3.91.3 \(\int \frac {-75-75 x-x^2+x^3+2 x^4+((75-x^2+x^3) \log (2)+(-75+x^2-x^3) \log (\frac {75-x^2+x^3}{x})) \log (\frac {-\log (2)+\log (\frac {75-x^2+x^3}{x})}{\log (2)})}{(-75 x^2+x^4-x^5) \log (2)+(75 x^2-x^4+x^5) \log (\frac {75-x^2+x^3}{x})} \, dx\)

Optimal. Leaf size=31 \[ 5+\frac {(1+x) \log \left (-1+\frac {\log \left (-x+x \left (\frac {75}{x^2}+x\right )\right )}{\log (2)}\right )}{x} \]

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Rubi [F]  time = 2.50, 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 {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-75 - 75*x - x^2 + x^3 + 2*x^4 + ((75 - x^2 + x^3)*Log[2] + (-75 + x^2 - x^3)*Log[(75 - x^2 + x^3)/x])*Lo
g[(-Log[2] + Log[(75 - x^2 + x^3)/x])/Log[2]])/((-75*x^2 + x^4 - x^5)*Log[2] + (75*x^2 - x^4 + x^5)*Log[(75 -
x^2 + x^3)/x]),x]

[Out]

Defer[Int][1/(x^2*(Log[2] - Log[75/x - x + x^2])), x] + Defer[Int][1/(x*(Log[2] - Log[75/x - x + x^2])), x] +
2*Defer[Int][1/((75 - x^2 + x^3)*(Log[2] - Log[75/x - x + x^2])), x] - Defer[Int][x/((75 - x^2 + x^3)*(Log[2]
- Log[75/x - x + x^2])), x] - 3*Defer[Int][x^2/((75 - x^2 + x^3)*(Log[2] - Log[75/x - x + x^2])), x] - Defer[I
nt][Log[-1 + Log[75/x - x + x^2]/Log[2]]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {75+75 x+x^2-x^3-2 x^4-\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{x^2 \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx\\ &=\int \left (\frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}+\frac {75}{x^2 \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}+\frac {75}{x \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}-\frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}-\frac {2 x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}-\frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx\right )+75 \int \frac {1}{x^2 \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+75 \int \frac {1}{x \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx\right )+75 \int \left (\frac {1}{75 x^2 \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}+\frac {1-x}{75 \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}\right ) \, dx+75 \int \left (\frac {1}{75 x \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}-\frac {(-1+x) x}{75 \left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}\right ) \, dx+\int \frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx\right )+\int \frac {1}{x^2 \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1}{x \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1-x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {(-1+x) x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx\right )+\int \left (\frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}-\frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}\right ) \, dx-\int \left (-\frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}+\frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )}\right ) \, dx+\int \frac {1}{x^2 \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1}{x \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x^2} \, dx\\ &=-\left (2 \int \frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx\right )+\int \frac {1}{x^2 \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+\int \frac {1}{x \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx+2 \int \frac {1}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {x}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {x^2}{\left (75-x^2+x^3\right ) \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )} \, dx-\int \frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 45, normalized size = 1.45 \begin {gather*} \log \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )+\frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-75 - 75*x - x^2 + x^3 + 2*x^4 + ((75 - x^2 + x^3)*Log[2] + (-75 + x^2 - x^3)*Log[(75 - x^2 + x^3)/
x])*Log[(-Log[2] + Log[(75 - x^2 + x^3)/x])/Log[2]])/((-75*x^2 + x^4 - x^5)*Log[2] + (75*x^2 - x^4 + x^5)*Log[
(75 - x^2 + x^3)/x]),x]

[Out]

Log[Log[2] - Log[75/x - x + x^2]] + Log[-1 + Log[75/x - x + x^2]/Log[2]]/x

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fricas [A]  time = 0.46, size = 34, normalized size = 1.10 \begin {gather*} \frac {{\left (x + 1\right )} \log \left (-\frac {\log \relax (2) - \log \left (\frac {x^{3} - x^{2} + 75}{x}\right )}{\log \relax (2)}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((log((x^3-x^2+75)/x)-log(2))/log(2))+2*
x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2)*log((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x, algorithm="fricas")

[Out]

(x + 1)*log(-(log(2) - log((x^3 - x^2 + 75)/x))/log(2))/x

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giac [A]  time = 0.34, size = 55, normalized size = 1.77 \begin {gather*} \frac {\log \left (-\log \relax (2) + \log \left (x^{3} - x^{2} + 75\right ) - \log \relax (x)\right )}{x} - \frac {\log \left (\log \relax (2)\right )}{x} + \log \left (-\log \relax (2) + \log \left (x^{3} - x^{2} + 75\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((log((x^3-x^2+75)/x)-log(2))/log(2))+2*
x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2)*log((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x, algorithm="giac")

[Out]

log(-log(2) + log(x^3 - x^2 + 75) - log(x))/x - log(log(2))/x + log(-log(2) + log(x^3 - x^2 + 75) - log(x))

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-x^{3}+x^{2}-75\right ) \ln \left (\frac {x^{3}-x^{2}+75}{x}\right )+\left (x^{3}-x^{2}+75\right ) \ln \relax (2)\right ) \ln \left (\frac {\ln \left (\frac {x^{3}-x^{2}+75}{x}\right )-\ln \relax (2)}{\ln \relax (2)}\right )+2 x^{4}+x^{3}-x^{2}-75 x -75}{\left (x^{5}-x^{4}+75 x^{2}\right ) \ln \left (\frac {x^{3}-x^{2}+75}{x}\right )+\left (-x^{5}+x^{4}-75 x^{2}\right ) \ln \relax (2)}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^3+x^2-75)*ln((x^3-x^2+75)/x)+(x^3-x^2+75)*ln(2))*ln((ln((x^3-x^2+75)/x)-ln(2))/ln(2))+2*x^4+x^3-x^2-
75*x-75)/((x^5-x^4+75*x^2)*ln((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*ln(2)),x)

[Out]

int((((-x^3+x^2-75)*ln((x^3-x^2+75)/x)+(x^3-x^2+75)*ln(2))*ln((ln((x^3-x^2+75)/x)-ln(2))/ln(2))+2*x^4+x^3-x^2-
75*x-75)/((x^5-x^4+75*x^2)*ln((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*ln(2)),x)

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maxima [A]  time = 0.48, size = 35, normalized size = 1.13 \begin {gather*} \frac {{\left (x + 1\right )} \log \left (-\log \relax (2) + \log \left (x^{3} - x^{2} + 75\right ) - \log \relax (x)\right ) - \log \left (\log \relax (2)\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((log((x^3-x^2+75)/x)-log(2))/log(2))+2*
x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2)*log((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x, algorithm="maxima")

[Out]

((x + 1)*log(-log(2) + log(x^3 - x^2 + 75) - log(x)) - log(log(2)))/x

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mupad [B]  time = 7.80, size = 55, normalized size = 1.77 \begin {gather*} \ln \left (\ln \left (\frac {x^3-x^2+75}{x}\right )-\ln \relax (2)\right )-\frac {\ln \left (\ln \relax (2)\right )}{x}+\frac {\ln \left (\ln \left (\frac {x^3-x^2+75}{x}\right )-\ln \relax (2)\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(75*x - log((log((x^3 - x^2 + 75)/x) - log(2))/log(2))*(log(2)*(x^3 - x^2 + 75) - log((x^3 - x^2 + 75)/x)
*(x^3 - x^2 + 75)) + x^2 - x^3 - 2*x^4 + 75)/(log((x^3 - x^2 + 75)/x)*(75*x^2 - x^4 + x^5) - log(2)*(75*x^2 -
x^4 + x^5)),x)

[Out]

log(log((x^3 - x^2 + 75)/x) - log(2)) - log(log(2))/x + log(log((x^3 - x^2 + 75)/x) - log(2))/x

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sympy [A]  time = 0.67, size = 37, normalized size = 1.19 \begin {gather*} \log {\left (\log {\left (\frac {x^{3} - x^{2} + 75}{x} \right )} - \log {\relax (2 )} \right )} + \frac {\log {\left (\frac {\log {\left (\frac {x^{3} - x^{2} + 75}{x} \right )} - \log {\relax (2 )}}{\log {\relax (2 )}} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**3+x**2-75)*ln((x**3-x**2+75)/x)+(x**3-x**2+75)*ln(2))*ln((ln((x**3-x**2+75)/x)-ln(2))/ln(2))+
2*x**4+x**3-x**2-75*x-75)/((x**5-x**4+75*x**2)*ln((x**3-x**2+75)/x)+(-x**5+x**4-75*x**2)*ln(2)),x)

[Out]

log(log((x**3 - x**2 + 75)/x) - log(2)) + log((log((x**3 - x**2 + 75)/x) - log(2))/log(2))/x

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