3.37.76 \(\int \frac {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+(-50+50 x-300 x \log ^2(x)) \log (x^2)+50 x \log ^2(x) \log ^2(x^2)}{(-9 x+9 x^2) \log ^2(x)+(6 x-6 x^2) \log ^2(x) \log (x^2)+(-x+x^2) \log ^2(x) \log ^2(x^2)} \, dx\)

Optimal. Leaf size=32 \[ 10 \left (-3+5 \left (\log \left (\frac {1-x}{3}\right )+\frac {1}{\log (x) \left (3-\log \left (x^2\right )\right )}\right )\right ) \]

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Rubi [F]  time = 0.20, 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 {150-150 x+(-100+100 x) \log (x)+450 x \log ^2(x)+\left (-50+50 x-300 x \log ^2(x)\right ) \log \left (x^2\right )+50 x \log ^2(x) \log ^2\left (x^2\right )}{\left (-9 x+9 x^2\right ) \log ^2(x)+\left (6 x-6 x^2\right ) \log ^2(x) \log \left (x^2\right )+\left (-x+x^2\right ) \log ^2(x) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(150 - 150*x + (-100 + 100*x)*Log[x] + 450*x*Log[x]^2 + (-50 + 50*x - 300*x*Log[x]^2)*Log[x^2] + 50*x*Log[
x]^2*Log[x^2]^2)/((-9*x + 9*x^2)*Log[x]^2 + (6*x - 6*x^2)*Log[x]^2*Log[x^2] + (-x + x^2)*Log[x]^2*Log[x^2]^2),
x]

[Out]

50*Log[1 - x] + 100*Defer[Int][1/(x*Log[x]*(-3 + Log[x^2])^2), x] + 50*Defer[Int][1/(x*Log[x]^2*(-3 + Log[x^2]
)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 50 \left (\frac {1}{-1+x}+\frac {2}{x \log (x) \left (-3+\log \left (x^2\right )\right )^2}+\frac {1}{x \log ^2(x) \left (-3+\log \left (x^2\right )\right )}\right ) \, dx\\ &=50 \int \left (\frac {1}{-1+x}+\frac {2}{x \log (x) \left (-3+\log \left (x^2\right )\right )^2}+\frac {1}{x \log ^2(x) \left (-3+\log \left (x^2\right )\right )}\right ) \, dx\\ &=50 \log (1-x)+50 \int \frac {1}{x \log ^2(x) \left (-3+\log \left (x^2\right )\right )} \, dx+100 \int \frac {1}{x \log (x) \left (-3+\log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 22, normalized size = 0.69 \begin {gather*} 50 \left (\log (-1+x)+\frac {1}{3 \log (x)-\log (x) \log \left (x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150 - 150*x + (-100 + 100*x)*Log[x] + 450*x*Log[x]^2 + (-50 + 50*x - 300*x*Log[x]^2)*Log[x^2] + 50*
x*Log[x]^2*Log[x^2]^2)/((-9*x + 9*x^2)*Log[x]^2 + (6*x - 6*x^2)*Log[x]^2*Log[x^2] + (-x + x^2)*Log[x]^2*Log[x^
2]^2),x]

[Out]

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

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fricas [A]  time = 0.58, size = 35, normalized size = 1.09 \begin {gather*} \frac {50 \, {\left (2 \, \log \left (x - 1\right ) \log \relax (x)^{2} - 3 \, \log \left (x - 1\right ) \log \relax (x) - 1\right )}}{2 \, \log \relax (x)^{2} - 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*log(x)^2*log(x^2)^2+(-300*x*log(x)^2+50*x-50)*log(x^2)+450*x*log(x)^2+(100*x-100)*log(x)-150*x
+150)/((x^2-x)*log(x)^2*log(x^2)^2+(-6*x^2+6*x)*log(x)^2*log(x^2)+(9*x^2-9*x)*log(x)^2),x, algorithm="fricas")

[Out]

50*(2*log(x - 1)*log(x)^2 - 3*log(x - 1)*log(x) - 1)/(2*log(x)^2 - 3*log(x))

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giac [A]  time = 0.21, size = 23, normalized size = 0.72 \begin {gather*} -\frac {100}{3 \, {\left (2 \, \log \relax (x) - 3\right )}} + \frac {50}{3 \, \log \relax (x)} + 50 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*log(x)^2*log(x^2)^2+(-300*x*log(x)^2+50*x-50)*log(x^2)+450*x*log(x)^2+(100*x-100)*log(x)-150*x
+150)/((x^2-x)*log(x)^2*log(x^2)^2+(-6*x^2+6*x)*log(x)^2*log(x^2)+(9*x^2-9*x)*log(x)^2),x, algorithm="giac")

[Out]

-100/3/(2*log(x) - 3) + 50/3/log(x) + 50*log(x - 1)

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maple [C]  time = 0.11, size = 69, normalized size = 2.16

method	result	size
risch	\(50 \ln \left (x -1\right )-\frac {100 i}{\left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )-6 i\right ) \ln \relax (x )}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x*ln(x)^2*ln(x^2)^2+(-300*x*ln(x)^2+50*x-50)*ln(x^2)+450*x*ln(x)^2+(100*x-100)*ln(x)-150*x+150)/((x^2-
x)*ln(x)^2*ln(x^2)^2+(-6*x^2+6*x)*ln(x)^2*ln(x^2)+(9*x^2-9*x)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

50*ln(x-1)-100*I/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*ln(x)-6*I)/ln(x
)

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maxima [A]  time = 0.44, size = 22, normalized size = 0.69 \begin {gather*} -\frac {50}{2 \, \log \relax (x)^{2} - 3 \, \log \relax (x)} + 50 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*log(x)^2*log(x^2)^2+(-300*x*log(x)^2+50*x-50)*log(x^2)+450*x*log(x)^2+(100*x-100)*log(x)-150*x
+150)/((x^2-x)*log(x)^2*log(x^2)^2+(-6*x^2+6*x)*log(x)^2*log(x^2)+(9*x^2-9*x)*log(x)^2),x, algorithm="maxima")

[Out]

-50/(2*log(x)^2 - 3*log(x)) + 50*log(x - 1)

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mupad [B]  time = 2.30, size = 21, normalized size = 0.66 \begin {gather*} 50\,\ln \left (x-1\right )-\frac {50}{\ln \relax (x)\,\left (\ln \left (x^2\right )-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(450*x*log(x)^2 - 150*x + log(x)*(100*x - 100) - log(x^2)*(300*x*log(x)^2 - 50*x + 50) + 50*x*log(x^2)^2*
log(x)^2 + 150)/(log(x)^2*(9*x - 9*x^2) - log(x^2)*log(x)^2*(6*x - 6*x^2) + log(x^2)^2*log(x)^2*(x - x^2)),x)

[Out]

50*log(x - 1) - 50/(log(x)*(log(x^2) - 3))

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sympy [A]  time = 0.27, size = 19, normalized size = 0.59 \begin {gather*} 50 \log {\left (x - 1 \right )} - \frac {50}{2 \log {\relax (x )}^{2} - 3 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*ln(x)**2*ln(x**2)**2+(-300*x*ln(x)**2+50*x-50)*ln(x**2)+450*x*ln(x)**2+(100*x-100)*ln(x)-150*x
+150)/((x**2-x)*ln(x)**2*ln(x**2)**2+(-6*x**2+6*x)*ln(x)**2*ln(x**2)+(9*x**2-9*x)*ln(x)**2),x)

[Out]

50*log(x - 1) - 50/(2*log(x)**2 - 3*log(x))

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