3.47.79 \(\int \frac {-18 x^2-15 x^3+(90+75 x) \log (4)-8 x^2 \log (\frac {-x^2+5 \log (4)}{\log (4)})+(6 x^2+5 x^3+(-30-25 x) \log (4)) \log ^2(\frac {-x^2+5 \log (4)}{\log (4)})+\log (x) (30 x^3-150 x \log (4)+20 x^3 \log (\frac {-x^2+5 \log (4)}{\log (4)})+(-10 x^3+50 x \log (4)) \log ^2(\frac {-x^2+5 \log (4)}{\log (4)}))}{-5 x^6+25 x^4 \log (4)} \, dx\)

Optimal. Leaf size=34 \[ 2+\frac {\left (-\frac {2}{5 x}+\log (x)\right ) \left (3-\log ^2\left (5-\frac {x^2}{\log (4)}\right )\right )}{x^2} \]

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Rubi [F]  time = 3.50, 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 {-18 x^2-15 x^3+(90+75 x) \log (4)-8 x^2 \log \left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\left (6 x^2+5 x^3+(-30-25 x) \log (4)\right ) \log ^2\left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\log (x) \left (30 x^3-150 x \log (4)+20 x^3 \log \left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\left (-10 x^3+50 x \log (4)\right ) \log ^2\left (\frac {-x^2+5 \log (4)}{\log (4)}\right )\right )}{-5 x^6+25 x^4 \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-18*x^2 - 15*x^3 + (90 + 75*x)*Log[4] - 8*x^2*Log[(-x^2 + 5*Log[4])/Log[4]] + (6*x^2 + 5*x^3 + (-30 - 25*
x)*Log[4])*Log[(-x^2 + 5*Log[4])/Log[4]]^2 + Log[x]*(30*x^3 - 150*x*Log[4] + 20*x^3*Log[(-x^2 + 5*Log[4])/Log[
4]] + (-10*x^3 + 50*x*Log[4])*Log[(-x^2 + 5*Log[4])/Log[4]]^2))/(-5*x^6 + 25*x^4*Log[4]),x]

[Out]

-6/(5*x^3) + (3*Log[x])/x^2 + (2*Log[5]*Log[x])/(5*Log[4]) + (2*Log[x]^2*Log[5 - x^2/Log[4]])/(5*Log[4]) + (2*
Log[5 - x^2/Log[4]]^2)/(5*x^3) + ((5 - x^2/Log[4])*Log[5 - x^2/Log[4]]^2)/(10*x^2) - (2*Log[x]^2*Log[1 - x^2/(
5*Log[4])])/(5*Log[4]) - (2*Log[x]*PolyLog[2, x^2/(5*Log[4])])/(5*Log[4]) - PolyLog[2, x^2/Log[1024]]/(5*Log[4
]) + PolyLog[3, x^2/(5*Log[4])]/(5*Log[4]) + (2*Defer[Int][(Log[x]*Log[5 - x^2/Log[4]])/(-x + Sqrt[5*Log[4]]),
x])/(5*Log[4]) - (2*Defer[Int][(Log[x]*Log[5 - x^2/Log[4]])/(x + Sqrt[5*Log[4]]), x])/(5*Log[4]) + 2*Defer[In
t][(Log[x]*Log[5 - x^2/Log[4]]^2)/x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-18 x^2-15 x^3+(90+75 x) \log (4)-8 x^2 \log \left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\left (6 x^2+5 x^3+(-30-25 x) \log (4)\right ) \log ^2\left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\log (x) \left (30 x^3-150 x \log (4)+20 x^3 \log \left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\left (-10 x^3+50 x \log (4)\right ) \log ^2\left (\frac {-x^2+5 \log (4)}{\log (4)}\right )\right )}{x^4 \left (-5 x^2+25 \log (4)\right )} \, dx\\ &=\int \left (-\frac {3 (-6-5 x+10 x \log (x))}{5 x^4}-\frac {4 (-2+5 x \log (x)) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 x^2 \left (x^2-5 \log (4)\right )}+\frac {(-6-5 x+10 x \log (x)) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^4}\right ) \, dx\\ &=\frac {1}{5} \int \frac {(-6-5 x+10 x \log (x)) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^4} \, dx-\frac {3}{5} \int \frac {-6-5 x+10 x \log (x)}{x^4} \, dx-\frac {4}{5} \int \frac {(-2+5 x \log (x)) \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2 \left (x^2-5 \log (4)\right )} \, dx\\ &=\frac {1}{5} \int \left (-\frac {6 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^4}-\frac {5 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3}+\frac {10 \log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3}\right ) \, dx-\frac {3}{5} \int \left (\frac {-6-5 x}{x^4}+\frac {10 \log (x)}{x^3}\right ) \, dx-\frac {4}{5} \int \left (-\frac {(-2+5 x \log (x)) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 x^2 \log (4)}+\frac {(-2+5 x \log (x)) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \left (x^2-5 \log (4)\right ) \log (4)}\right ) \, dx\\ &=-\left (\frac {3}{5} \int \frac {-6-5 x}{x^4} \, dx\right )-\frac {6}{5} \int \frac {\log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^4} \, dx+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx-6 \int \frac {\log (x)}{x^3} \, dx+\frac {4 \int \frac {(-2+5 x \log (x)) \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2} \, dx}{25 \log (4)}-\frac {4 \int \frac {(-2+5 x \log (x)) \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2-5 \log (4)} \, dx}{25 \log (4)}-\int \frac {\log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx\\ &=\frac {3}{2 x^2}+\frac {3 \log (x)}{x^2}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log ^2\left (5-\frac {x}{\log (4)}\right )}{x^2} \, dx,x,x^2\right )-\frac {3}{5} \int \left (-\frac {6}{x^4}-\frac {5}{x^3}\right ) \, dx+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {4 \int \left (-\frac {2 \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2}+\frac {5 \log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x}\right ) \, dx}{25 \log (4)}-\frac {4 \int \left (-\frac {2 \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2-5 \log (4)}+\frac {5 x \log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2-5 \log (4)}\right ) \, dx}{25 \log (4)}+\frac {8 \int \frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{x^2 \left (5-\frac {x^2}{\log (4)}\right )} \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {3 \log (x)}{x^2}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {\operatorname {Subst}\left (\int \frac {\log \left (5-\frac {x}{\log (4)}\right )}{x} \, dx,x,x^2\right )}{5 \log (4)}-\frac {8 \int \frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{x^2} \, dx}{25 \log (4)}+\frac {8 \int \frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{x^2-5 \log (4)} \, dx}{25 \log (4)}+\frac {4 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x} \, dx}{5 \log (4)}-\frac {4 \int \frac {x \log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x^2-5 \log (4)} \, dx}{5 \log (4)}+\frac {8 \int \left (\frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{5 x^2}-\frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{5 \left (x^2-5 \log (4)\right )}\right ) \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}-\frac {8 \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right ) \log \left (5-\frac {x^2}{\log (4)}\right )}{25 \sqrt {5} \log ^{\frac {3}{2}}(4)}+\frac {8 \log \left (5-\frac {x^2}{\log (4)}\right )}{25 x \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {16 \int \frac {1}{5-\frac {x^2}{\log (4)}} \, dx}{25 \log ^2(4)}-\frac {16 \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )}{\left (5-\frac {x^2}{\log (4)}\right ) \sqrt {5 \log (4)}} \, dx}{25 \log ^2(4)}+\frac {4 \int \frac {x \log ^2(x)}{5-\frac {x^2}{\log (4)}} \, dx}{5 \log ^2(4)}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{5 \log (4)}\right )}{x} \, dx,x,x^2\right )}{5 \log (4)}+\frac {8 \int \frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{x^2} \, dx}{25 \log (4)}-\frac {8 \int \frac {\log \left (5-\frac {x^2}{\log (4)}\right )}{x^2-5 \log (4)} \, dx}{25 \log (4)}-\frac {4 \int \left (-\frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{2 \left (-x+\sqrt {5 \log (4)}\right )}+\frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{2 \left (x+\sqrt {5 \log (4)}\right )}\right ) \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {16 \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right ) \sqrt {\frac {2 \log (2)}{5}}}{25 \log ^2(4)}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}-\frac {2 \log ^2(x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {\text {Li}_2\left (\frac {x^2}{\log (1024)}\right )}{5 \log (4)}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx-\frac {16 \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )}{5-\frac {x^2}{\log (4)}} \, dx}{25 \sqrt {5} \log ^{\frac {5}{2}}(4)}-\frac {16 \int \frac {1}{5-\frac {x^2}{\log (4)}} \, dx}{25 \log ^2(4)}+\frac {16 \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )}{\left (5-\frac {x^2}{\log (4)}\right ) \sqrt {5 \log (4)}} \, dx}{25 \log ^2(4)}+\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{-x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}-\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}+\frac {4 \int \frac {\log (x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{x} \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {8 \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )^2}{25 \sqrt {5} \log ^{\frac {3}{2}}(4)}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}-\frac {2 \log ^2(x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {2 \log (x) \text {Li}_2\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {\text {Li}_2\left (\frac {x^2}{\log (1024)}\right )}{5 \log (4)}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {16 \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )}{5-\frac {x^2}{\log (4)}} \, dx}{25 \sqrt {5} \log ^{\frac {5}{2}}(4)}-\frac {16 \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )}{1-\frac {x}{\sqrt {5 \log (4)}}} \, dx}{125 \log ^2(4)}+\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{-x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}-\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}+\frac {2 \int \frac {\text {Li}_2\left (\frac {x^2}{5 \log (4)}\right )}{x} \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}-\frac {2 \log ^2(x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {16 \tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right ) \log \left (\frac {2}{1-\frac {x}{\sqrt {5 \log (4)}}}\right )}{25 \sqrt {5} \log ^{\frac {3}{2}}(4)}-\frac {2 \log (x) \text {Li}_2\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {\text {Li}_2\left (\frac {x^2}{\log (1024)}\right )}{5 \log (4)}+\frac {\text {Li}_3\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {16 \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {5 \log (4)}}\right )}{1-\frac {x}{\sqrt {5 \log (4)}}} \, dx}{125 \log ^2(4)}+\frac {16 \int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {5 \log (4)}}}\right )}{1-\frac {x^2}{5 \log (4)}} \, dx}{125 \log ^2(4)}+\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{-x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}-\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}-\frac {2 \log ^2(x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {2 \log (x) \text {Li}_2\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {\text {Li}_2\left (\frac {x^2}{\log (1024)}\right )}{5 \log (4)}+\frac {\text {Li}_3\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx-\frac {16 \int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {5 \log (4)}}}\right )}{1-\frac {x^2}{5 \log (4)}} \, dx}{125 \log ^2(4)}-\frac {16 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {x}{\sqrt {5 \log (4)}}}\right )}{25 \sqrt {5} \log ^{\frac {3}{2}}(4)}+\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{-x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}-\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}-\frac {2 \log ^2(x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {2 \log (x) \text {Li}_2\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {8 \text {Li}_2\left (1-\frac {2}{1-\frac {x}{\sqrt {5 \log (4)}}}\right )}{25 \sqrt {5} \log ^{\frac {3}{2}}(4)}-\frac {\text {Li}_2\left (\frac {x^2}{\log (1024)}\right )}{5 \log (4)}+\frac {\text {Li}_3\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {16 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {x}{\sqrt {5 \log (4)}}}\right )}{25 \sqrt {5} \log ^{\frac {3}{2}}(4)}+\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{-x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}-\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}\\ &=-\frac {6}{5 x^3}+\frac {3 \log (x)}{x^2}+\frac {2 \log (5) \log (x)}{5 \log (4)}+\frac {2 \log ^2(x) \log \left (5-\frac {x^2}{\log (4)}\right )}{5 \log (4)}+\frac {2 \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{5 x^3}+\frac {\left (5-\frac {x^2}{\log (4)}\right ) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{10 x^2}-\frac {2 \log ^2(x) \log \left (1-\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {2 \log (x) \text {Li}_2\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}-\frac {\text {Li}_2\left (\frac {x^2}{\log (1024)}\right )}{5 \log (4)}+\frac {\text {Li}_3\left (\frac {x^2}{5 \log (4)}\right )}{5 \log (4)}+2 \int \frac {\log (x) \log ^2\left (5-\frac {x^2}{\log (4)}\right )}{x^3} \, dx+\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{-x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}-\frac {2 \int \frac {\log (x) \log \left (5-\frac {x^2}{\log (4)}\right )}{x+\sqrt {5 \log (4)}} \, dx}{5 \log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 5.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-18 x^2-15 x^3+(90+75 x) \log (4)-8 x^2 \log \left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\left (6 x^2+5 x^3+(-30-25 x) \log (4)\right ) \log ^2\left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\log (x) \left (30 x^3-150 x \log (4)+20 x^3 \log \left (\frac {-x^2+5 \log (4)}{\log (4)}\right )+\left (-10 x^3+50 x \log (4)\right ) \log ^2\left (\frac {-x^2+5 \log (4)}{\log (4)}\right )\right )}{-5 x^6+25 x^4 \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-18*x^2 - 15*x^3 + (90 + 75*x)*Log[4] - 8*x^2*Log[(-x^2 + 5*Log[4])/Log[4]] + (6*x^2 + 5*x^3 + (-30
- 25*x)*Log[4])*Log[(-x^2 + 5*Log[4])/Log[4]]^2 + Log[x]*(30*x^3 - 150*x*Log[4] + 20*x^3*Log[(-x^2 + 5*Log[4]
)/Log[4]] + (-10*x^3 + 50*x*Log[4])*Log[(-x^2 + 5*Log[4])/Log[4]]^2))/(-5*x^6 + 25*x^4*Log[4]),x]

[Out]

Integrate[(-18*x^2 - 15*x^3 + (90 + 75*x)*Log[4] - 8*x^2*Log[(-x^2 + 5*Log[4])/Log[4]] + (6*x^2 + 5*x^3 + (-30
- 25*x)*Log[4])*Log[(-x^2 + 5*Log[4])/Log[4]]^2 + Log[x]*(30*x^3 - 150*x*Log[4] + 20*x^3*Log[(-x^2 + 5*Log[4]
)/Log[4]] + (-10*x^3 + 50*x*Log[4])*Log[(-x^2 + 5*Log[4])/Log[4]]^2))/(-5*x^6 + 25*x^4*Log[4]), x]

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fricas [A]  time = 0.89, size = 53, normalized size = 1.56 \begin {gather*} -\frac {5 \, {\left (x \log \left (-\frac {x^{2} - 10 \, \log \relax (2)}{2 \, \log \relax (2)}\right )^{2} - 3 \, x\right )} \log \relax (x) - 2 \, \log \left (-\frac {x^{2} - 10 \, \log \relax (2)}{2 \, \log \relax (2)}\right )^{2} + 6}{5 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((100*x*log(2)-10*x^3)*log(1/2*(10*log(2)-x^2)/log(2))^2+20*x^3*log(1/2*(10*log(2)-x^2)/log(2))-300
*x*log(2)+30*x^3)*log(x)+(2*(-25*x-30)*log(2)+5*x^3+6*x^2)*log(1/2*(10*log(2)-x^2)/log(2))^2-8*x^2*log(1/2*(10
*log(2)-x^2)/log(2))+2*(75*x+90)*log(2)-15*x^3-18*x^2)/(50*x^4*log(2)-5*x^6),x, algorithm="fricas")

[Out]

-1/5*(5*(x*log(-1/2*(x^2 - 10*log(2))/log(2))^2 - 3*x)*log(x) - 2*log(-1/2*(x^2 - 10*log(2))/log(2))^2 + 6)/x^
3

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giac [B]  time = 0.30, size = 115, normalized size = 3.38 \begin {gather*} -\frac {1}{5} \, {\left (\frac {5 \, \log \relax (x)}{x^{2}} - \frac {2}{x^{3}}\right )} \log \left (-x^{2} + 10 \, \log \relax (2)\right )^{2} + \frac {2}{5} \, {\left (\frac {5 \, {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \log \relax (x)}{x^{2}} - \frac {2 \, {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )}}{x^{3}}\right )} \log \left (-x^{2} + 10 \, \log \relax (2)\right ) - \frac {{\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (\log \relax (2)\right ) + \log \left (\log \relax (2)\right )^{2} - 3\right )} \log \relax (x)}{x^{2}} + \frac {2 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (\log \relax (2)\right ) + \log \left (\log \relax (2)\right )^{2} - 3\right )}}{5 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((100*x*log(2)-10*x^3)*log(1/2*(10*log(2)-x^2)/log(2))^2+20*x^3*log(1/2*(10*log(2)-x^2)/log(2))-300
*x*log(2)+30*x^3)*log(x)+(2*(-25*x-30)*log(2)+5*x^3+6*x^2)*log(1/2*(10*log(2)-x^2)/log(2))^2-8*x^2*log(1/2*(10
*log(2)-x^2)/log(2))+2*(75*x+90)*log(2)-15*x^3-18*x^2)/(50*x^4*log(2)-5*x^6),x, algorithm="giac")

[Out]

-1/5*(5*log(x)/x^2 - 2/x^3)*log(-x^2 + 10*log(2))^2 + 2/5*(5*(log(2) + log(log(2)))*log(x)/x^2 - 2*(log(2) + l
og(log(2)))/x^3)*log(-x^2 + 10*log(2)) - (log(2)^2 + 2*log(2)*log(log(2)) + log(log(2))^2 - 3)*log(x)/x^2 + 2/
5*(log(2)^2 + 2*log(2)*log(log(2)) + log(log(2))^2 - 3)/x^3

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maple [A]  time = 0.19, size = 45, normalized size = 1.32

method	result	size
risch	\(-\frac {\left (5 x \ln \relax (x )-2\right ) \ln \left (\frac {10 \ln \relax (2)-x^{2}}{2 \ln \relax (2)}\right )^{2}}{5 x^{3}}+\frac {3 x \ln \relax (x )-\frac {6}{5}}{x^{3}}\)	\(45\)
default	error in gcdex: invalid arguments\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((100*x*ln(2)-10*x^3)*ln(1/2*(10*ln(2)-x^2)/ln(2))^2+20*x^3*ln(1/2*(10*ln(2)-x^2)/ln(2))-300*x*ln(2)+30*x
^3)*ln(x)+(2*(-25*x-30)*ln(2)+5*x^3+6*x^2)*ln(1/2*(10*ln(2)-x^2)/ln(2))^2-8*x^2*ln(1/2*(10*ln(2)-x^2)/ln(2))+2
*(75*x+90)*ln(2)-15*x^3-18*x^2)/(50*x^4*ln(2)-5*x^6),x,method=_RETURNVERBOSE)

[Out]

-1/5*(5*x*ln(x)-2)/x^3*ln(1/2*(10*ln(2)-x^2)/ln(2))^2+3/5*(5*x*ln(x)-2)/x^3

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maxima [B]  time = 0.50, size = 273, normalized size = 8.03 \begin {gather*} -\frac {3}{20} \, {\left (\frac {\log \left (x^{2} - 10 \, \log \relax (2)\right )}{\log \relax (2)^{2}} - \frac {\log \left (x^{2}\right )}{\log \relax (2)^{2}} + \frac {10}{x^{2} \log \relax (2)}\right )} \log \relax (2) - \frac {3}{500} \, {\left (\frac {3 \, \sqrt {10} \log \left (\frac {x - \sqrt {10} \sqrt {\log \relax (2)}}{x + \sqrt {10} \sqrt {\log \relax (2)}}\right )}{\log \relax (2)^{\frac {5}{2}}} + \frac {20 \, {\left (3 \, x^{2} + 10 \, \log \relax (2)\right )}}{x^{3} \log \relax (2)^{2}}\right )} \log \relax (2) + \frac {3 \, \log \left (x^{2} - 10 \, \log \relax (2)\right )}{20 \, \log \relax (2)} - \frac {3 \, \log \left (x^{2}\right )}{20 \, \log \relax (2)} + \frac {9 \, \sqrt {10} \log \left (\frac {x - \sqrt {10} \sqrt {\log \relax (2)}}{x + \sqrt {10} \sqrt {\log \relax (2)}}\right )}{500 \, \log \relax (2)^{\frac {3}{2}}} - \frac {2 \, {\left (5 \, x \log \relax (x) - 2\right )} \log \left (-x^{2} + 10 \, \log \relax (2)\right )^{2} + 10 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (\log \relax (2)\right ) + \log \left (\log \relax (2)\right )^{2} - 3\right )} x \log \relax (x) - 4 \, \log \relax (2)^{2} - 4 \, {\left (5 \, x {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \log \relax (x) - 2 \, \log \relax (2) - 2 \, \log \left (\log \relax (2)\right )\right )} \log \left (-x^{2} + 10 \, \log \relax (2)\right ) - 8 \, \log \relax (2) \log \left (\log \relax (2)\right ) - 4 \, \log \left (\log \relax (2)\right )^{2} - 15 \, x}{10 \, x^{3}} + \frac {9}{25 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((100*x*log(2)-10*x^3)*log(1/2*(10*log(2)-x^2)/log(2))^2+20*x^3*log(1/2*(10*log(2)-x^2)/log(2))-300
*x*log(2)+30*x^3)*log(x)+(2*(-25*x-30)*log(2)+5*x^3+6*x^2)*log(1/2*(10*log(2)-x^2)/log(2))^2-8*x^2*log(1/2*(10
*log(2)-x^2)/log(2))+2*(75*x+90)*log(2)-15*x^3-18*x^2)/(50*x^4*log(2)-5*x^6),x, algorithm="maxima")

[Out]

-3/20*(log(x^2 - 10*log(2))/log(2)^2 - log(x^2)/log(2)^2 + 10/(x^2*log(2)))*log(2) - 3/500*(3*sqrt(10)*log((x
- sqrt(10)*sqrt(log(2)))/(x + sqrt(10)*sqrt(log(2))))/log(2)^(5/2) + 20*(3*x^2 + 10*log(2))/(x^3*log(2)^2))*lo
g(2) + 3/20*log(x^2 - 10*log(2))/log(2) - 3/20*log(x^2)/log(2) + 9/500*sqrt(10)*log((x - sqrt(10)*sqrt(log(2))
)/(x + sqrt(10)*sqrt(log(2))))/log(2)^(3/2) - 1/10*(2*(5*x*log(x) - 2)*log(-x^2 + 10*log(2))^2 + 10*(log(2)^2
+ 2*log(2)*log(log(2)) + log(log(2))^2 - 3)*x*log(x) - 4*log(2)^2 - 4*(5*x*(log(2) + log(log(2)))*log(x) - 2*l
og(2) - 2*log(log(2)))*log(-x^2 + 10*log(2)) - 8*log(2)*log(log(2)) - 4*log(log(2))^2 - 15*x)/x^3 + 9/25/(x*lo
g(2))

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mupad [B]  time = 3.48, size = 53, normalized size = 1.56 \begin {gather*} \frac {3\,\ln \relax (x)}{x^2}-{\ln \left (\frac {\ln \left (32\right )-\frac {x^2}{2}}{\ln \relax (2)}\right )}^2\,\left (\frac {\ln \relax (x)}{x^2}-\frac {\frac {x}{2}+\frac {2}{5}}{x^3}+\frac {1}{2\,x^2}\right )-\frac {6}{5\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(75*x + 90) - 8*x^2*log((5*log(2) - x^2/2)/log(2)) + log((5*log(2) - x^2/2)/log(2))^2*(6*x^2 - 2
*log(2)*(25*x + 30) + 5*x^3) - 18*x^2 - 15*x^3 + log(x)*(20*x^3*log((5*log(2) - x^2/2)/log(2)) - 300*x*log(2)
+ log((5*log(2) - x^2/2)/log(2))^2*(100*x*log(2) - 10*x^3) + 30*x^3))/(50*x^4*log(2) - 5*x^6),x)

[Out]

(3*log(x))/x^2 - log((log(32) - x^2/2)/log(2))^2*(log(x)/x^2 - (x/2 + 2/5)/x^3 + 1/(2*x^2)) - 6/(5*x^3)

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sympy [A]  time = 0.86, size = 44, normalized size = 1.29 \begin {gather*} \frac {3 \log {\relax (x )}}{x^{2}} + \frac {\left (- 5 x \log {\relax (x )} + 2\right ) \log {\left (\frac {- \frac {x^{2}}{2} + 5 \log {\relax (2 )}}{\log {\relax (2 )}} \right )}^{2}}{5 x^{3}} - \frac {6}{5 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((100*x*ln(2)-10*x**3)*ln(1/2*(10*ln(2)-x**2)/ln(2))**2+20*x**3*ln(1/2*(10*ln(2)-x**2)/ln(2))-300*x
*ln(2)+30*x**3)*ln(x)+(2*(-25*x-30)*ln(2)+5*x**3+6*x**2)*ln(1/2*(10*ln(2)-x**2)/ln(2))**2-8*x**2*ln(1/2*(10*ln
(2)-x**2)/ln(2))+2*(75*x+90)*ln(2)-15*x**3-18*x**2)/(50*x**4*ln(2)-5*x**6),x)

[Out]

3*log(x)/x**2 + (-5*x*log(x) + 2)*log((-x**2/2 + 5*log(2))/log(2))**2/(5*x**3) - 6/(5*x**3)

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