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

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

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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 = {} $$\begin{array}{c}\int \frac{-4x^3+x^{\frac{2(-3+3x^2)}{x}}(-6-x+6x^2+(6+6x^2)\log (x))}{-4x^4+x^{2+\frac{2(-3+3x^2)}{x}}}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{2x^{-1+\frac{3}{x}}(-3-x+3x^2+3\log (x)+3x^2\log (x))}{2x^{1+\frac{3}{x}}-x^{3x}}-\frac{2x^{-1+\frac{3}{x}}(-3-x+3x^2+3\log (x)+3x^2\log (x))}{2x^{1+\frac{3}{x}}+x^{3x}}+\frac{-6-x+6x^2+6\log (x)+6x^2\log (x)}{x^2})dx\\ & =-(2\int \frac{x^{-1+\frac{3}{x}}(-3-x+3x^2+3\log (x)+3x^2\log (x))}{2x^{1+\frac{3}{x}}-x^{3x}}dx)-2\int \frac{x^{-1+\frac{3}{x}}(-3-x+3x^2+3\log (x)+3x^2\log (x))}{2x^{1+\frac{3}{x}}+x^{3x}}dx+\int \frac{-6-x+6x^2+6\log (x)+6x^2\log (x)}{x^2}dx\\ & =-(2\int (\frac{3x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}-x^{3x}}-\frac{x^{3/x}}{2x^{1+\frac{3}{x}}-x^{3x}}+\frac{3x^{-1+\frac{3}{x}}}{-2x^{1+\frac{3}{x}}+x^{3x}}+\frac{3x^{1+\frac{3}{x}}\log (x)}{2x^{1+\frac{3}{x}}-x^{3x}}-\frac{3x^{-1+\frac{3}{x}}\log (x)}{-2x^{1+\frac{3}{x}}+x^{3x}})dx)-2\int (-\frac{3x^{-1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}+\frac{3x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}-\frac{x^{3/x}}{2x^{1+\frac{3}{x}}+x^{3x}}+\frac{3x^{-1+\frac{3}{x}}\log (x)}{2x^{1+\frac{3}{x}}+x^{3x}}+\frac{3x^{1+\frac{3}{x}}\log (x)}{2x^{1+\frac{3}{x}}+x^{3x}})dx+\int (\frac{-6-x+6x^2}{x^2}+\frac{6(1+x^2)\log (x)}{x^2})dx\\ & =2\int \frac{x^{3/x}}{2x^{1+\frac{3}{x}}-x^{3x}}dx+2\int \frac{x^{3/x}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}-x^{3x}}dx-6\int \frac{x^{-1+\frac{3}{x}}}{-2x^{1+\frac{3}{x}}+x^{3x}}dx+6\int \frac{x^{-1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx+6\int \frac{(1+x^2)\log (x)}{x^2}dx-6\int \frac{x^{1+\frac{3}{x}}\log (x)}{2x^{1+\frac{3}{x}}-x^{3x}}dx+6\int \frac{x^{-1+\frac{3}{x}}\log (x)}{-2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{-1+\frac{3}{x}}\log (x)}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{1+\frac{3}{x}}\log (x)}{2x^{1+\frac{3}{x}}+x^{3x}}dx+\int \frac{-6-x+6x^2}{x^2}dx\\ & =-6(\frac{1}{x}-x)\log (x)+2\int \frac{x^{3/x}}{2x^{1+\frac{3}{x}}-x^{3x}}dx+2\int \frac{x^{3/x}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int (1-\frac{1}{x^2})dx-6\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}-x^{3x}}dx+6\int \frac{x^{-1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{1}{x(-2x+x^{-\frac{3}{x}+3x})}dx+6\int \frac{\int \frac{x^{1+\frac{3}{x}}}{-x^{3x}+2x^{\frac{3+x}{x}}}dx}{x}dx+6\int \frac{\int \frac{x^{-1+\frac{3}{x}}}{x^{3x}+2x^{\frac{3+x}{x}}}dx}{x}dx+6\int \frac{\int \frac{x^{1+\frac{3}{x}}}{x^{3x}+2x^{\frac{3+x}{x}}}dx}{x}dx-6\int \frac{\int \frac{1}{x(-2x+x^{-\frac{3}{x}+3x})}dx}{x}dx-(6\log (x))\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}-x^{3x}}dx-(6\log (x))\int \frac{x^{-1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-(6\log (x))\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx+(6\log (x))\int \frac{1}{x(-2x+x^{-\frac{3}{x}+3x})}dx+\int (6-\frac{6}{x^2}-\frac{1}{x})dx\\ & =-\log (x)-6(\frac{1}{x}-x)\log (x)+2\int \frac{x^{3/x}}{2x^{1+\frac{3}{x}}-x^{3x}}dx+2\int \frac{x^{3/x}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}-x^{3x}}dx+6\int \frac{x^{-1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-6\int \frac{1}{x(-2x+x^{-\frac{3}{x}+3x})}dx+6\int \frac{\int \frac{x^{1+\frac{3}{x}}}{-x^{3x}+2x^{\frac{3+x}{x}}}dx}{x}dx+6\int \frac{\int \frac{x^{-1+\frac{3}{x}}}{x^{3x}+2x^{\frac{3+x}{x}}}dx}{x}dx+6\int \frac{\int \frac{x^{1+\frac{3}{x}}}{x^{3x}+2x^{\frac{3+x}{x}}}dx}{x}dx-6\int \frac{\int \frac{1}{x(-2x+x^{-\frac{3}{x}+3x})}dx}{x}dx-(6\log (x))\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}-x^{3x}}dx-(6\log (x))\int \frac{x^{-1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx-(6\log (x))\int \frac{x^{1+\frac{3}{x}}}{2x^{1+\frac{3}{x}}+x^{3x}}dx+(6\log (x))\int \frac{1}{x(-2x+x^{-\frac{3}{x}+3x})}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}\log (-4x^2+x^{\frac{6(x^2-1)}{x}})-\log (x)\end{array}$$

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 $$\begin{array}{c}\log (-\frac{(2x{x}^{\frac{3}{x}}+x^{3x})(2x{x}^{\frac{3}{x}}-x^{3x})}{x^{\frac{6}{x}}})-\log (x)\end{array}$$

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	$\frac{6(x^2-1)\ln (x)}{x}-\ln (x)-\frac{2(3x^2-3)\ln (x)}{x}+\ln (x^{\frac{6x^2-6}{x}}-4x^2)$	$53$

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 $$\begin{array}{c}-\frac{6\log (x)}{x}+\log (x)+\log \left(\frac{2x{x}^{\frac{3}{x}}+x^{3x}}{2x}\right)+\log \left(\frac{2x{x}^{\frac{3}{x}}-x^{3x}}{2x}\right)\end{array}$$

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 $$\begin{array}{c}\ln (x^2-\frac{x^{6x}}{4x^{6/x}})-\ln (x)\end{array}$$

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 $$\begin{array}{c}-\log (x)+\log (-4x^2+e^{\frac{2(3x^2-3)\log (x)}{x}})\end{array}$$

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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