3.90.57 \(\int \frac {(-7+5 x^2) \log (\frac {1}{5} (7-5 x^2))+(10 x^2+(7-5 x^2) \log (\frac {1}{5} (7-5 x^2))+\log (\frac {5}{x}) (10 x^2+(7-5 x^2) \log (\frac {1}{5} (7-5 x^2)))) \log (1+\log (\frac {5}{x}))+(-7 x+5 x^3+(-7 x+5 x^3) \log (\frac {5}{x})) \log ^2(1+\log (\frac {5}{x}))}{(-7 x^2+5 x^4+(-7 x^2+5 x^4) \log (\frac {5}{x})) \log ^2(1+\log (\frac {5}{x}))} \, dx\)

Optimal. Leaf size=28 \[ \log (x)+\frac {\log \left (\frac {7}{5}-x^2\right )}{x \log \left (1+\log \left (\frac {5}{x}\right )\right )} \]

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Rubi [F]  time = 3.85, 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 {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-7 + 5*x^2)*Log[(7 - 5*x^2)/5] + (10*x^2 + (7 - 5*x^2)*Log[(7 - 5*x^2)/5] + Log[5/x]*(10*x^2 + (7 - 5*x^
2)*Log[(7 - 5*x^2)/5]))*Log[1 + Log[5/x]] + (-7*x + 5*x^3 + (-7*x + 5*x^3)*Log[5/x])*Log[1 + Log[5/x]]^2)/((-7
*x^2 + 5*x^4 + (-7*x^2 + 5*x^4)*Log[5/x])*Log[1 + Log[5/x]]^2),x]

[Out]

Log[x] + Defer[Int][Log[7/5 - x^2]/(x^2*(1 + Log[5/x])*Log[1 + Log[5/x]]^2), x] - (5*Defer[Int][1/((Sqrt[7] -
Sqrt[5]*x)*Log[1 + Log[5/x]]), x])/Sqrt[7] - (5*Defer[Int][1/((Sqrt[7] + Sqrt[5]*x)*Log[1 + Log[5/x]]), x])/Sq
rt[7] - Defer[Int][Log[7/5 - x^2]/(x^2*Log[1 + Log[5/x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log \left (\frac {7}{5}-x^2\right ) \left (\frac {1}{1+\log \left (\frac {5}{x}\right )}-\log \left (1+\log \left (\frac {5}{x}\right )\right )\right )+x \log \left (1+\log \left (\frac {5}{x}\right )\right ) \left (\frac {10 x}{-7+5 x^2}+\log \left (1+\log \left (\frac {5}{x}\right )\right )\right )}{x^2 \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {10 x^2+7 \log \left (\frac {7}{5}-x^2\right )-5 x^2 \log \left (\frac {7}{5}-x^2\right )}{x^2 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx\\ &=\log (x)+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {10 x^2+7 \log \left (\frac {7}{5}-x^2\right )-5 x^2 \log \left (\frac {7}{5}-x^2\right )}{x^2 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\int \left (\frac {-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )}{7 x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {5 \left (-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )\right )}{7 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {1}{7} \int \frac {-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {5}{7} \int \frac {-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {1}{7} \int \frac {-10 x^2+\left (-7+5 x^2\right ) \log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {5}{7} \int \left (-\frac {10 x^2}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {7 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {5 x^2 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {1}{7} \int \left (-\frac {10}{\log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {5 \log \left (\frac {7}{5}-x^2\right )}{\log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {7 \log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx-\frac {25}{7} \int \frac {x^2 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\frac {50}{7} \int \frac {x^2}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {5}{7} \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {10}{7} \int \frac {1}{\log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {25}{7} \int \left (\frac {\log \left (\frac {7}{5}-x^2\right )}{5 \log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {7 \log \left (\frac {7}{5}-x^2\right )}{5 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+5 \int \left (-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\frac {50}{7} \int \left (\frac {1}{5 \log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {7}{5 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)-5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+10 \int \frac {1}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)-5 \int \left (-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+10 \int \left (-\frac {1}{2 \sqrt {7} \left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {1}{2 \sqrt {7} \left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)-\frac {5 \int \frac {1}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{\sqrt {7}}-\frac {5 \int \frac {1}{\left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{\sqrt {7}}+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 28, normalized size = 1.00 \begin {gather*} \log (x)+\frac {\log \left (\frac {7}{5}-x^2\right )}{x \log \left (1+\log \left (\frac {5}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-7 + 5*x^2)*Log[(7 - 5*x^2)/5] + (10*x^2 + (7 - 5*x^2)*Log[(7 - 5*x^2)/5] + Log[5/x]*(10*x^2 + (7
- 5*x^2)*Log[(7 - 5*x^2)/5]))*Log[1 + Log[5/x]] + (-7*x + 5*x^3 + (-7*x + 5*x^3)*Log[5/x])*Log[1 + Log[5/x]]^2
)/((-7*x^2 + 5*x^4 + (-7*x^2 + 5*x^4)*Log[5/x])*Log[1 + Log[5/x]]^2),x]

[Out]

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

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fricas [A]  time = 0.89, size = 44, normalized size = 1.57 \begin {gather*} -\frac {x \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right ) + 1\right ) - \log \left (-x^{2} + \frac {7}{5}\right )}{x \log \left (\log \left (\frac {5}{x}\right ) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(5/x)+(-5*
x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5/x)+1)+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log
(log(5/x)+1)^2,x, algorithm="fricas")

[Out]

-(x*log(5/x)*log(log(5/x) + 1) - log(-x^2 + 7/5))/(x*log(log(5/x) + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{3} + {\left (5 \, x^{3} - 7 \, x\right )} \log \left (\frac {5}{x}\right ) - 7 \, x\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )^{2} + {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right ) + {\left (10 \, x^{2} - {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right ) + {\left (10 \, x^{2} - {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right )\right )} \log \left (\frac {5}{x}\right )\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )}{{\left (5 \, x^{4} - 7 \, x^{2} + {\left (5 \, x^{4} - 7 \, x^{2}\right )} \log \left (\frac {5}{x}\right )\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(5/x)+(-5*
x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5/x)+1)+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log
(log(5/x)+1)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.27, size = 27, normalized size = 0.96

method	result	size
risch	\(\ln \relax (x )+\frac {\ln \left (-x^{2}+\frac {7}{5}\right )}{x \ln \left (\ln \relax (5)-\ln \relax (x )+1\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^3-7*x)*ln(5/x)+5*x^3-7*x)*ln(ln(5/x)+1)^2+(((-5*x^2+7)*ln(-x^2+7/5)+10*x^2)*ln(5/x)+(-5*x^2+7)*ln(-
x^2+7/5)+10*x^2)*ln(ln(5/x)+1)+(5*x^2-7)*ln(-x^2+7/5))/((5*x^4-7*x^2)*ln(5/x)+5*x^4-7*x^2)/ln(ln(5/x)+1)^2,x,m
ethod=_RETURNVERBOSE)

[Out]

ln(x)+ln(-x^2+7/5)/x/ln(ln(5)-ln(x)+1)

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maxima [A]  time = 0.50, size = 32, normalized size = 1.14 \begin {gather*} -\frac {\log \relax (5) - \log \left (-5 \, x^{2} + 7\right )}{x \log \left (\log \relax (5) - \log \relax (x) + 1\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(5/x)+(-5*
x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5/x)+1)+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log
(log(5/x)+1)^2,x, algorithm="maxima")

[Out]

-(log(5) - log(-5*x^2 + 7))/(x*log(log(5) - log(x) + 1)) + log(x)

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mupad [B]  time = 7.43, size = 139, normalized size = 4.96 \begin {gather*} \ln \relax (x)+\ln \left (\frac {5}{x}\right )\,\left (\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}-\frac {2\,x}{x^2-\frac {7}{5}}\right )+\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}-\frac {2\,x}{x^2-\frac {7}{5}}+\frac {\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}+\frac {\ln \left (\ln \left (\frac {5}{x}\right )+1\right )\,\left (\ln \left (\frac {5}{x}\right )+1\right )\,\left (7\,\ln \left (\frac {7}{5}-x^2\right )-5\,x^2\,\ln \left (\frac {7}{5}-x^2\right )+10\,x^2\right )}{x\,\left (5\,x^2-7\right )}}{\ln \left (\ln \left (\frac {5}{x}\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.42, size = 20, normalized size = 0.71 \begin {gather*} \log {\relax (x )} + \frac {\log {\left (\frac {7}{5} - x^{2} \right )}}{x \log {\left (\log {\left (\frac {5}{x} \right )} + 1 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**3-7*x)*ln(5/x)+5*x**3-7*x)*ln(ln(5/x)+1)**2+(((-5*x**2+7)*ln(-x**2+7/5)+10*x**2)*ln(5/x)+(-5
*x**2+7)*ln(-x**2+7/5)+10*x**2)*ln(ln(5/x)+1)+(5*x**2-7)*ln(-x**2+7/5))/((5*x**4-7*x**2)*ln(5/x)+5*x**4-7*x**2
)/ln(ln(5/x)+1)**2,x)

[Out]

log(x) + log(7/5 - x**2)/(x*log(log(5/x) + 1))

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