3.19.2 \(\int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{(-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+(64+16 x^2-8 x^3+x^4) \log (\frac {1}{16} (320+80 x^2-40 x^3+5 x^4))) \log (x-x^3-\log (\frac {1}{16} (320+80 x^2-40 x^3+5 x^4)))} \, dx\)

Optimal. Leaf size=27 \[ \log \left (\log \left (x-x^3-\log \left (5 \left (4+\left (x-\frac {x^2}{4}\right )^2\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 31, normalized size of antiderivative = 1.15, number of steps used = 1, number of rules used = 1, integrand size = 132, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6684} \begin {gather*} \log \left (\log \left (-x^3-\log \left (\frac {5}{16} \left (x^4-8 x^3+16 x^2+64\right )\right )+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-64 + 32*x + 152*x^2 + 12*x^3 + 47*x^4 - 24*x^5 + 3*x^6)/((-64*x + 48*x^3 + 8*x^4 + 15*x^5 - 8*x^6 + x^7
+ (64 + 16*x^2 - 8*x^3 + x^4)*Log[(320 + 80*x^2 - 40*x^3 + 5*x^4)/16])*Log[x - x^3 - Log[(320 + 80*x^2 - 40*x^
3 + 5*x^4)/16]]),x]

[Out]

Log[Log[x - x^3 - Log[(5*(64 + 16*x^2 - 8*x^3 + x^4))/16]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (x-x^3-\log \left (\frac {5}{16} \left (64+16 x^2-8 x^3+x^4\right )\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 31, normalized size = 1.15 \begin {gather*} \log \left (\log \left (x-x^3-\log \left (\frac {5}{16} \left (64+16 x^2-8 x^3+x^4\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64 + 32*x + 152*x^2 + 12*x^3 + 47*x^4 - 24*x^5 + 3*x^6)/((-64*x + 48*x^3 + 8*x^4 + 15*x^5 - 8*x^6
+ x^7 + (64 + 16*x^2 - 8*x^3 + x^4)*Log[(320 + 80*x^2 - 40*x^3 + 5*x^4)/16])*Log[x - x^3 - Log[(320 + 80*x^2 -
40*x^3 + 5*x^4)/16]]),x]

[Out]

Log[Log[x - x^3 - Log[(5*(64 + 16*x^2 - 8*x^3 + x^4))/16]]]

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fricas [A]  time = 0.73, size = 29, normalized size = 1.07 \begin {gather*} \log \left (\log \left (-x^{3} + x - \log \left (\frac {5}{16} \, x^{4} - \frac {5}{2} \, x^{3} + 5 \, x^{2} + 20\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+64)*log(5/16*x^4-5/2*x^3+5*x^2+20)+x
^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log(-log(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x, algorithm="fricas")

[Out]

log(log(-x^3 + x - log(5/16*x^4 - 5/2*x^3 + 5*x^2 + 20)))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+64)*log(5/16*x^4-5/2*x^3+5*x^2+20)+x
^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log(-log(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%{poly1[30468481522675443878296491324449711363425260500
24030208,-1

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maple [A]  time = 0.04, size = 30, normalized size = 1.11

method	result	size
risch	\(\ln \left (\ln \left (-\ln \left (\frac {5}{16} x^{4}-\frac {5}{2} x^{3}+5 x^{2}+20\right )-x^{3}+x \right )\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+64)*ln(5/16*x^4-5/2*x^3+5*x^2+20)+x^7-8*x^
6+15*x^5+8*x^4+48*x^3-64*x)/ln(-ln(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-ln(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x))

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maxima [A]  time = 0.92, size = 35, normalized size = 1.30 \begin {gather*} \log \left (\log \left (-x^{3} + x - \log \relax (5) + 4 \, \log \relax (2) - \log \left (x^{4} - 8 \, x^{3} + 16 \, x^{2} + 64\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+64)*log(5/16*x^4-5/2*x^3+5*x^2+20)+x
^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log(-log(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x, algorithm="maxima")

[Out]

log(log(-x^3 + x - log(5) + 4*log(2) - log(x^4 - 8*x^3 + 16*x^2 + 64)))

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mupad [B]  time = 1.72, size = 29, normalized size = 1.07 \begin {gather*} \ln \left (\ln \left (x-\ln \left (\frac {5\,x^4}{16}-\frac {5\,x^3}{2}+5\,x^2+20\right )-x^3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x + 152*x^2 + 12*x^3 + 47*x^4 - 24*x^5 + 3*x^6 - 64)/(log(x - log(5*x^2 - (5*x^3)/2 + (5*x^4)/16 + 20)
- x^3)*(log(5*x^2 - (5*x^3)/2 + (5*x^4)/16 + 20)*(16*x^2 - 8*x^3 + x^4 + 64) - 64*x + 48*x^3 + 8*x^4 + 15*x^5
- 8*x^6 + x^7)),x)

[Out]

log(log(x - log(5*x^2 - (5*x^3)/2 + (5*x^4)/16 + 20) - x^3))

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sympy [A]  time = 1.58, size = 29, normalized size = 1.07 \begin {gather*} \log {\left (\log {\left (- x^{3} + x - \log {\left (\frac {5 x^{4}}{16} - \frac {5 x^{3}}{2} + 5 x^{2} + 20 \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**6-24*x**5+47*x**4+12*x**3+152*x**2+32*x-64)/((x**4-8*x**3+16*x**2+64)*ln(5/16*x**4-5/2*x**3+5*
x**2+20)+x**7-8*x**6+15*x**5+8*x**4+48*x**3-64*x)/ln(-ln(5/16*x**4-5/2*x**3+5*x**2+20)-x**3+x),x)

[Out]

log(log(-x**3 + x - log(5*x**4/16 - 5*x**3/2 + 5*x**2 + 20)))

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