3.17.45 \(\int \frac {-4 x^3+x^{\frac {2 (-3+3 x^2)}{x}} (-6-x+6 x^2+(6+6 x^2) \log (x))}{-4 x^4+x^{2+\frac {2 (-3+3 x^2)}{x}}} \, dx\)

Optimal. Leaf size=22 \[ \log \left (-x+\frac {1}{4} x^{-1+6 \left (-\frac {1}{x}+x\right )}\right ) \]

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Rubi [F]  time = 1.98, 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 x^3+x^{\frac {2 \left (-3+3 x^2\right )}{x}} \left (-6-x+6 x^2+\left (6+6 x^2\right ) \log (x)\right )}{-4 x^4+x^{2+\frac {2 \left (-3+3 x^2\right )}{x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-Log[x] - 6*(x^(-1) - x)*Log[x] - 6*Defer[Int][x^(1 + 3/x)/(2*x^(1 + 3/x) - x^(3*x)), x] - 6*Log[x]*Defer[Int]
[x^(1 + 3/x)/(2*x^(1 + 3/x) - x^(3*x)), x] + 2*Defer[Int][x^(3/x)/(2*x^(1 + 3/x) - x^(3*x)), x] + 6*Defer[Int]
[x^(-1 + 3/x)/(2*x^(1 + 3/x) + x^(3*x)), x] - 6*Log[x]*Defer[Int][x^(-1 + 3/x)/(2*x^(1 + 3/x) + x^(3*x)), x] -
6*Defer[Int][x^(1 + 3/x)/(2*x^(1 + 3/x) + x^(3*x)), x] - 6*Log[x]*Defer[Int][x^(1 + 3/x)/(2*x^(1 + 3/x) + x^(
3*x)), x] + 2*Defer[Int][x^(3/x)/(2*x^(1 + 3/x) + x^(3*x)), x] - 6*Defer[Int][1/(x*(-2*x + x^(-3/x + 3*x))), x
] + 6*Log[x]*Defer[Int][1/(x*(-2*x + x^(-3/x + 3*x))), x] + 6*Defer[Int][Defer[Int][x^(1 + 3/x)/(-x^(3*x) + 2*
x^((3 + x)/x)), x]/x, x] + 6*Defer[Int][Defer[Int][x^(-1 + 3/x)/(x^(3*x) + 2*x^((3 + x)/x)), x]/x, x] + 6*Defe
r[Int][Defer[Int][x^(1 + 3/x)/(x^(3*x) + 2*x^((3 + x)/x)), x]/x, x] - 6*Defer[Int][Defer[Int][1/(x*(-2*x + x^(
-3/x + 3*x))), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 x^{-1+\frac {3}{x}} \left (-3-x+3 x^2+3 \log (x)+3 x^2 \log (x)\right )}{2 x^{1+\frac {3}{x}}-x^{3 x}}-\frac {2 x^{-1+\frac {3}{x}} \left (-3-x+3 x^2+3 \log (x)+3 x^2 \log (x)\right )}{2 x^{1+\frac {3}{x}}+x^{3 x}}+\frac {-6-x+6 x^2+6 \log (x)+6 x^2 \log (x)}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^{-1+\frac {3}{x}} \left (-3-x+3 x^2+3 \log (x)+3 x^2 \log (x)\right )}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx\right )-2 \int \frac {x^{-1+\frac {3}{x}} \left (-3-x+3 x^2+3 \log (x)+3 x^2 \log (x)\right )}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx+\int \frac {-6-x+6 x^2+6 \log (x)+6 x^2 \log (x)}{x^2} \, dx\\ &=-\left (2 \int \left (\frac {3 x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}-x^{3 x}}-\frac {x^{3/x}}{2 x^{1+\frac {3}{x}}-x^{3 x}}+\frac {3 x^{-1+\frac {3}{x}}}{-2 x^{1+\frac {3}{x}}+x^{3 x}}+\frac {3 x^{1+\frac {3}{x}} \log (x)}{2 x^{1+\frac {3}{x}}-x^{3 x}}-\frac {3 x^{-1+\frac {3}{x}} \log (x)}{-2 x^{1+\frac {3}{x}}+x^{3 x}}\right ) \, dx\right )-2 \int \left (-\frac {3 x^{-1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}}+\frac {3 x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}}-\frac {x^{3/x}}{2 x^{1+\frac {3}{x}}+x^{3 x}}+\frac {3 x^{-1+\frac {3}{x}} \log (x)}{2 x^{1+\frac {3}{x}}+x^{3 x}}+\frac {3 x^{1+\frac {3}{x}} \log (x)}{2 x^{1+\frac {3}{x}}+x^{3 x}}\right ) \, dx+\int \left (\frac {-6-x+6 x^2}{x^2}+\frac {6 \left (1+x^2\right ) \log (x)}{x^2}\right ) \, dx\\ &=2 \int \frac {x^{3/x}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx+2 \int \frac {x^{3/x}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx-6 \int \frac {x^{-1+\frac {3}{x}}}{-2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx+6 \int \frac {x^{-1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx+6 \int \frac {\left (1+x^2\right ) \log (x)}{x^2} \, dx-6 \int \frac {x^{1+\frac {3}{x}} \log (x)}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx+6 \int \frac {x^{-1+\frac {3}{x}} \log (x)}{-2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{-1+\frac {3}{x}} \log (x)}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{1+\frac {3}{x}} \log (x)}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx+\int \frac {-6-x+6 x^2}{x^2} \, dx\\ &=-6 \left (\frac {1}{x}-x\right ) \log (x)+2 \int \frac {x^{3/x}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx+2 \int \frac {x^{3/x}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \left (1-\frac {1}{x^2}\right ) \, dx-6 \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx+6 \int \frac {x^{-1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {1}{x \left (-2 x+x^{-\frac {3}{x}+3 x}\right )} \, dx+6 \int \frac {\int \frac {x^{1+\frac {3}{x}}}{-x^{3 x}+2 x^{\frac {3+x}{x}}} \, dx}{x} \, dx+6 \int \frac {\int \frac {x^{-1+\frac {3}{x}}}{x^{3 x}+2 x^{\frac {3+x}{x}}} \, dx}{x} \, dx+6 \int \frac {\int \frac {x^{1+\frac {3}{x}}}{x^{3 x}+2 x^{\frac {3+x}{x}}} \, dx}{x} \, dx-6 \int \frac {\int \frac {1}{x \left (-2 x+x^{-\frac {3}{x}+3 x}\right )} \, dx}{x} \, dx-(6 \log (x)) \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx-(6 \log (x)) \int \frac {x^{-1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-(6 \log (x)) \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx+(6 \log (x)) \int \frac {1}{x \left (-2 x+x^{-\frac {3}{x}+3 x}\right )} \, dx+\int \left (6-\frac {6}{x^2}-\frac {1}{x}\right ) \, dx\\ &=-\log (x)-6 \left (\frac {1}{x}-x\right ) \log (x)+2 \int \frac {x^{3/x}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx+2 \int \frac {x^{3/x}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx+6 \int \frac {x^{-1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-6 \int \frac {1}{x \left (-2 x+x^{-\frac {3}{x}+3 x}\right )} \, dx+6 \int \frac {\int \frac {x^{1+\frac {3}{x}}}{-x^{3 x}+2 x^{\frac {3+x}{x}}} \, dx}{x} \, dx+6 \int \frac {\int \frac {x^{-1+\frac {3}{x}}}{x^{3 x}+2 x^{\frac {3+x}{x}}} \, dx}{x} \, dx+6 \int \frac {\int \frac {x^{1+\frac {3}{x}}}{x^{3 x}+2 x^{\frac {3+x}{x}}} \, dx}{x} \, dx-6 \int \frac {\int \frac {1}{x \left (-2 x+x^{-\frac {3}{x}+3 x}\right )} \, dx}{x} \, dx-(6 \log (x)) \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}-x^{3 x}} \, dx-(6 \log (x)) \int \frac {x^{-1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx-(6 \log (x)) \int \frac {x^{1+\frac {3}{x}}}{2 x^{1+\frac {3}{x}}+x^{3 x}} \, dx+(6 \log (x)) \int \frac {1}{x \left (-2 x+x^{-\frac {3}{x}+3 x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.54, size = 24, normalized size = 1.09 \begin {gather*} -\log (x)+\log \left (-4 x^2+x^{\frac {6 \left (-1+x^2\right )}{x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[-4*x^2 + x^((6*(-1 + x^2))/x)]

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fricas [A]  time = 0.95, size = 24, normalized size = 1.09 \begin {gather*} \log \left (-4 \, x^{2} + x^{\frac {6 \, {\left (x^{2} - 1\right )}}{x}}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^2+6)*log(x)+6*x^2-x-6)*exp((3*x^2-3)*log(x)/x)^2-4*x^3)/(x^2*exp((3*x^2-3)*log(x)/x)^2-4*x^4)
,x, algorithm="fricas")

[Out]

log(-4*x^2 + x^(6*(x^2 - 1)/x)) - log(x)

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giac [B]  time = 0.57, size = 51, normalized size = 2.32 \begin {gather*} \log \left (-\frac {{\left (2 \, x x^{\frac {3}{x}} + x^{3 \, x}\right )} {\left (2 \, x x^{\frac {3}{x}} - x^{3 \, x}\right )}}{x^{\frac {6}{x}}}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^2+6)*log(x)+6*x^2-x-6)*exp((3*x^2-3)*log(x)/x)^2-4*x^3)/(x^2*exp((3*x^2-3)*log(x)/x)^2-4*x^4)
,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 53, normalized size = 2.41

method	result	size
risch	\(\frac {6 \left (x^{2}-1\right ) \ln \relax (x )}{x}-\ln \relax (x )-\frac {2 \left (3 x^{2}-3\right ) \ln \relax (x )}{x}+\ln \left (x^{\frac {6 x^{2}-6}{x}}-4 x^{2}\right )\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.56, size = 56, normalized size = 2.55 \begin {gather*} -\frac {6 \, \log \relax (x)}{x} + \log \relax (x) + \log \left (\frac {2 \, x x^{\frac {3}{x}} + x^{3 \, x}}{2 \, x}\right ) + \log \left (\frac {2 \, x x^{\frac {3}{x}} - x^{3 \, x}}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^2+6)*log(x)+6*x^2-x-6)*exp((3*x^2-3)*log(x)/x)^2-4*x^3)/(x^2*exp((3*x^2-3)*log(x)/x)^2-4*x^4)
,x, algorithm="maxima")

[Out]

-6*log(x)/x + log(x) + log(1/2*(2*x*x^(3/x) + x^(3*x))/x) + log(1/2*(2*x*x^(3/x) - x^(3*x))/x)

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mupad [B]  time = 1.39, size = 26, normalized size = 1.18 \begin {gather*} \ln \left (x^2-\frac {x^{6\,x}}{4\,x^{6/x}}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - x^(6*x)/(4*x^(6/x))) - log(x)

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sympy [A]  time = 0.37, size = 24, normalized size = 1.09 \begin {gather*} - \log {\relax (x )} + \log {\left (- 4 x^{2} + e^{\frac {2 \left (3 x^{2} - 3\right ) \log {\relax (x )}}{x}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x**2+6)*ln(x)+6*x**2-x-6)*exp((3*x**2-3)*ln(x)/x)**2-4*x**3)/(x**2*exp((3*x**2-3)*ln(x)/x)**2-4
*x**4),x)

[Out]

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

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