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

Optimal. Leaf size=20 \[ 5+\left (e^{\frac {3}{4}-\frac {x^2}{e}}\right )^{\frac {1}{x}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.35, 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 {\left (e^{\frac {3}{4}-\frac {x^2}{e}}\right )^{\frac {1}{x}} \left (-2 x^2-e \log \left (e^{\frac {3}{4}-\frac {x^2}{e}}\right )\right )}{e x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 18, normalized size = 0.90 \begin {gather*} \left (e^{\frac {3}{4}-\frac {x^2}{e}}\right )^{\frac {1}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(3/4 - x^2/E))^x^(-1)

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (-\frac {{\left (4 \, x^{2} - 3 \, e\right )} e^{\left (-1\right )}}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1/4*(4*x^2 - 3*e)*e^(-1)/x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 12, normalized size = 0.60 \begin {gather*} e^{\left (-x e^{\left (-1\right )} + \frac {3}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x*e^(-1) + 3/4/x)

________________________________________________________________________________________

maple [A]  time = 0.24, size = 19, normalized size = 0.95

method	result	size
risch	\({\mathrm e}^{-\frac {-3+4 \ln \left ({\mathrm e}^{x^{2} {\mathrm e}^{-1}}\right )}{4 x}}\)	\(19\)
default	\({\mathrm e}^{\frac {\ln \left ({\mathrm e}^{\frac {3}{4}} {\mathrm e}^{-x^{2} {\mathrm e}^{-1}}\right )}{x}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 12, normalized size = 0.60 \begin {gather*} e^{\left (-x e^{\left (-1\right )} + \frac {3}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x*e^(-1) + 3/4/x)

________________________________________________________________________________________

mupad [B]  time = 4.36, size = 12, normalized size = 0.60 \begin {gather*} {\mathrm {e}}^{\frac {3}{4\,x}-x\,{\mathrm {e}}^{-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(3/(4*x) - x*exp(-1))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________