3.43.11 \(\int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} (3 x^2+x^4)+e^{\frac {-7+x}{x^2}} (-42+3 x-14 x^2+x^3-2 x^4) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ 4-\frac {3+x^2}{2+e^{-\frac {-7+x}{x^2}} \log (x)} \]

________________________________________________________________________________________

Rubi [F]  time = 20.64, 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 {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*E^((2*(-7 + x))/x^2)*x^4 + E^((-7 + x)/x^2)*(3*x^2 + x^4) + E^((-7 + x)/x^2)*(-42 + 3*x - 14*x^2 + x^3
- 2*x^4)*Log[x])/(4*E^((2*(-7 + x))/x^2)*x^3 + 4*E^((-7 + x)/x^2)*x^3*Log[x] + x^3*Log[x]^2),x]

[Out]

-1/2*x^2 - (3*Defer[Int][(E^(14/x^2)*Log[x])/(x*(2*E^x^(-1) + E^(7/x^2)*Log[x])^2), x])/2 - Defer[Int][(E^(14/
x^2)*x*Log[x])/(2*E^x^(-1) + E^(7/x^2)*Log[x])^2, x]/2 - Defer[Int][(E^(14/x^2)*Log[x]^2)/(2*E^x^(-1) + E^(7/x
^2)*Log[x])^2, x]/2 + 21*Defer[Int][(E^(14/x^2)*Log[x]^2)/(x^3*(2*E^x^(-1) + E^(7/x^2)*Log[x])^2), x] - (3*Def
er[Int][(E^(14/x^2)*Log[x]^2)/(x^2*(2*E^x^(-1) + E^(7/x^2)*Log[x])^2), x])/2 + 7*Defer[Int][(E^(14/x^2)*Log[x]
^2)/(x*(2*E^x^(-1) + E^(7/x^2)*Log[x])^2), x] + (3*Defer[Int][E^(7/x^2)/(x*(2*E^x^(-1) + E^(7/x^2)*Log[x])), x
])/2 + Defer[Int][(E^(7/x^2)*x)/(2*E^x^(-1) + E^(7/x^2)*Log[x]), x]/2 + Defer[Int][(E^(7/x^2)*Log[x])/(2*E^x^(
-1) + E^(7/x^2)*Log[x]), x]/2 - 21*Defer[Int][(E^(7/x^2)*Log[x])/(x^3*(2*E^x^(-1) + E^(7/x^2)*Log[x])), x] + (
3*Defer[Int][(E^(7/x^2)*Log[x])/(x^2*(2*E^x^(-1) + E^(7/x^2)*Log[x])), x])/2 - 7*Defer[Int][(E^(7/x^2)*Log[x])
/(x*(2*E^x^(-1) + E^(7/x^2)*Log[x])), x] + Defer[Int][(E^(7/x^2)*x*Log[x])/(2*E^x^(-1) + E^(7/x^2)*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {14}{x^2}} \left (-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)\right )}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx\\ &=\int \left (-x-\frac {e^{\frac {14}{x^2}} \left (3+x^2\right ) \log (x) \left (x^2-14 \log (x)+x \log (x)\right )}{2 x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}+\frac {e^{\frac {7}{x^2}} \left (3 x^2+x^4-42 \log (x)+3 x \log (x)-14 x^2 \log (x)+x^3 \log (x)+2 x^4 \log (x)\right )}{2 x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )}\right ) \, dx\\ &=-\frac {x^2}{2}-\frac {1}{2} \int \frac {e^{\frac {14}{x^2}} \left (3+x^2\right ) \log (x) \left (x^2-14 \log (x)+x \log (x)\right )}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} \left (3 x^2+x^4-42 \log (x)+3 x \log (x)-14 x^2 \log (x)+x^3 \log (x)+2 x^4 \log (x)\right )}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx\\ &=-\frac {x^2}{2}+\frac {1}{2} \int \left (\frac {3 e^{\frac {7}{x^2}}}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )}+\frac {e^{\frac {7}{x^2}} x}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)}+\frac {e^{\frac {7}{x^2}} \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)}-\frac {42 e^{\frac {7}{x^2}} \log (x)}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )}+\frac {3 e^{\frac {7}{x^2}} \log (x)}{x^2 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )}-\frac {14 e^{\frac {7}{x^2}} \log (x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )}+\frac {2 e^{\frac {7}{x^2}} x \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)}\right ) \, dx-\frac {1}{2} \int \left (\frac {3 e^{\frac {14}{x^2}} \log (x) \left (x^2-14 \log (x)+x \log (x)\right )}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}+\frac {e^{\frac {14}{x^2}} \log (x) \left (x^2-14 \log (x)+x \log (x)\right )}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}\right ) \, dx\\ &=-\frac {x^2}{2}+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} x}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx-\frac {1}{2} \int \frac {e^{\frac {14}{x^2}} \log (x) \left (x^2-14 \log (x)+x \log (x)\right )}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx+\frac {3}{2} \int \frac {e^{\frac {7}{x^2}}}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+\frac {3}{2} \int \frac {e^{\frac {7}{x^2}} \log (x)}{x^2 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx-\frac {3}{2} \int \frac {e^{\frac {14}{x^2}} \log (x) \left (x^2-14 \log (x)+x \log (x)\right )}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx-7 \int \frac {e^{\frac {7}{x^2}} \log (x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx-21 \int \frac {e^{\frac {7}{x^2}} \log (x)}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+\int \frac {e^{\frac {7}{x^2}} x \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx\\ &=-\frac {x^2}{2}+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} x}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx-\frac {1}{2} \int \left (\frac {e^{\frac {14}{x^2}} x \log (x)}{\left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}+\frac {e^{\frac {14}{x^2}} \log ^2(x)}{\left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}-\frac {14 e^{\frac {14}{x^2}} \log ^2(x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}\right ) \, dx+\frac {3}{2} \int \frac {e^{\frac {7}{x^2}}}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+\frac {3}{2} \int \frac {e^{\frac {7}{x^2}} \log (x)}{x^2 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx-\frac {3}{2} \int \left (\frac {e^{\frac {14}{x^2}} \log (x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}-\frac {14 e^{\frac {14}{x^2}} \log ^2(x)}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}+\frac {e^{\frac {14}{x^2}} \log ^2(x)}{x^2 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2}\right ) \, dx-7 \int \frac {e^{\frac {7}{x^2}} \log (x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx-21 \int \frac {e^{\frac {7}{x^2}} \log (x)}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+\int \frac {e^{\frac {7}{x^2}} x \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx\\ &=-\frac {x^2}{2}-\frac {1}{2} \int \frac {e^{\frac {14}{x^2}} x \log (x)}{\left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx-\frac {1}{2} \int \frac {e^{\frac {14}{x^2}} \log ^2(x)}{\left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} x}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx+\frac {1}{2} \int \frac {e^{\frac {7}{x^2}} \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx-\frac {3}{2} \int \frac {e^{\frac {14}{x^2}} \log (x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx-\frac {3}{2} \int \frac {e^{\frac {14}{x^2}} \log ^2(x)}{x^2 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx+\frac {3}{2} \int \frac {e^{\frac {7}{x^2}}}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+\frac {3}{2} \int \frac {e^{\frac {7}{x^2}} \log (x)}{x^2 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+7 \int \frac {e^{\frac {14}{x^2}} \log ^2(x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx-7 \int \frac {e^{\frac {7}{x^2}} \log (x)}{x \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+21 \int \frac {e^{\frac {14}{x^2}} \log ^2(x)}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )^2} \, dx-21 \int \frac {e^{\frac {7}{x^2}} \log (x)}{x^3 \left (2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)\right )} \, dx+\int \frac {e^{\frac {7}{x^2}} x \log (x)}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 32, normalized size = 1.23 \begin {gather*} -\frac {e^{\frac {1}{x}} \left (3+x^2\right )}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^((2*(-7 + x))/x^2)*x^4 + E^((-7 + x)/x^2)*(3*x^2 + x^4) + E^((-7 + x)/x^2)*(-42 + 3*x - 14*x^2
+ x^3 - 2*x^4)*Log[x])/(4*E^((2*(-7 + x))/x^2)*x^3 + 4*E^((-7 + x)/x^2)*x^3*Log[x] + x^3*Log[x]^2),x]

[Out]

-((E^x^(-1)*(3 + x^2))/(2*E^x^(-1) + E^(7/x^2)*Log[x]))

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 30, normalized size = 1.15 \begin {gather*} -\frac {{\left (x^{2} + 3\right )} e^{\left (\frac {x - 7}{x^{2}}\right )}}{2 \, e^{\left (\frac {x - 7}{x^{2}}\right )} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^3-14*x^2+3*x-42)*exp((x-7)/x^2)*log(x)-4*x^4*exp((x-7)/x^2)^2+(x^4+3*x^2)*exp((x-7)/x^2))
/(x^3*log(x)^2+4*x^3*exp((x-7)/x^2)*log(x)+4*x^3*exp((x-7)/x^2)^2),x, algorithm="fricas")

[Out]

-(x^2 + 3)*e^((x - 7)/x^2)/(2*e^((x - 7)/x^2) + log(x))

________________________________________________________________________________________

giac [A]  time = 0.39, size = 35, normalized size = 1.35 \begin {gather*} -\frac {2 \, x^{2} e^{\left (\frac {x - 7}{x^{2}}\right )} - 3 \, \log \relax (x)}{2 \, {\left (2 \, e^{\left (\frac {x - 7}{x^{2}}\right )} + \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^3-14*x^2+3*x-42)*exp((x-7)/x^2)*log(x)-4*x^4*exp((x-7)/x^2)^2+(x^4+3*x^2)*exp((x-7)/x^2))
/(x^3*log(x)^2+4*x^3*exp((x-7)/x^2)*log(x)+4*x^3*exp((x-7)/x^2)^2),x, algorithm="giac")

[Out]

-1/2*(2*x^2*e^((x - 7)/x^2) - 3*log(x))/(2*e^((x - 7)/x^2) + log(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 31, normalized size = 1.19

method	result	size
risch	\(-\frac {\left (x^{2}+3\right ) {\mathrm e}^{\frac {x -7}{x^{2}}}}{\ln \relax (x )+2 \,{\mathrm e}^{\frac {x -7}{x^{2}}}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^4+x^3-14*x^2+3*x-42)*exp((x-7)/x^2)*ln(x)-4*x^4*exp((x-7)/x^2)^2+(x^4+3*x^2)*exp((x-7)/x^2))/(x^3*l
n(x)^2+4*x^3*exp((x-7)/x^2)*ln(x)+4*x^3*exp((x-7)/x^2)^2),x,method=_RETURNVERBOSE)

[Out]

-(x^2+3)*exp((x-7)/x^2)/(ln(x)+2*exp((x-7)/x^2))

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 29, normalized size = 1.12 \begin {gather*} -\frac {{\left (x^{2} + 3\right )} e^{\frac {1}{x}}}{e^{\left (\frac {7}{x^{2}}\right )} \log \relax (x) + 2 \, e^{\frac {1}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^3-14*x^2+3*x-42)*exp((x-7)/x^2)*log(x)-4*x^4*exp((x-7)/x^2)^2+(x^4+3*x^2)*exp((x-7)/x^2))
/(x^3*log(x)^2+4*x^3*exp((x-7)/x^2)*log(x)+4*x^3*exp((x-7)/x^2)^2),x, algorithm="maxima")

[Out]

-(x^2 + 3)*e^(1/x)/(e^(7/x^2)*log(x) + 2*e^(1/x))

________________________________________________________________________________________

mupad [B]  time = 3.31, size = 36, normalized size = 1.38 \begin {gather*} \frac {\frac {3\,\ln \relax (x)}{2}+\frac {x^2\,\ln \relax (x)}{2}}{2\,{\mathrm {e}}^{\frac {1}{x}-\frac {7}{x^2}}+\ln \relax (x)}-\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x^4*exp((2*(x - 7))/x^2) - exp((x - 7)/x^2)*(3*x^2 + x^4) + exp((x - 7)/x^2)*log(x)*(14*x^2 - 3*x - x^
3 + 2*x^4 + 42))/(x^3*log(x)^2 + 4*x^3*exp((2*(x - 7))/x^2) + 4*x^3*exp((x - 7)/x^2)*log(x)),x)

[Out]

((3*log(x))/2 + (x^2*log(x))/2)/(2*exp(1/x - 7/x^2) + log(x)) - x^2/2

________________________________________________________________________________________

sympy [A]  time = 0.33, size = 31, normalized size = 1.19 \begin {gather*} - \frac {x^{2}}{2} + \frac {x^{2} \log {\relax (x )} + 3 \log {\relax (x )}}{4 e^{\frac {x - 7}{x^{2}}} + 2 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**4+x**3-14*x**2+3*x-42)*exp((x-7)/x**2)*ln(x)-4*x**4*exp((x-7)/x**2)**2+(x**4+3*x**2)*exp((x-
7)/x**2))/(x**3*ln(x)**2+4*x**3*exp((x-7)/x**2)*ln(x)+4*x**3*exp((x-7)/x**2)**2),x)

[Out]

-x**2/2 + (x**2*log(x) + 3*log(x))/(4*exp((x - 7)/x**2) + 2*log(x))

________________________________________________________________________________________