3.30.75 \(\int \frac {x^{-6/x} (-x^2-2 x^4+(x+2 x^3) \log (9)+(-30 x+5 x^2+6 x^3-x^4+(30-6 x^2) \log (9)+(36 x-x^2-6 x^3+(-36+6 x^2) \log (9)) \log (x)+(-6 x+6 \log (9)) \log ^2(x)) \log (-5+x^2+\log (x)))}{-5 x^2+x^4+x^2 \log (x)} \, dx\)

Optimal. Leaf size=22 \[ x^{-6/x} (-x+\log (9)) \log \left (-5+x^2+\log (x)\right ) \]

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Rubi [F]  time = 1.50, 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 {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 - 2*x^4 + (x + 2*x^3)*Log[9] + (-30*x + 5*x^2 + 6*x^3 - x^4 + (30 - 6*x^2)*Log[9] + (36*x - x^2 - 6*
x^3 + (-36 + 6*x^2)*Log[9])*Log[x] + (-6*x + 6*Log[9])*Log[x]^2)*Log[-5 + x^2 + Log[x]])/(x^(6/x)*(-5*x^2 + x^
4 + x^2*Log[x])),x]

[Out]

Log[9]*Defer[Int][x^(-1 - 6/x)/(-5 + x^2 + Log[x]), x] + 2*Log[9]*Defer[Int][x^(1 - 6/x)/(-5 + x^2 + Log[x]),
x] - 2*Defer[Int][x^(2 - 6/x)/(-5 + x^2 + Log[x]), x] - Defer[Int][1/(x^(6/x)*(-5 + x^2 + Log[x])), x] - 6*Log
[9]*Defer[Int][x^(-2 - 6/x)*Log[-5 + x^2 + Log[x]], x] + 6*Defer[Int][x^(-1 - 6/x)*Log[-5 + x^2 + Log[x]], x]
- Defer[Int][Log[-5 + x^2 + Log[x]]/x^(6/x), x] + 6*Log[9]*Defer[Int][x^(-2 - 6/x)*Log[x]*Log[-5 + x^2 + Log[x
]], x] - 6*Defer[Int][x^(-1 - 6/x)*Log[x]*Log[-5 + x^2 + Log[x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^{-2-\frac {6}{x}} \left (x \left (1+2 x^2\right ) (x-\log (9))+\left (-5+x^2+\log (x)\right ) \left (-6 x+x^2+6 \log (9)+6 (x-\log (9)) \log (x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{5-x^2-\log (x)} \, dx\\ &=\int \left (-\frac {x^{-1-\frac {6}{x}} \left (1+2 x^2\right ) (x-\log (9))}{-5+x^2+\log (x)}-x^{-2-\frac {6}{x}} \left (-6 x+x^2+6 \log (9)+6 x \log (x)-6 \log (9) \log (x)\right ) \log \left (-5+x^2+\log (x)\right )\right ) \, dx\\ &=-\int \frac {x^{-1-\frac {6}{x}} \left (1+2 x^2\right ) (x-\log (9))}{-5+x^2+\log (x)} \, dx-\int x^{-2-\frac {6}{x}} \left (-6 x+x^2+6 \log (9)+6 x \log (x)-6 \log (9) \log (x)\right ) \log \left (-5+x^2+\log (x)\right ) \, dx\\ &=-\int \left (\frac {2 x^{2-\frac {6}{x}}}{-5+x^2+\log (x)}+\frac {x^{-6/x}}{-5+x^2+\log (x)}-\frac {x^{-1-\frac {6}{x}} \log (9)}{-5+x^2+\log (x)}-\frac {2 x^{1-\frac {6}{x}} \log (9)}{-5+x^2+\log (x)}\right ) \, dx-\int \left (-6 x^{-1-\frac {6}{x}} \log \left (-5+x^2+\log (x)\right )+x^{-6/x} \log \left (-5+x^2+\log (x)\right )+6 x^{-2-\frac {6}{x}} \log (9) \log \left (-5+x^2+\log (x)\right )+6 x^{-1-\frac {6}{x}} \log (x) \log \left (-5+x^2+\log (x)\right )-6 x^{-2-\frac {6}{x}} \log (9) \log (x) \log \left (-5+x^2+\log (x)\right )\right ) \, dx\\ &=-\left (2 \int \frac {x^{2-\frac {6}{x}}}{-5+x^2+\log (x)} \, dx\right )+6 \int x^{-1-\frac {6}{x}} \log \left (-5+x^2+\log (x)\right ) \, dx-6 \int x^{-1-\frac {6}{x}} \log (x) \log \left (-5+x^2+\log (x)\right ) \, dx+\log (9) \int \frac {x^{-1-\frac {6}{x}}}{-5+x^2+\log (x)} \, dx+(2 \log (9)) \int \frac {x^{1-\frac {6}{x}}}{-5+x^2+\log (x)} \, dx-(6 \log (9)) \int x^{-2-\frac {6}{x}} \log \left (-5+x^2+\log (x)\right ) \, dx+(6 \log (9)) \int x^{-2-\frac {6}{x}} \log (x) \log \left (-5+x^2+\log (x)\right ) \, dx-\int \frac {x^{-6/x}}{-5+x^2+\log (x)} \, dx-\int x^{-6/x} \log \left (-5+x^2+\log (x)\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 23, normalized size = 1.05 \begin {gather*} -x^{-6/x} (x-\log (9)) \log \left (-5+x^2+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 - 2*x^4 + (x + 2*x^3)*Log[9] + (-30*x + 5*x^2 + 6*x^3 - x^4 + (30 - 6*x^2)*Log[9] + (36*x - x^
2 - 6*x^3 + (-36 + 6*x^2)*Log[9])*Log[x] + (-6*x + 6*Log[9])*Log[x]^2)*Log[-5 + x^2 + Log[x]])/(x^(6/x)*(-5*x^
2 + x^4 + x^2*Log[x])),x]

[Out]

-(((x - Log[9])*Log[-5 + x^2 + Log[x]])/x^(6/x))

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fricas [A]  time = 0.72, size = 25, normalized size = 1.14 \begin {gather*} -\frac {{\left (x - 2 \, \log \relax (3)\right )} \log \left (x^{2} + \log \relax (x) - 5\right )}{x^{\frac {6}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*log(3)-6*x)*log(x)^2+(2*(6*x^2-36)*log(3)-6*x^3-x^2+36*x)*log(x)+2*(-6*x^2+30)*log(3)-x^4+6*x^
3+5*x^2-30*x)*log(log(x)+x^2-5)+2*(2*x^3+x)*log(3)-2*x^4-x^2)/(x^2*log(x)+x^4-5*x^2)/exp(6*log(x)/x),x, algori
thm="fricas")

[Out]

-(x - 2*log(3))*log(x^2 + log(x) - 5)/x^(6/x)

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giac [A]  time = 1.94, size = 42, normalized size = 1.91 \begin {gather*} -\frac {x \log \left (x^{2} + \log \relax (x) - 5\right )}{x^{\frac {6}{x}}} + \frac {2 \, \log \relax (3) \log \left (x^{2} + \log \relax (x) - 5\right )}{x^{\frac {6}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*log(3)-6*x)*log(x)^2+(2*(6*x^2-36)*log(3)-6*x^3-x^2+36*x)*log(x)+2*(-6*x^2+30)*log(3)-x^4+6*x^
3+5*x^2-30*x)*log(log(x)+x^2-5)+2*(2*x^3+x)*log(3)-2*x^4-x^2)/(x^2*log(x)+x^4-5*x^2)/exp(6*log(x)/x),x, algori
thm="giac")

[Out]

-x*log(x^2 + log(x) - 5)/x^(6/x) + 2*log(3)*log(x^2 + log(x) - 5)/x^(6/x)

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maple [A]  time = 0.06, size = 27, normalized size = 1.23

method	result	size
risch	\(\ln \left (\ln \relax (x )+x^{2}-5\right ) x^{-\frac {6}{x}} \left (2 \ln \relax (3)-x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((12*ln(3)-6*x)*ln(x)^2+(2*(6*x^2-36)*ln(3)-6*x^3-x^2+36*x)*ln(x)+2*(-6*x^2+30)*ln(3)-x^4+6*x^3+5*x^2-30*
x)*ln(ln(x)+x^2-5)+2*(2*x^3+x)*ln(3)-2*x^4-x^2)/(x^2*ln(x)+x^4-5*x^2)/exp(6*ln(x)/x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x)+x^2-5)/(x^(6/x))*(2*ln(3)-x)

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maxima [A]  time = 0.67, size = 25, normalized size = 1.14 \begin {gather*} -\frac {{\left (x - 2 \, \log \relax (3)\right )} \log \left (x^{2} + \log \relax (x) - 5\right )}{x^{\frac {6}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*log(3)-6*x)*log(x)^2+(2*(6*x^2-36)*log(3)-6*x^3-x^2+36*x)*log(x)+2*(-6*x^2+30)*log(3)-x^4+6*x^
3+5*x^2-30*x)*log(log(x)+x^2-5)+2*(2*x^3+x)*log(3)-2*x^4-x^2)/(x^2*log(x)+x^4-5*x^2)/exp(6*log(x)/x),x, algori
thm="maxima")

[Out]

-(x - 2*log(3))*log(x^2 + log(x) - 5)/x^(6/x)

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mupad [B]  time = 2.10, size = 25, normalized size = 1.14 \begin {gather*} -\frac {\ln \left (\ln \relax (x)+x^2-5\right )\,\left (x-\ln \relax (9)\right )}{x^{6/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(6*log(x))/x)*(log(log(x) + x^2 - 5)*(30*x - log(x)*(36*x + 2*log(3)*(6*x^2 - 36) - x^2 - 6*x^3) +
2*log(3)*(6*x^2 - 30) + log(x)^2*(6*x - 12*log(3)) - 5*x^2 - 6*x^3 + x^4) + x^2 + 2*x^4 - 2*log(3)*(x + 2*x^3)
))/(x^2*log(x) - 5*x^2 + x^4),x)

[Out]

-(log(log(x) + x^2 - 5)*(x - log(9)))/x^(6/x)

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sympy [A]  time = 26.39, size = 34, normalized size = 1.55 \begin {gather*} \left (- x \log {\left (x^{2} + \log {\relax (x )} - 5 \right )} + 2 \log {\relax (3 )} \log {\left (x^{2} + \log {\relax (x )} - 5 \right )}\right ) e^{- \frac {6 \log {\relax (x )}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*ln(3)-6*x)*ln(x)**2+(2*(6*x**2-36)*ln(3)-6*x**3-x**2+36*x)*ln(x)+2*(-6*x**2+30)*ln(3)-x**4+6*x
**3+5*x**2-30*x)*ln(ln(x)+x**2-5)+2*(2*x**3+x)*ln(3)-2*x**4-x**2)/(x**2*ln(x)+x**4-5*x**2)/exp(6*ln(x)/x),x)

[Out]

(-x*log(x**2 + log(x) - 5) + 2*log(3)*log(x**2 + log(x) - 5))*exp(-6*log(x)/x)

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