3.95.70 \(\int \frac {-4 \log ^2(5) \log (x^2)+2 \log ^2(5) \log ^2(x^2)+(-34+3 x) \log ^4(x^2)}{x^3 \log ^4(5)+(-34 x^3+2 x^4) \log ^2(5) \log ^2(x^2)+(289 x^3-34 x^4+x^5) \log ^4(x^2)} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{x^2 \left (17-x-\frac {\log ^2(5)}{\log ^2\left (x^2\right )}\right )} \]

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Rubi [F]  time = 19.33, 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 {-4 \log ^2(5) \log \left (x^2\right )+2 \log ^2(5) \log ^2\left (x^2\right )+(-34+3 x) \log ^4\left (x^2\right )}{x^3 \log ^4(5)+\left (-34 x^3+2 x^4\right ) \log ^2(5) \log ^2\left (x^2\right )+\left (289 x^3-34 x^4+x^5\right ) \log ^4\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*Log[5]^2*Log[x^2] + 2*Log[5]^2*Log[x^2]^2 + (-34 + 3*x)*Log[x^2]^4)/(x^3*Log[5]^4 + (-34*x^3 + 2*x^4)*
Log[5]^2*Log[x^2]^2 + (289*x^3 - 34*x^4 + x^5)*Log[x^2]^4),x]

[Out]

1/((17 - x)*x^2) - (2*Log[5]^4*Defer[Int][1/((-17 + x)*(Log[5]^2 + (-17 + x)*Log[x^2]^2)^2), x])/4913 + (Log[5
]^4*Defer[Int][1/(x^2*(Log[5]^2 + (-17 + x)*Log[x^2]^2)^2), x])/289 + (2*Log[5]^4*Defer[Int][1/(x*(Log[5]^2 +
(-17 + x)*Log[x^2]^2)^2), x])/4913 - 4*Log[5]^2*Defer[Int][Log[x^2]/(x^3*(Log[5]^2 + (-17 + x)*Log[x^2]^2)^2),
x] + (2*Log[5]^2*Defer[Int][1/((-17 + x)*(Log[5]^2 + (-17 + x)*Log[x^2]^2)), x])/4913 + (2*Log[5]^2*Defer[Int
][1/(x^3*(Log[5]^2 + (-17 + x)*Log[x^2]^2)), x])/17 - (2*Log[5]^2*Defer[Int][1/(x*(Log[5]^2 + (-17 + x)*Log[x^
2]^2)), x])/4913 + (Log[5]^4*Defer[Int][1/((-17 + x)^2*(Log[5]^2 - 17*Log[x^2]^2 + x*Log[x^2]^2)^2), x])/289 -
(2*Log[5]^2*Defer[Int][1/((-17 + x)^2*(Log[5]^2 - 17*Log[x^2]^2 + x*Log[x^2]^2)), x])/289

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log \left (x^2\right ) \left (-4 \log ^2(5)+2 \log ^2(5) \log \left (x^2\right )+(-34+3 x) \log ^3\left (x^2\right )\right )}{x^3 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx\\ &=\int \left (\frac {-34+3 x}{(-17+x)^2 x^3}-\frac {\log ^2(5) \left (-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )\right )}{(-17+x)^2 x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2}-\frac {2 (-17+2 x) \log ^2(5)}{(-17+x)^2 x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )}\right ) \, dx\\ &=-\left (\log ^2(5) \int \frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{(-17+x)^2 x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2} \, dx\right )-\left (2 \log ^2(5)\right ) \int \frac {-17+2 x}{(-17+x)^2 x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx+\int \frac {-34+3 x}{(-17+x)^2 x^3} \, dx\\ &=\frac {1}{(17-x) x^2}-\log ^2(5) \int \frac {-x \log ^2(5)+4 (-17+x)^2 \log \left (x^2\right )}{(17-x)^2 x^3 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx-\left (2 \log ^2(5)\right ) \int \left (\frac {1}{289 (-17+x)^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )}-\frac {1}{4913 (-17+x) \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )}-\frac {1}{17 x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )}+\frac {1}{4913 x \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )}\right ) \, dx\\ &=\frac {1}{(17-x) x^2}+\frac {\left (2 \log ^2(5)\right ) \int \frac {1}{(-17+x) \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx}{4913}-\frac {\left (2 \log ^2(5)\right ) \int \frac {1}{x \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx}{4913}-\frac {1}{289} \left (2 \log ^2(5)\right ) \int \frac {1}{(-17+x)^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx+\frac {1}{17} \left (2 \log ^2(5)\right ) \int \frac {1}{x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx-\log ^2(5) \int \left (\frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{4913 (-17+x)^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2}-\frac {3 \left (-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )\right )}{83521 (-17+x) \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2}+\frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{289 x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2}+\frac {2 \left (-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )\right )}{4913 x^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2}+\frac {3 \left (-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )\right )}{83521 x \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2}\right ) \, dx\\ &=\frac {1}{(17-x) x^2}+\frac {\left (3 \log ^2(5)\right ) \int \frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{(-17+x) \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2} \, dx}{83521}-\frac {\left (3 \log ^2(5)\right ) \int \frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{x \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2} \, dx}{83521}-\frac {\log ^2(5) \int \frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{(-17+x)^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2} \, dx}{4913}+\frac {\left (2 \log ^2(5)\right ) \int \frac {1}{(-17+x) \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \, dx}{4913}-\frac {\left (2 \log ^2(5)\right ) \int \frac {1}{x \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \, dx}{4913}-\frac {\left (2 \log ^2(5)\right ) \int \frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{x^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2} \, dx}{4913}-\frac {1}{289} \log ^2(5) \int \frac {-x \log ^2(5)+1156 \log \left (x^2\right )-136 x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )}{x^3 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )^2} \, dx-\frac {1}{289} \left (2 \log ^2(5)\right ) \int \frac {1}{(-17+x)^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx+\frac {1}{17} \left (2 \log ^2(5)\right ) \int \frac {1}{x^3 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \, dx\\ &=\frac {1}{(17-x) x^2}+\frac {\left (3 \log ^2(5)\right ) \int \frac {x \log ^2(5)-4 (-17+x)^2 \log \left (x^2\right )}{(17-x) \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx}{83521}-\frac {\left (3 \log ^2(5)\right ) \int \frac {-x \log ^2(5)+4 (-17+x)^2 \log \left (x^2\right )}{x \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx}{83521}-\frac {\log ^2(5) \int \frac {-x \log ^2(5)+4 (-17+x)^2 \log \left (x^2\right )}{(17-x)^2 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx}{4913}-\frac {\left (2 \log ^2(5)\right ) \int \frac {-x \log ^2(5)+4 (-17+x)^2 \log \left (x^2\right )}{x^2 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx}{4913}+\frac {\left (2 \log ^2(5)\right ) \int \frac {1}{(-17+x) \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \, dx}{4913}-\frac {\left (2 \log ^2(5)\right ) \int \frac {1}{x \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \, dx}{4913}-\frac {1}{289} \log ^2(5) \int \frac {-x \log ^2(5)+4 (-17+x)^2 \log \left (x^2\right )}{x^3 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )^2} \, dx-\frac {1}{289} \left (2 \log ^2(5)\right ) \int \frac {1}{(-17+x)^2 \left (\log ^2(5)-17 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )\right )} \, dx+\frac {1}{17} \left (2 \log ^2(5)\right ) \int \frac {1}{x^3 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.70, size = 28, normalized size = 1.22 \begin {gather*} -\frac {\log ^2\left (x^2\right )}{x^2 \left (\log ^2(5)+(-17+x) \log ^2\left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*Log[5]^2*Log[x^2] + 2*Log[5]^2*Log[x^2]^2 + (-34 + 3*x)*Log[x^2]^4)/(x^3*Log[5]^4 + (-34*x^3 + 2
*x^4)*Log[5]^2*Log[x^2]^2 + (289*x^3 - 34*x^4 + x^5)*Log[x^2]^4),x]

[Out]

-(Log[x^2]^2/(x^2*(Log[5]^2 + (-17 + x)*Log[x^2]^2)))

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fricas [A]  time = 0.54, size = 35, normalized size = 1.52 \begin {gather*} -\frac {\log \left (x^{2}\right )^{2}}{x^{2} \log \relax (5)^{2} + {\left (x^{3} - 17 \, x^{2}\right )} \log \left (x^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/((x^5-34*x^4+289*x^3)*log(x^2)^4+(2*
x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*log(5)^4),x, algorithm="fricas")

[Out]

-log(x^2)^2/(x^2*log(5)^2 + (x^3 - 17*x^2)*log(x^2)^2)

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giac [B]  time = 0.49, size = 73, normalized size = 3.17 \begin {gather*} \frac {\log \relax (5)^{2}}{x^{4} \log \left (x^{2}\right )^{2} + x^{3} \log \relax (5)^{2} - 34 \, x^{3} \log \left (x^{2}\right )^{2} - 17 \, x^{2} \log \relax (5)^{2} + 289 \, x^{2} \log \left (x^{2}\right )^{2}} - \frac {1}{289 \, {\left (x - 17\right )}} + \frac {x + 17}{289 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/((x^5-34*x^4+289*x^3)*log(x^2)^4+(2*
x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*log(5)^4),x, algorithm="giac")

[Out]

log(5)^2/(x^4*log(x^2)^2 + x^3*log(5)^2 - 34*x^3*log(x^2)^2 - 17*x^2*log(5)^2 + 289*x^2*log(x^2)^2) - 1/289/(x
- 17) + 1/289*(x + 17)/x^2

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maple [B]  time = 0.04, size = 48, normalized size = 2.09

method	result	size
risch	\(-\frac {1}{x^{2} \left (x -17\right )}+\frac {\ln \relax (5)^{2}}{x^{2} \left (x -17\right ) \left (x \ln \left (x^{2}\right )^{2}+\ln \relax (5)^{2}-17 \ln \left (x^{2}\right )^{2}\right )}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x-34)*ln(x^2)^4+2*ln(5)^2*ln(x^2)^2-4*ln(5)^2*ln(x^2))/((x^5-34*x^4+289*x^3)*ln(x^2)^4+(2*x^4-34*x^3)*
ln(5)^2*ln(x^2)^2+x^3*ln(5)^4),x,method=_RETURNVERBOSE)

[Out]

-1/x^2/(x-17)+ln(5)^2/x^2/(x-17)/(x*ln(x^2)^2+ln(5)^2-17*ln(x^2)^2)

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maxima [A]  time = 0.50, size = 32, normalized size = 1.39 \begin {gather*} -\frac {4 \, \log \relax (x)^{2}}{x^{2} \log \relax (5)^{2} + 4 \, {\left (x^{3} - 17 \, x^{2}\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-34)*log(x^2)^4+2*log(5)^2*log(x^2)^2-4*log(5)^2*log(x^2))/((x^5-34*x^4+289*x^3)*log(x^2)^4+(2*
x^4-34*x^3)*log(5)^2*log(x^2)^2+x^3*log(5)^4),x, algorithm="maxima")

[Out]

-4*log(x)^2/(x^2*log(5)^2 + 4*(x^3 - 17*x^2)*log(x)^2)

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mupad [B]  time = 7.41, size = 34, normalized size = 1.48 \begin {gather*} -\frac {{\ln \left (x^2\right )}^2}{x^2\,\left (x\,{\ln \left (x^2\right )}^2-17\,{\ln \left (x^2\right )}^2+{\ln \relax (5)}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2)^4*(3*x - 34) - 4*log(x^2)*log(5)^2 + 2*log(x^2)^2*log(5)^2)/(x^3*log(5)^4 + log(x^2)^4*(289*x^3
- 34*x^4 + x^5) - log(x^2)^2*log(5)^2*(34*x^3 - 2*x^4)),x)

[Out]

-log(x^2)^2/(x^2*(x*log(x^2)^2 - 17*log(x^2)^2 + log(5)^2))

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sympy [B]  time = 0.24, size = 53, normalized size = 2.30 \begin {gather*} \frac {\log {\relax (5 )}^{2}}{x^{3} \log {\relax (5 )}^{2} - 17 x^{2} \log {\relax (5 )}^{2} + \left (x^{4} - 34 x^{3} + 289 x^{2}\right ) \log {\left (x^{2} \right )}^{2}} - \frac {1}{x^{3} - 17 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-34)*ln(x**2)**4+2*ln(5)**2*ln(x**2)**2-4*ln(5)**2*ln(x**2))/((x**5-34*x**4+289*x**3)*ln(x**2)*
*4+(2*x**4-34*x**3)*ln(5)**2*ln(x**2)**2+x**3*ln(5)**4),x)

[Out]

log(5)**2/(x**3*log(5)**2 - 17*x**2*log(5)**2 + (x**4 - 34*x**3 + 289*x**2)*log(x**2)**2) - 1/(x**3 - 17*x**2)

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