{2} \operatorname {Subst}\left (\int e^{-3+9 x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} e^{-3+\frac {9}{x^2}} x+\frac {3 \sqrt {\pi } \text {erfi}\left (\frac {3}{x}\right )}{4 e^3}+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx-\frac {9}{2} \operatorname {Subst}\left (\int e^{-3+9 x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} e^{-3+\frac {9}{x^2}} x+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{16} e^{-3+\frac {9}{x^2}} \left (4+e^x\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((9 - 3*x^2 - x^2*Log[16])/x^2)*(-72 + 4*x^2) + E^(x + (9 - 3*x^2 - x^2*Log[16])/x^2)*(-18 + x^2
+ x^3))/x^2,x]

[Out]

(E^(-3 + 9/x^2)*(4 + E^x)*x)/16

________________________________________________________________________________________

fricas [B]  time = 1.22, size = 48, normalized size = 2.53 \begin {gather*} x e^{\left (\frac {x^{3} - 4 \, x^{2} \log \relax (2) - 3 \, x^{2} + 9}{x^{2}}\right )} + 4 \, x e^{\left (-\frac {4 \, x^{2} \log \relax (2) + 3 \, x^{2} - 9}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72)*exp((-4*x^2*log(2)-3*x^2+9)/x^2))/x
^2,x, algorithm="fricas")

[Out]

x*e^((x^3 - 4*x^2*log(2) - 3*x^2 + 9)/x^2) + 4*x*e^(-(4*x^2*log(2) + 3*x^2 - 9)/x^2)

________________________________________________________________________________________

giac [A]  time = 0.34, size = 33, normalized size = 1.74 \begin {gather*} \frac {1}{16} \, x e^{\left (\frac {x^{3} - 3 \, x^{2} + 9}{x^{2}}\right )} + \frac {1}{4} \, x e^{\left (-\frac {3 \, {\left (x^{2} - 3\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72)*exp((-4*x^2*log(2)-3*x^2+9)/x^2))/x
^2,x, algorithm="giac")

[Out]

1/16*x*e^((x^3 - 3*x^2 + 9)/x^2) + 1/4*x*e^(-3*(x^2 - 3)/x^2)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 22, normalized size = 1.16

method	result	size
risch	\(\frac {\left ({\mathrm e}^{x} x +4 x \right ) {\mathrm e}^{-\frac {3 \left (x^{2}-3\right )}{x^{2}}}}{16}\)	\(22\)
norman	\(\frac {{\mathrm e}^{x} x^{2} {\mathrm e}^{\frac {-4 x^{2} \ln \relax (2)-3 x^{2}+9}{x^{2}}}+4 x^{2} {\mathrm e}^{\frac {-4 x^{2} \ln \relax (2)-3 x^{2}+9}{x^{2}}}}{x}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+x^2-18)*exp((-4*x^2*ln(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72)*exp((-4*x^2*ln(2)-3*x^2+9)/x^2))/x^2,x,met
hod=_RETURNVERBOSE)

[Out]

1/16*(exp(x)*x+4*x)*exp(-3*(x^2-3)/x^2)

________________________________________________________________________________________

maxima [C]  time = 0.70, size = 61, normalized size = 3.21 \begin {gather*} \frac {3}{8} \, x \sqrt {-\frac {1}{x^{2}}} e^{\left (-3\right )} \Gamma \left (-\frac {1}{2}, -\frac {9}{x^{2}}\right ) + \frac {1}{16} \, x e^{\left (x + \frac {9}{x^{2}} - 3\right )} + \frac {3 \, \sqrt {\pi } {\left (\operatorname {erf}\left (3 \, \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e^{\left (-3\right )}}{4 \, x \sqrt {-\frac {1}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72)*exp((-4*x^2*log(2)-3*x^2+9)/x^2))/x
^2,x, algorithm="maxima")

[Out]

3/8*x*sqrt(-1/x^2)*e^(-3)*gamma(-1/2, -9/x^2) + 1/16*x*e^(x + 9/x^2 - 3) + 3/4*sqrt(pi)*(erf(3*sqrt(-1/x^2)) -
1)*e^(-3)/(x*sqrt(-1/x^2))

________________________________________________________________________________________

mupad [B]  time = 0.89, size = 15, normalized size = 0.79 \begin {gather*} \frac {x\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{\frac {9}{x^2}}\,\left ({\mathrm {e}}^x+4\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(4*x^2*log(2) + 3*x^2 - 9)/x^2)*(4*x^2 - 72) + exp(-(4*x^2*log(2) + 3*x^2 - 9)/x^2)*exp(x)*(x^2 + x^
3 - 18))/x^2,x)

[Out]

(x*exp(-3)*exp(9/x^2)*(exp(x) + 4))/16

________________________________________________________________________________________

sympy [A]  time = 1.48, size = 27, normalized size = 1.42 \begin {gather*} \left (x e^{x} + 4 x\right ) e^{\frac {- 3 x^{2} - 4 x^{2} \log {\relax (2 )} + 9}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+x**2-18)*exp((-4*x**2*ln(2)-3*x**2+9)/x**2)*exp(x)+(4*x**2-72)*exp((-4*x**2*ln(2)-3*x**2+9)/x
**2))/x**2,x)

[Out]

(x*exp(x) + 4*x)*exp((-3*x**2 - 4*x**2*log(2) + 9)/x**2)

________________________________________________________________________________________