Integrand size = 92, antiderivative size = 24 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 x (4+x) \log ^2\left (\frac {5}{x}-x^2 \log ^2(4)\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
Time = 1.93 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=8 \left (2 x \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )+\frac {1}{2} x^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )\right ) \]
Integrate[((160 + 40*x + (64*x^3 + 16*x^4)*Log[4]^2)*Log[(5 - x^3*Log[4]^2 )/x] + (-80 - 40*x + (16*x^3 + 8*x^4)*Log[4]^2)*Log[(5 - x^3*Log[4]^2)/x]^ 2)/(-5 + x^3*Log[4]^2),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 6.44 (sec) , antiderivative size = 3127, normalized size of antiderivative = 130.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (\left (8 x^4+16 x^3\right ) \log ^2(4)-40 x-80\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (\left (16 x^4+64 x^3\right ) \log ^2(4)+40 x+160\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{x^3 \log ^2(4)-5} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (8 (x+2) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )+\frac {8 (x+4) \left (2 x^3 \log ^2(4)+5\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{x^3 \log ^2(4)-5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \log ^2(x)+32 \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \log (x)-32 \log \left (1-\frac {1}{5} x^3 \log ^2(4)\right ) \log (x)+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \log (5) \log (x)}{3 \log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log (5) \log (x)}{3 \log ^{\frac {2}{3}}(4)}-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \log (5) \log (x)}{3 \log ^{\frac {2}{3}}(4)}-\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \log (5) \log (x)}{3 \log ^{\frac {4}{3}}(4)}-\frac {8}{3} \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log (5) \log (x)+\frac {8}{3} \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log (5) \log (x)-\frac {4 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \log ^2\left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {4 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \log ^2\left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {4 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log ^2\left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{\log ^{\frac {2}{3}}(4)}+\frac {4 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \log ^2\left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{\log ^{\frac {4}{3}}(4)}+4 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log ^2\left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )-4 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log ^2\left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )+4 (x+2)^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (\frac {\left (i+\sqrt {3}\right ) \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{\sqrt [3]{5} \left (3 i+\sqrt {3}\right )}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (\frac {\left (i+\sqrt {3}\right ) \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{\sqrt [3]{5} \left (3 i+\sqrt {3}\right )}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {\log ^{\frac {2}{3}}(4) x+\sqrt [3]{-5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {\log ^{\frac {2}{3}}(4) x+\sqrt [3]{-5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {4}{3}}(4)}+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (\frac {\sqrt [3]{-\frac {1}{5}} \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{1+\sqrt [3]{-1}}\right ) \log \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )-8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (\frac {\sqrt [3]{-\frac {1}{5}} \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{1+\sqrt [3]{-1}}\right ) \log \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (-\frac {\sqrt [3]{-\frac {1}{5}} \left (\log ^{\frac {2}{3}}(4) x+\sqrt [3]{-5}\right )}{1-(-1)^{2/3}}\right ) \log \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )-8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (-\frac {\sqrt [3]{-\frac {1}{5}} \left (\log ^{\frac {2}{3}}(4) x+\sqrt [3]{-5}\right )}{1-(-1)^{2/3}}\right ) \log \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )-\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (-\frac {2 (-1)^{2/3} \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (-\frac {2 (-1)^{2/3} \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )}{\log ^{\frac {4}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\log ^{\frac {4}{3}}(4)}-8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}{\sqrt [3]{5} \left (3-i \sqrt {3}\right )}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x (-\log (4))^{2/3}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x (-\log (4))^{2/3}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}-\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )}{\log ^{\frac {4}{3}}(4)}+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )-8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}}{\sqrt [3]{5} \left (1-(-1)^{2/3}\right )}\right )+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \log ^{\frac {2}{3}}(4) x+\sqrt [3]{5}}{\sqrt [3]{-5}+\sqrt [3]{5}}\right )-\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \log ^{\frac {2}{3}}(4)\right ) \operatorname {PolyLog}\left (2,\frac {x (-\log (4))^{2/3}}{\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{-5}-\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {x (-\log (4))^{2/3}}{\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {1}{5}} x \log ^{\frac {2}{3}}(4)\right )-8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {1}{5}} x \log ^{\frac {2}{3}}(4)\right )-\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {x \log ^{\frac {2}{3}}(4)}{\sqrt [3]{5}}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \operatorname {PolyLog}\left (2,\frac {x \log ^{\frac {2}{3}}(4)}{\sqrt [3]{5}}\right )}{\log ^{\frac {4}{3}}(4)}-\frac {32}{3} \operatorname {PolyLog}\left (2,\frac {1}{5} x^3 \log ^2(4)\right )-\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \operatorname {PolyLog}\left (2,\frac {2 \left (5-x (5 \log (4))^{2/3}\right )}{5 \left (3-i \sqrt {3}\right )}\right )}{\log ^{\frac {2}{3}}(4)}+\frac {8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2 \operatorname {PolyLog}\left (2,\frac {2 \left (5-x (5 \log (4))^{2/3}\right )}{5 \left (3-i \sqrt {3}\right )}\right )}{\log ^{\frac {4}{3}}(4)}\) |
Int[((160 + 40*x + (64*x^3 + 16*x^4)*Log[4]^2)*Log[(5 - x^3*Log[4]^2)/x] + (-80 - 40*x + (16*x^3 + 8*x^4)*Log[4]^2)*Log[(5 - x^3*Log[4]^2)/x]^2)/(-5 + x^3*Log[4]^2),x]
(8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3)*Log[5]*Log[x])/3 - (8*(4*5^(1/3) + (-5/Log[4])^(2/3) - 4*(-1)^(1/3)*Log[4]^(2/3))*(-Log[4]^( -1))^(2/3)*Log[5]*Log[x])/3 - (8*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5]*Log[x ])/(3*Log[4]^(4/3)) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*Log[5]*Log[x]) /(3*Log[4]^(2/3)) + (8*(4*5^(1/3) + (5/Log[4])^(2/3))*Log[5]*Log[x])/(3*Lo g[4]^(2/3)) + (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*Log[5 ]*Log[x])/(3*Log[4]^(2/3)) + 16*Log[x]^2 + (4*(4*(-5)^(1/3) - (-5/Log[4])^ (2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3)]^2)/Log[4]^(2/3) - (4*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3)]^2) /Log[4]^(2/3) + (4*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5^(1/3) - x*Log[4]^(2/ 3)]^2)/Log[4]^(4/3) - (4*(4*5^(1/3) + (5/Log[4])^(2/3))*Log[5^(1/3) - x*Lo g[4]^(2/3)]^2)/Log[4]^(2/3) + (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*Log[5^ (1/3) - x*(-Log[4])^(2/3)]*Log[((I + Sqrt[3])*(5^(1/3) - x*Log[4]^(2/3)))/ (5^(1/3)*(3*I + Sqrt[3]))])/Log[4]^(2/3) - (8*(4*(-5)^(1/3) - (-5/Log[4])^ (2/3) - 4*Log[4]^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3)]*Log[((I + Sqrt[3] )*(5^(1/3) - x*Log[4]^(2/3)))/(5^(1/3)*(3*I + Sqrt[3]))])/Log[4]^(2/3) + ( 8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3)]*Log[ -(((-1)^(2/3)*((-1)^(2/3)*5^(1/3) - x*Log[4]^(2/3)))/((-5)^(1/3) + 5^(1/3) ))])/Log[4]^(2/3) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3)) *Log[5^(1/3) - x*(-Log[4])^(2/3)]*Log[-(((-1)^(2/3)*((-1)^(2/3)*5^(1/3)...
3.4.100.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(\left (4 x^{2}+16 x \right ) \ln \left (\frac {-4 x^{3} \ln \left (2\right )^{2}+5}{x}\right )^{2}\) | \(29\) |
| norman | \(16 x \ln \left (\frac {-4 x^{3} \ln \left (2\right )^{2}+5}{x}\right )^{2}+4 x^{2} \ln \left (\frac {-4 x^{3} \ln \left (2\right )^{2}+5}{x}\right )^{2}\) | \(46\) |
| parallelrisch | \(4 x^{2} \ln \left (-\frac {4 x^{3} \ln \left (2\right )^{2}-5}{x}\right )^{2}+16 \ln \left (-\frac {4 x^{3} \ln \left (2\right )^{2}-5}{x}\right )^{2} x\) | \(48\) |
int(((4*(8*x^4+16*x^3)*ln(2)^2-40*x-80)*ln((-4*x^3*ln(2)^2+5)/x)^2+(4*(16* x^4+64*x^3)*ln(2)^2+40*x+160)*ln((-4*x^3*ln(2)^2+5)/x))/(4*x^3*ln(2)^2-5), x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-\frac {4 \, x^{3} \log \left (2\right )^{2} - 5}{x}\right )^{2} \]
integrate(((4*(8*x^4+16*x^3)*log(2)^2-40*x-80)*log((-4*x^3*log(2)^2+5)/x)^ 2+(4*(16*x^4+64*x^3)*log(2)^2+40*x+160)*log((-4*x^3*log(2)^2+5)/x))/(4*x^3 *log(2)^2-5),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=\left (4 x^{2} + 16 x\right ) \log {\left (\frac {- 4 x^{3} \log {\left (2 \right )}^{2} + 5}{x} \right )}^{2} \]
integrate(((4*(8*x**4+16*x**3)*ln(2)**2-40*x-80)*ln((-4*x**3*ln(2)**2+5)/x )**2+(4*(16*x**4+64*x**3)*ln(2)**2+40*x+160)*ln((-4*x**3*ln(2)**2+5)/x))/( 4*x**3*ln(2)**2-5),x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.79 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right )^{2} - 8 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right ) \log \left (x\right ) + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} \]
integrate(((4*(8*x^4+16*x^3)*log(2)^2-40*x-80)*log((-4*x^3*log(2)^2+5)/x)^ 2+(4*(16*x^4+64*x^3)*log(2)^2+40*x+160)*log((-4*x^3*log(2)^2+5)/x))/(4*x^3 *log(2)^2-5),x, algorithm=\
4*(x^2 + 4*x)*log(-4*x^3*log(2)^2 + 5)^2 - 8*(x^2 + 4*x)*log(-4*x^3*log(2) ^2 + 5)*log(x) + 4*(x^2 + 4*x)*log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.49 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right )^{2} - 8 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right ) \log \left (x\right ) + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} \]
integrate(((4*(8*x^4+16*x^3)*log(2)^2-40*x-80)*log((-4*x^3*log(2)^2+5)/x)^ 2+(4*(16*x^4+64*x^3)*log(2)^2+40*x+160)*log((-4*x^3*log(2)^2+5)/x))/(4*x^3 *log(2)^2-5),x, algorithm=\
4*(x^2 + 4*x)*log(-4*x^3*log(2)^2 + 5)^2 - 8*(x^2 + 4*x)*log(-4*x^3*log(2) ^2 + 5)*log(x) + 4*(x^2 + 4*x)*log(x)^2
Time = 13.77 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4\,x\,{\ln \left (-\frac {4\,x^3\,{\ln \left (2\right )}^2-5}{x}\right )}^2\,\left (x+4\right ) \]