3.24.31 \(\int \frac {32 x^4-16 x^5+2 x^6+(240 x^2-120 x^3-x^4+8 x^5-x^6) \log (\frac {1}{5} (-15+x^2))+(-15 x^2+x^4+(-240+120 x+x^2-8 x^3+x^4) \log (3)) \log ^2(\frac {1}{5} (-15+x^2))}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+(2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+(-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6) \log (3)) \log (\frac {1}{5} (-15+x^2))+(-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+(2400 x-1080 x^2-40 x^3+72 x^4-8 x^5) \log (3)+(-960+480 x+4 x^2-32 x^3+4 x^4) \log ^2(3)) \log ^2(\frac {1}{5} (-15+x^2))} \, dx\) [2331]

Optimal. Leaf size=42 \[ \frac {1}{4 \left (\frac {-x+\frac {x}{-4+x}+\log (3)}{x}+\frac {x}{\log \left (\left (-\frac {3}{x}+\frac {x}{5}\right ) x\right )}\right )} \]

[Out]

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

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Rubi [F]
time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32*x^4 - 16*x^5 + 2*x^6 + (240*x^2 - 120*x^3 - x^4 + 8*x^5 - x^6)*Log[(-15 + x^2)/5] + (-15*x^2 + x^4 + (
-240 + 120*x + x^2 - 8*x^3 + x^4)*Log[3])*Log[(-15 + x^2)/5]^2)/(-960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x^8 +
(2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7 + (-1920*x^2 + 960*x^3 + 8*x^4 - 64*x^5 + 8*x^6)*Log[3])*Log[(
-15 + x^2)/5] + (-1500*x^2 + 600*x^3 + 40*x^4 - 40*x^5 + 4*x^6 + (2400*x - 1080*x^2 - 40*x^3 + 72*x^4 - 8*x^5)
*Log[3] + (-960 + 480*x + 4*x^2 - 32*x^3 + 4*x^4)*Log[3]^2)*Log[(-15 + x^2)/5]^2),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]
time = 0.09, size = 70, normalized size = 1.67 \begin {gather*} \frac {(-4+x) x^2+(x-4 \log (3)+x \log (3)) \log \left (-3+\frac {x^2}{5}\right )}{4 \left ((-4+x) x^2+\left (-x^2-4 \log (3)+x (5+\log (3))\right ) \log \left (-3+\frac {x^2}{5}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32*x^4 - 16*x^5 + 2*x^6 + (240*x^2 - 120*x^3 - x^4 + 8*x^5 - x^6)*Log[(-15 + x^2)/5] + (-15*x^2 + x
^4 + (-240 + 120*x + x^2 - 8*x^3 + x^4)*Log[3])*Log[(-15 + x^2)/5]^2)/(-960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4
*x^8 + (2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7 + (-1920*x^2 + 960*x^3 + 8*x^4 - 64*x^5 + 8*x^6)*Log[3])
*Log[(-15 + x^2)/5] + (-1500*x^2 + 600*x^3 + 40*x^4 - 40*x^5 + 4*x^6 + (2400*x - 1080*x^2 - 40*x^3 + 72*x^4 -
8*x^5)*Log[3] + (-960 + 480*x + 4*x^2 - 32*x^3 + 4*x^4)*Log[3]^2)*Log[(-15 + x^2)/5]^2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(38)=76\).
time = 0.76, size = 124, normalized size = 2.95

method	result	size
risch	\(\frac {x \ln \left (3\right )-4 \ln \left (3\right )+x}{4 x \ln \left (3\right )-4 x^{2}-16 \ln \left (3\right )+20 x}-\frac {x^{3} \left (x^{2}-8 x +16\right )}{4 \left (x \ln \left (3\right )-x^{2}-4 \ln \left (3\right )+5 x \right ) \left (\ln \left (3\right ) x \ln \left (\frac {x^{2}}{5}-3\right )-x^{2} \ln \left (\frac {x^{2}}{5}-3\right )+x^{3}-4 \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )+5 \ln \left (\frac {x^{2}}{5}-3\right ) x -4 x^{2}\right )}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4-8*x^3+x^2+120*x-240)*ln(3)+x^4-15*x^2)*ln(1/5*x^2-3)^2+(-x^6+8*x^5-x^4-120*x^3+240*x^2)*ln(1/5*x^2-
3)+2*x^6-16*x^5+32*x^4)/(((4*x^4-32*x^3+4*x^2+480*x-960)*ln(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400*x)*ln(3)+
4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*ln(1/5*x^2-3)^2+((8*x^6-64*x^5+8*x^4+960*x^3-1920*x^2)*ln(3)-8*x^7+72*x^
6-40*x^5-1080*x^4+2400*x^3)*ln(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x,method=_RETURNVERBOSE)

[Out]

1/4*(x*ln(3)-4*ln(3)+x)/(x*ln(3)-x^2-4*ln(3)+5*x)-1/4*x^3*(x^2-8*x+16)/(x*ln(3)-x^2-4*ln(3)+5*x)/(ln(3)*x*ln(1
/5*x^2-3)-x^2*ln(1/5*x^2-3)+x^3-4*ln(3)*ln(1/5*x^2-3)+5*ln(1/5*x^2-3)*x-4*x^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (39) = 78\).
time = 0.58, size = 102, normalized size = 2.43 \begin {gather*} \frac {x^{3} - {\left (\log \left (5\right ) \log \left (3\right ) + \log \left (5\right )\right )} x - 4 \, x^{2} + 4 \, \log \left (5\right ) \log \left (3\right ) + {\left (x {\left (\log \left (3\right ) + 1\right )} - 4 \, \log \left (3\right )\right )} \log \left (x^{2} - 15\right )}{4 \, {\left (x^{3} + x^{2} {\left (\log \left (5\right ) - 4\right )} - {\left (\log \left (5\right ) \log \left (3\right ) + 5 \, \log \left (5\right )\right )} x + 4 \, \log \left (5\right ) \log \left (3\right ) - {\left (x^{2} - x {\left (\log \left (3\right ) + 5\right )} + 4 \, \log \left (3\right )\right )} \log \left (x^{2} - 15\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+(-x^6+8*x^5-x^4-120*x^3+240*x^2)*log
(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4*x^4-32*x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400
*x)*log(3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64*x^5+8*x^4+960*x^3-1920*x^2)*log(3
)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3)*log(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x, algorithm="maxi
ma")

[Out]

1/4*(x^3 - (log(5)*log(3) + log(5))*x - 4*x^2 + 4*log(5)*log(3) + (x*(log(3) + 1) - 4*log(3))*log(x^2 - 15))/(
x^3 + x^2*(log(5) - 4) - (log(5)*log(3) + 5*log(5))*x + 4*log(5)*log(3) - (x^2 - x*(log(3) + 5) + 4*log(3))*lo
g(x^2 - 15))

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Fricas [A]
time = 0.39, size = 63, normalized size = 1.50 \begin {gather*} \frac {x^{3} - 4 \, x^{2} + {\left ({\left (x - 4\right )} \log \left (3\right ) + x\right )} \log \left (\frac {1}{5} \, x^{2} - 3\right )}{4 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - {\left (x - 4\right )} \log \left (3\right ) - 5 \, x\right )} \log \left (\frac {1}{5} \, x^{2} - 3\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+(-x^6+8*x^5-x^4-120*x^3+240*x^2)*log
(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4*x^4-32*x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400
*x)*log(3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64*x^5+8*x^4+960*x^3-1920*x^2)*log(3
)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3)*log(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x, algorithm="fric
as")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (31) = 62\).
time = 0.77, size = 160, normalized size = 3.81 \begin {gather*} \frac {x \left (- \log {\left (3 \right )} - 1\right ) + 4 \log {\left (3 \right )}}{4 x^{2} + x \left (-20 - 4 \log {\left (3 \right )}\right ) + 16 \log {\left (3 \right )}} + \frac {- x^{5} + 8 x^{4} - 16 x^{3}}{- 4 x^{5} + 4 x^{4} \log {\left (3 \right )} + 36 x^{4} - 80 x^{3} - 32 x^{3} \log {\left (3 \right )} + 64 x^{2} \log {\left (3 \right )} + \left (4 x^{4} - 40 x^{3} - 8 x^{3} \log {\left (3 \right )} + 4 x^{2} \log {\left (3 \right )}^{2} + 72 x^{2} \log {\left (3 \right )} + 100 x^{2} - 160 x \log {\left (3 \right )} - 32 x \log {\left (3 \right )}^{2} + 64 \log {\left (3 \right )}^{2}\right ) \log {\left (\frac {x^{2}}{5} - 3 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4-8*x**3+x**2+120*x-240)*ln(3)+x**4-15*x**2)*ln(1/5*x**2-3)**2+(-x**6+8*x**5-x**4-120*x**3+240
*x**2)*ln(1/5*x**2-3)+2*x**6-16*x**5+32*x**4)/(((4*x**4-32*x**3+4*x**2+480*x-960)*ln(3)**2+(-8*x**5+72*x**4-40
*x**3-1080*x**2+2400*x)*ln(3)+4*x**6-40*x**5+40*x**4+600*x**3-1500*x**2)*ln(1/5*x**2-3)**2+((8*x**6-64*x**5+8*
x**4+960*x**3-1920*x**2)*ln(3)-8*x**7+72*x**6-40*x**5-1080*x**4+2400*x**3)*ln(1/5*x**2-3)+4*x**8-32*x**7+4*x**
6+480*x**5-960*x**4),x)

[Out]

(x*(-log(3) - 1) + 4*log(3))/(4*x**2 + x*(-20 - 4*log(3)) + 16*log(3)) + (-x**5 + 8*x**4 - 16*x**3)/(-4*x**5 +
4*x**4*log(3) + 36*x**4 - 80*x**3 - 32*x**3*log(3) + 64*x**2*log(3) + (4*x**4 - 40*x**3 - 8*x**3*log(3) + 4*x
**2*log(3)**2 + 72*x**2*log(3) + 100*x**2 - 160*x*log(3) - 32*x*log(3)**2 + 64*log(3)**2)*log(x**2/5 - 3))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (39) = 78\).
time = 0.92, size = 266, normalized size = 6.33 \begin {gather*} \frac {x^{5} - 8 \, x^{4} + 16 \, x^{3}}{4 \, {\left (x^{5} + x^{4} \log \left (5\right ) - x^{4} \log \left (3\right ) - 2 \, x^{3} \log \left (5\right ) \log \left (3\right ) + x^{2} \log \left (5\right ) \log \left (3\right )^{2} - x^{4} \log \left (x^{2} - 15\right ) + 2 \, x^{3} \log \left (3\right ) \log \left (x^{2} - 15\right ) - x^{2} \log \left (3\right )^{2} \log \left (x^{2} - 15\right ) - 9 \, x^{4} - 10 \, x^{3} \log \left (5\right ) + 8 \, x^{3} \log \left (3\right ) + 18 \, x^{2} \log \left (5\right ) \log \left (3\right ) - 8 \, x \log \left (5\right ) \log \left (3\right )^{2} + 10 \, x^{3} \log \left (x^{2} - 15\right ) - 18 \, x^{2} \log \left (3\right ) \log \left (x^{2} - 15\right ) + 8 \, x \log \left (3\right )^{2} \log \left (x^{2} - 15\right ) + 20 \, x^{3} + 25 \, x^{2} \log \left (5\right ) - 16 \, x^{2} \log \left (3\right ) - 40 \, x \log \left (5\right ) \log \left (3\right ) + 16 \, \log \left (5\right ) \log \left (3\right )^{2} - 25 \, x^{2} \log \left (x^{2} - 15\right ) + 40 \, x \log \left (3\right ) \log \left (x^{2} - 15\right ) - 16 \, \log \left (3\right )^{2} \log \left (x^{2} - 15\right )\right )}} - \frac {x \log \left (3\right ) + x - 4 \, \log \left (3\right )}{4 \, {\left (x^{2} - x \log \left (3\right ) - 5 \, x + 4 \, \log \left (3\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+(-x^6+8*x^5-x^4-120*x^3+240*x^2)*log
(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4*x^4-32*x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400
*x)*log(3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64*x^5+8*x^4+960*x^3-1920*x^2)*log(3
)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3)*log(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x, algorithm="giac
")

[Out]

1/4*(x^5 - 8*x^4 + 16*x^3)/(x^5 + x^4*log(5) - x^4*log(3) - 2*x^3*log(5)*log(3) + x^2*log(5)*log(3)^2 - x^4*lo
g(x^2 - 15) + 2*x^3*log(3)*log(x^2 - 15) - x^2*log(3)^2*log(x^2 - 15) - 9*x^4 - 10*x^3*log(5) + 8*x^3*log(3) +
18*x^2*log(5)*log(3) - 8*x*log(5)*log(3)^2 + 10*x^3*log(x^2 - 15) - 18*x^2*log(3)*log(x^2 - 15) + 8*x*log(3)^
2*log(x^2 - 15) + 20*x^3 + 25*x^2*log(5) - 16*x^2*log(3) - 40*x*log(5)*log(3) + 16*log(5)*log(3)^2 - 25*x^2*lo
g(x^2 - 15) + 40*x*log(3)*log(x^2 - 15) - 16*log(3)^2*log(x^2 - 15)) - 1/4*(x*log(3) + x - 4*log(3))/(x^2 - x*
log(3) - 5*x + 4*log(3))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\ln \left (\frac {x^2}{5}-3\right )}^2\,\left (x^4-15\,x^2+\ln \left (3\right )\,\left (x^4-8\,x^3+x^2+120\,x-240\right )\right )-\ln \left (\frac {x^2}{5}-3\right )\,\left (x^6-8\,x^5+x^4+120\,x^3-240\,x^2\right )+32\,x^4-16\,x^5+2\,x^6}{\ln \left (\frac {x^2}{5}-3\right )\,\left (\ln \left (3\right )\,\left (8\,x^6-64\,x^5+8\,x^4+960\,x^3-1920\,x^2\right )+2400\,x^3-1080\,x^4-40\,x^5+72\,x^6-8\,x^7\right )+{\ln \left (\frac {x^2}{5}-3\right )}^2\,\left ({\ln \left (3\right )}^2\,\left (4\,x^4-32\,x^3+4\,x^2+480\,x-960\right )-1500\,x^2+600\,x^3+40\,x^4-40\,x^5+4\,x^6-\ln \left (3\right )\,\left (8\,x^5-72\,x^4+40\,x^3+1080\,x^2-2400\,x\right )\right )-960\,x^4+480\,x^5+4\,x^6-32\,x^7+4\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2/5 - 3)^2*(x^4 - 15*x^2 + log(3)*(120*x + x^2 - 8*x^3 + x^4 - 240)) - log(x^2/5 - 3)*(120*x^3 - 24
0*x^2 + x^4 - 8*x^5 + x^6) + 32*x^4 - 16*x^5 + 2*x^6)/(log(x^2/5 - 3)*(log(3)*(960*x^3 - 1920*x^2 + 8*x^4 - 64
*x^5 + 8*x^6) + 2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7) + log(x^2/5 - 3)^2*(log(3)^2*(480*x + 4*x^2 - 3
2*x^3 + 4*x^4 - 960) - 1500*x^2 + 600*x^3 + 40*x^4 - 40*x^5 + 4*x^6 - log(3)*(1080*x^2 - 2400*x + 40*x^3 - 72*
x^4 + 8*x^5)) - 960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x^8),x)

[Out]

int((log(x^2/5 - 3)^2*(x^4 - 15*x^2 + log(3)*(120*x + x^2 - 8*x^3 + x^4 - 240)) - log(x^2/5 - 3)*(120*x^3 - 24
0*x^2 + x^4 - 8*x^5 + x^6) + 32*x^4 - 16*x^5 + 2*x^6)/(log(x^2/5 - 3)*(log(3)*(960*x^3 - 1920*x^2 + 8*x^4 - 64
*x^5 + 8*x^6) + 2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7) + log(x^2/5 - 3)^2*(log(3)^2*(480*x + 4*x^2 - 3
2*x^3 + 4*x^4 - 960) - 1500*x^2 + 600*x^3 + 40*x^4 - 40*x^5 + 4*x^6 - log(3)*(1080*x^2 - 2400*x + 40*x^3 - 72*
x^4 + 8*x^5)) - 960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x^8), x)

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