3.53.67 \(\int \frac {-36+320 x \log (25)-575 x^2 \log ^2(25)+(8-80 x \log (25)+150 x^2 \log ^2(25)) \log (x)+(-28+280 x \log (25)-525 x^2 \log ^2(25)+(8-80 x \log (25)+150 x^2 \log ^2(25)) \log (x)) \log (\frac {1}{2} (-7+2 \log (x)))}{-7+2 \log (x)+(-14+4 \log (x)) \log (\frac {1}{2} (-7+2 \log (x)))+(-7+2 \log (x)) \log ^2(\frac {1}{2} (-7+2 \log (x)))} \, dx\)

Optimal. Leaf size=22 \[ \frac {x (-2+5 x \log (25))^2}{1+\log \left (-\frac {7}{2}+\log (x)\right )} \]

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Rubi [F]  time = 1.13, 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 {-36+320 x \log (25)-575 x^2 \log ^2(25)+\left (8-80 x \log (25)+150 x^2 \log ^2(25)\right ) \log (x)+\left (-28+280 x \log (25)-525 x^2 \log ^2(25)+\left (8-80 x \log (25)+150 x^2 \log ^2(25)\right ) \log (x)\right ) \log \left (\frac {1}{2} (-7+2 \log (x))\right )}{-7+2 \log (x)+(-14+4 \log (x)) \log \left (\frac {1}{2} (-7+2 \log (x))\right )+(-7+2 \log (x)) \log ^2\left (\frac {1}{2} (-7+2 \log (x))\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-36 + 320*x*Log[25] - 575*x^2*Log[25]^2 + (8 - 80*x*Log[25] + 150*x^2*Log[25]^2)*Log[x] + (-28 + 280*x*Lo
g[25] - 525*x^2*Log[25]^2 + (8 - 80*x*Log[25] + 150*x^2*Log[25]^2)*Log[x])*Log[(-7 + 2*Log[x])/2])/(-7 + 2*Log
[x] + (-14 + 4*Log[x])*Log[(-7 + 2*Log[x])/2] + (-7 + 2*Log[x])*Log[(-7 + 2*Log[x])/2]^2),x]

[Out]

-8*Defer[Int][1/((-7 + 2*Log[x])*(1 + Log[-7/2 + Log[x]])^2), x] + 40*Log[25]*Defer[Int][x/((-7 + 2*Log[x])*(1
+ Log[-7/2 + Log[x]])^2), x] - 50*Log[25]^2*Defer[Int][x^2/((-7 + 2*Log[x])*(1 + Log[-7/2 + Log[x]])^2), x] +
4*Defer[Int][(1 + Log[-7/2 + Log[x]])^(-1), x] - 40*Log[25]*Defer[Int][x/(1 + Log[-7/2 + Log[x]]), x] + 75*Lo
g[25]^2*Defer[Int][x^2/(1 + Log[-7/2 + Log[x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(2-5 x \log (25)) \left (18-115 x \log (25)-7 (-2+15 x \log (25)) \log \left (-\frac {7}{2}+\log (x)\right )+2 (-2+15 x \log (25)) \log (x) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )\right )}{(7-2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2} \, dx\\ &=\int \left (-\frac {2 (-2+5 x \log (25))^2}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2}+\frac {4-40 x \log (25)+75 x^2 \log ^2(25)}{1+\log \left (-\frac {7}{2}+\log (x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {(-2+5 x \log (25))^2}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2} \, dx\right )+\int \frac {4-40 x \log (25)+75 x^2 \log ^2(25)}{1+\log \left (-\frac {7}{2}+\log (x)\right )} \, dx\\ &=-\left (2 \int \left (\frac {4}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2}-\frac {20 x \log (25)}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2}+\frac {25 x^2 \log ^2(25)}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2}\right ) \, dx\right )+\int \left (\frac {4}{1+\log \left (-\frac {7}{2}+\log (x)\right )}-\frac {40 x \log (25)}{1+\log \left (-\frac {7}{2}+\log (x)\right )}+\frac {75 x^2 \log ^2(25)}{1+\log \left (-\frac {7}{2}+\log (x)\right )}\right ) \, dx\\ &=4 \int \frac {1}{1+\log \left (-\frac {7}{2}+\log (x)\right )} \, dx-8 \int \frac {1}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2} \, dx+(40 \log (25)) \int \frac {x}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2} \, dx-(40 \log (25)) \int \frac {x}{1+\log \left (-\frac {7}{2}+\log (x)\right )} \, dx-\left (50 \log ^2(25)\right ) \int \frac {x^2}{(-7+2 \log (x)) \left (1+\log \left (-\frac {7}{2}+\log (x)\right )\right )^2} \, dx+\left (75 \log ^2(25)\right ) \int \frac {x^2}{1+\log \left (-\frac {7}{2}+\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 22, normalized size = 1.00 \begin {gather*} \frac {x (2-5 x \log (25))^2}{1+\log \left (-\frac {7}{2}+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 + 320*x*Log[25] - 575*x^2*Log[25]^2 + (8 - 80*x*Log[25] + 150*x^2*Log[25]^2)*Log[x] + (-28 + 28
0*x*Log[25] - 525*x^2*Log[25]^2 + (8 - 80*x*Log[25] + 150*x^2*Log[25]^2)*Log[x])*Log[(-7 + 2*Log[x])/2])/(-7 +
2*Log[x] + (-14 + 4*Log[x])*Log[(-7 + 2*Log[x])/2] + (-7 + 2*Log[x])*Log[(-7 + 2*Log[x])/2]^2),x]

[Out]

(x*(2 - 5*x*Log[25])^2)/(1 + Log[-7/2 + Log[x]])

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fricas [A]  time = 0.45, size = 29, normalized size = 1.32 \begin {gather*} \frac {4 \, {\left (25 \, x^{3} \log \relax (5)^{2} - 10 \, x^{2} \log \relax (5) + x\right )}}{\log \left (\log \relax (x) - \frac {7}{2}\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((600*x^2*log(5)^2-160*x*log(5)+8)*log(x)-2100*x^2*log(5)^2+560*x*log(5)-28)*log(log(x)-7/2)+(600*x
^2*log(5)^2-160*x*log(5)+8)*log(x)-2300*x^2*log(5)^2+640*x*log(5)-36)/((2*log(x)-7)*log(log(x)-7/2)^2+(4*log(x
)-14)*log(log(x)-7/2)+2*log(x)-7),x, algorithm="fricas")

[Out]

4*(25*x^3*log(5)^2 - 10*x^2*log(5) + x)/(log(log(x) - 7/2) + 1)

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giac [B]  time = 1.64, size = 454, normalized size = 20.64 \begin {gather*} -\frac {100 \, x^{3} \log \relax (5)^{2} \log \relax (2)}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} + \frac {100 \, x^{3} \log \relax (5)^{2} \log \left (2 \, \log \relax (x) - 7\right )}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} + \frac {100 \, x^{3} \log \relax (5)^{2}}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} + \frac {40 \, x^{2} \log \relax (5) \log \relax (2)}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} - \frac {40 \, x^{2} \log \relax (5) \log \left (2 \, \log \relax (x) - 7\right )}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} - \frac {40 \, x^{2} \log \relax (5)}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} - \frac {4 \, x \log \relax (2)}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} + \frac {4 \, x \log \left (2 \, \log \relax (x) - 7\right )}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} + \frac {4 \, x}{\log \relax (2)^{2} - 2 \, \log \relax (2) \log \left (2 \, \log \relax (x) - 7\right ) + \log \left (2 \, \log \relax (x) - 7\right )^{2} - 2 \, \log \relax (2) + 2 \, \log \left (2 \, \log \relax (x) - 7\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((600*x^2*log(5)^2-160*x*log(5)+8)*log(x)-2100*x^2*log(5)^2+560*x*log(5)-28)*log(log(x)-7/2)+(600*x
^2*log(5)^2-160*x*log(5)+8)*log(x)-2300*x^2*log(5)^2+640*x*log(5)-36)/((2*log(x)-7)*log(log(x)-7/2)^2+(4*log(x
)-14)*log(log(x)-7/2)+2*log(x)-7),x, algorithm="giac")

[Out]

-100*x^3*log(5)^2*log(2)/(log(2)^2 - 2*log(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log
(x) - 7) + 1) + 100*x^3*log(5)^2*log(2*log(x) - 7)/(log(2)^2 - 2*log(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^
2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) + 100*x^3*log(5)^2/(log(2)^2 - 2*log(2)*log(2*log(x) - 7) + log(2*log(
x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) + 40*x^2*log(5)*log(2)/(log(2)^2 - 2*log(2)*log(2*log(x) - 7)
+ log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) - 40*x^2*log(5)*log(2*log(x) - 7)/(log(2)^2 - 2*lo
g(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) - 40*x^2*log(5)/(log(2)^2 -
2*log(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) - 4*x*log(2)/(log(2)^2
- 2*log(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) + 4*x*log(2*log(x) -
7)/(log(2)^2 - 2*log(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1) + 4*x/(
log(2)^2 - 2*log(2)*log(2*log(x) - 7) + log(2*log(x) - 7)^2 - 2*log(2) + 2*log(2*log(x) - 7) + 1)

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maple [A]  time = 0.39, size = 29, normalized size = 1.32

method	result	size
risch	\(\frac {4 x \left (25 x^{2} \ln \relax (5)^{2}-10 x \ln \relax (5)+1\right )}{1+\ln \left (\ln \relax (x )-\frac {7}{2}\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((600*x^2*ln(5)^2-160*x*ln(5)+8)*ln(x)-2100*x^2*ln(5)^2+560*x*ln(5)-28)*ln(ln(x)-7/2)+(600*x^2*ln(5)^2-16
0*x*ln(5)+8)*ln(x)-2300*x^2*ln(5)^2+640*x*ln(5)-36)/((2*ln(x)-7)*ln(ln(x)-7/2)^2+(4*ln(x)-14)*ln(ln(x)-7/2)+2*
ln(x)-7),x,method=_RETURNVERBOSE)

[Out]

4*x*(25*x^2*ln(5)^2-10*x*ln(5)+1)/(1+ln(ln(x)-7/2))

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maxima [A]  time = 0.50, size = 35, normalized size = 1.59 \begin {gather*} -\frac {4 \, {\left (25 \, x^{3} \log \relax (5)^{2} - 10 \, x^{2} \log \relax (5) + x\right )}}{\log \relax (2) - \log \left (2 \, \log \relax (x) - 7\right ) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((600*x^2*log(5)^2-160*x*log(5)+8)*log(x)-2100*x^2*log(5)^2+560*x*log(5)-28)*log(log(x)-7/2)+(600*x
^2*log(5)^2-160*x*log(5)+8)*log(x)-2300*x^2*log(5)^2+640*x*log(5)-36)/((2*log(x)-7)*log(log(x)-7/2)^2+(4*log(x
)-14)*log(log(x)-7/2)+2*log(x)-7),x, algorithm="maxima")

[Out]

-4*(25*x^3*log(5)^2 - 10*x^2*log(5) + x)/(log(2) - log(2*log(x) - 7) - 1)

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mupad [B]  time = 3.99, size = 112, normalized size = 5.09 \begin {gather*} \ln \relax (x)\,\left (300\,{\ln \relax (5)}^2\,x^3-80\,\ln \relax (5)\,x^2+4\,x\right )-1050\,x^3\,{\ln \relax (5)}^2-14\,x+\frac {2\,x\,\left (5\,x\,\ln \relax (5)-1\right )\,\left (2\,\ln \relax (x)+115\,x\,\ln \relax (5)-30\,x\,\ln \relax (5)\,\ln \relax (x)-9\right )-2\,x\,\ln \left (\ln \relax (x)-\frac {7}{2}\right )\,\left (2\,\ln \relax (x)-7\right )\,\left (75\,{\ln \relax (5)}^2\,x^2-20\,\ln \relax (5)\,x+1\right )}{\ln \left (\ln \relax (x)-\frac {7}{2}\right )+1}+280\,x^2\,\ln \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2300*x^2*log(5)^2 - 640*x*log(5) + log(log(x) - 7/2)*(2100*x^2*log(5)^2 - 560*x*log(5) - log(x)*(600*x^2
*log(5)^2 - 160*x*log(5) + 8) + 28) - log(x)*(600*x^2*log(5)^2 - 160*x*log(5) + 8) + 36)/(2*log(x) + log(log(x
) - 7/2)^2*(2*log(x) - 7) + log(log(x) - 7/2)*(4*log(x) - 14) - 7),x)

[Out]

log(x)*(4*x + 300*x^3*log(5)^2 - 80*x^2*log(5)) - 1050*x^3*log(5)^2 - 14*x + (2*x*(5*x*log(5) - 1)*(2*log(x) +
115*x*log(5) - 30*x*log(5)*log(x) - 9) - 2*x*log(log(x) - 7/2)*(2*log(x) - 7)*(75*x^2*log(5)^2 - 20*x*log(5)
+ 1))/(log(log(x) - 7/2) + 1) + 280*x^2*log(5)

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sympy [A]  time = 0.33, size = 31, normalized size = 1.41 \begin {gather*} \frac {100 x^{3} \log {\relax (5 )}^{2} - 40 x^{2} \log {\relax (5 )} + 4 x}{\log {\left (\log {\relax (x )} - \frac {7}{2} \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((600*x**2*ln(5)**2-160*x*ln(5)+8)*ln(x)-2100*x**2*ln(5)**2+560*x*ln(5)-28)*ln(ln(x)-7/2)+(600*x**2
*ln(5)**2-160*x*ln(5)+8)*ln(x)-2300*x**2*ln(5)**2+640*x*ln(5)-36)/((2*ln(x)-7)*ln(ln(x)-7/2)**2+(4*ln(x)-14)*l
n(ln(x)-7/2)+2*ln(x)-7),x)

[Out]

(100*x**3*log(5)**2 - 40*x**2*log(5) + 4*x)/(log(log(x) - 7/2) + 1)

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