∫ 4−3x2−3x3+6x5−2x6+(4x−4x3+2x4) log(x)+(−x2+x5−x3 log(x)) log(1−x3+x log(x) x ) ________________________________________________________________________________________________________________________________________________________________________________________ (4x+x2+4x3−5x4−x5−4x6+x7+(4x2+x3+4x4−x5) log(x)+(x3−x6+x4 log(x)) log(1−x3+x log(x) x )) log2(4+x+4x2−x3+x2 log(1−x3+x log(x) x ) x ) dx

3.43.43 \(\int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+(4 x-4 x^3+2 x^4) \log (x)+(-x^2+x^5-x^3 \log (x)) \log (\frac {1-x^3+x \log (x)}{x})}{(4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+(4 x^2+x^3+4 x^4-x^5) \log (x)+(x^3-x^6+x^4 \log (x)) \log (\frac {1-x^3+x \log (x)}{x})) \log ^2(\frac {4+x+4 x^2-x^3+x^2 \log (\frac {1-x^3+x \log (x)}{x})}{x})} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{\log \left (\frac {4+x}{x}+x \left (4-x+\log \left (\frac {1}{x}-x^2+\log (x)\right )\right )\right )} \]

________________________________________________________________________________________

Rubi [A] time = 0.32, antiderivative size = 40, normalized size of antiderivative = 1.33, number of steps used = 1, number of rules used = 1, integrand size = 199, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {6686} \begin {gather*} \frac {1}{\log \left (\frac {-x^3+4 x^2+x^2 \log \left (\frac {-x^3+x \log (x)+1}{x}\right )+x+4}{x}\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(4 - 3*x^2 - 3*x^3 + 6*x^5 - 2*x^6 + (4*x - 4*x^3 + 2*x^4)*Log[x] + (-x^2 + x^5 - x^3*Log[x])*Log[(1 - x^3

+ x*Log[x])/x])/((4*x + x^2 + 4*x^3 - 5*x^4 - x^5 - 4*x^6 + x^7 + (4*x^2 + x^3 + 4*x^4 - x^5)*Log[x] + (x^3 -

x^6 + x^4*Log[x])*Log[(1 - x^3 + x*Log[x])/x])*Log[(4 + x + 4*x^2 - x^3 + x^2*Log[(1 - x^3 + x*Log[x])/x])/x]

^2),x]

[Out]

Log[(4 + x + 4*x^2 - x^3 + x^2*Log[(1 - x^3 + x*Log[x])/x])/x]^(-1)

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /; !F

alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A] time = 0.11, size = 32, normalized size = 1.07 \begin {gather*} \frac {1}{\log \left (1+\frac {4}{x}+4 x-x^2+x \log \left (\frac {1}{x}-x^2+\log (x)\right )\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(4 - 3*x^2 - 3*x^3 + 6*x^5 - 2*x^6 + (4*x - 4*x^3 + 2*x^4)*Log[x] + (-x^2 + x^5 - x^3*Log[x])*Log[(1

- x^3 + x*Log[x])/x])/((4*x + x^2 + 4*x^3 - 5*x^4 - x^5 - 4*x^6 + x^7 + (4*x^2 + x^3 + 4*x^4 - x^5)*Log[x] +

(x^3 - x^6 + x^4*Log[x])*Log[(1 - x^3 + x*Log[x])/x])*Log[(4 + x + 4*x^2 - x^3 + x^2*Log[(1 - x^3 + x*Log[x])/

x])/x]^2),x]

[Out]

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

________________________________________________________________________________________

fricas [A] time = 0.64, size = 42, normalized size = 1.40 \begin {gather*} \frac {1}{\log \left (-\frac {x^{3} - x^{2} \log \left (-\frac {x^{3} - x \log \relax (x) - 1}{x}\right ) - 4 \, x^{2} - x - 4}{x}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3*log(x)+x^5-x^2)*log((x*log(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*log(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/(

(x^4*log(x)-x^6+x^3)*log((x*log(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*log(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/

log((x^2*log((x*log(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x, algorithm="fricas")

[Out]

1/log(-(x^3 - x^2*log(-(x^3 - x*log(x) - 1)/x) - 4*x^2 - x - 4)/x)

________________________________________________________________________________________

giac [A] time = 1.44, size = 44, normalized size = 1.47 \begin {gather*} \frac {1}{\log \left (-x^{3} + x^{2} \log \left (-x^{3} + x \log \relax (x) + 1\right ) - x^{2} \log \relax (x) + 4 \, x^{2} + x + 4\right ) - \log \relax (x)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3*log(x)+x^5-x^2)*log((x*log(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*log(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/(

(x^4*log(x)-x^6+x^3)*log((x*log(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*log(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/

log((x^2*log((x*log(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [C] time = 0.87, size = 3013, normalized size = 100.43


method	result	size

risch	\(\text {Expression too large to display}\)	\(3013\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3*ln(x)+x^5-x^2)*ln((x*ln(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*ln(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/((x^4*ln(x)

-x^6+x^3)*ln((x*ln(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*ln(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/ln((x^2*ln((x*

ln(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x,method=_RETURNVERBOSE)

[Out]

2*I/(Pi*csgn(I/x)*csgn(I*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(

I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3

+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-

1)*x^2+2*x^2*Pi))*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csg

n(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x

^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^

3-1)*x^2+2*x^2*Pi))-Pi*csgn(I/x)*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/

x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*cs

gn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I

*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))^2+Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-

x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^

2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln

(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x

)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-

x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*

ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))+Pi*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x

*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/

x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I

*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))^2-Pi*csgn(I*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn

(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^

3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x

)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*cs

gn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-

x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)

/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))^2+Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(

I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x

*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)

-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))^3-Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*P

i*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*cs

gn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(

x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2

*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*

csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I

*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))^2-Pi*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*

x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*

x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))

*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))^3-Pi-2*I*ln(2)-2*I*ln(x)+2*I*ln

(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I

*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-x^2*Pi*csgn(I/x)

*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)-2*I*ln(-x*ln(x)+x^3-1)*x^2+2*x^2*Pi))

________________________________________________________________________________________

maxima [A] time = 0.43, size = 41, normalized size = 1.37 \begin {gather*} \frac {1}{\log \left (-x^{3} + x^{2} {\left (\log \left (-x^{3} + x \log \relax (x) + 1\right ) + 4\right )} - x^{2} \log \relax (x) + x + 4\right ) - \log \relax (x)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3*log(x)+x^5-x^2)*log((x*log(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*log(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/(

(x^4*log(x)-x^6+x^3)*log((x*log(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*log(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/

log((x^2*log((x*log(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B] time = 5.01, size = 36, normalized size = 1.20 \begin {gather*} \frac {1}{\ln \left (4\,x+x\,\ln \left (\frac {x\,\ln \relax (x)-x^3+1}{x}\right )+\frac {4}{x}-x^2+1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 + 3*x^3 - 6*x^5 + 2*x^6 + log((x*log(x) - x^3 + 1)/x)*(x^3*log(x) + x^2 - x^5) - log(x)*(4*x - 4*x

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

+ 4*x^4 - x^5) + x^2 + 4*x^3 - 5*x^4 - x^5 - 4*x^6 + x^7 + log((x*log(x) - x^3 + 1)/x)*(x^4*log(x) + x^3 - x^6

))),x)

[Out]

1/log(4*x + x*log((x*log(x) - x^3 + 1)/x) + 4/x - x^2 + 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(((-x**3*ln(x)+x**5-x**2)*ln((x*ln(x)-x**3+1)/x)+(2*x**4-4*x**3+4*x)*ln(x)-2*x**6+6*x**5-3*x**3-3*x**

2+4)/((x**4*ln(x)-x**6+x**3)*ln((x*ln(x)-x**3+1)/x)+(-x**5+4*x**4+x**3+4*x**2)*ln(x)+x**7-4*x**6-x**5-5*x**4+4

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

[Out]

Timed out

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]