3.98.29 \(\int \frac {(2 x^2+8 x^3) \log ^3(2) \log (\frac {2}{-2+x+2 x^2})+(4 x-2 x^2-4 x^3) \log ^3(2) \log ^2(\frac {2}{-2+x+2 x^2})}{(2-x-2 x^2) \log ^3(2)+(-6 x+3 x^2+6 x^3) \log ^2(2) \log (\frac {2}{-2+x+2 x^2})+(6 x^2-3 x^3-6 x^4) \log (2) \log ^2(\frac {2}{-2+x+2 x^2})+(-2 x^3+x^4+2 x^5) \log ^3(\frac {2}{-2+x+2 x^2})} \, dx\)

Optimal. Leaf size=35 \[ -3+\frac {x^2 \log ^2(2)}{\left (-x+\frac {\log (2)}{\log \left (\frac {2}{x+2 \left (-1+x^2\right )}\right )}\right )^2} \]

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Rubi [F]  time = 6.70, 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 (2 x^2+8 x^3\right ) \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (4 x-2 x^2-4 x^3\right ) \log ^3(2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )}{\left (2-x-2 x^2\right ) \log ^3(2)+\left (-6 x+3 x^2+6 x^3\right ) \log ^2(2) \log \left (\frac {2}{-2+x+2 x^2}\right )+\left (6 x^2-3 x^3-6 x^4\right ) \log (2) \log ^2\left (\frac {2}{-2+x+2 x^2}\right )+\left (-2 x^3+x^4+2 x^5\right ) \log ^3\left (\frac {2}{-2+x+2 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2*x^2 + 8*x^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)] + (4*x - 2*x^2 - 4*x^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)]
^2)/((2 - x - 2*x^2)*Log[2]^3 + (-6*x + 3*x^2 + 6*x^3)*Log[2]^2*Log[2/(-2 + x + 2*x^2)] + (6*x^2 - 3*x^3 - 6*x
^4)*Log[2]*Log[2/(-2 + x + 2*x^2)]^2 + (-2*x^3 + x^4 + 2*x^5)*Log[2/(-2 + x + 2*x^2)]^3),x]

[Out]

-4*Log[2]^4*Defer[Int][(Log[2] - x*Log[2/(-2 + x + 2*x^2)])^(-3), x] + 4*Log[2]^3*Defer[Int][(Log[2] - x*Log[2
/(-2 + x + 2*x^2)])^(-2), x] - (32*Log[2]^4*Defer[Int][1/((-1 + Sqrt[17] - 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2
*x^2)])^3), x])/Sqrt[17] - Log[2]^4*Log[4]*Defer[Int][1/(x*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^3), x] - (2*(
17 - Sqrt[17])*Log[2]^4*Defer[Int][1/((1 - Sqrt[17] + 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^3), x])/17 -
(32*Log[2]^4*Defer[Int][1/((1 + Sqrt[17] + 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^3), x])/Sqrt[17] - (2*(1
7 + Sqrt[17])*Log[2]^4*Defer[Int][1/((1 + Sqrt[17] + 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^3), x])/17 - (
32*Log[2]^3*Defer[Int][1/((-1 + Sqrt[17] - 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^2), x])/Sqrt[17] - Log[2
]^3*Log[16]*Defer[Int][1/(x*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^2), x] - (2*(17 - Sqrt[17])*Log[2]^3*Defer[I
nt][1/((1 - Sqrt[17] + 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^2), x])/17 - (32*Log[2]^3*Defer[Int][1/((1 +
Sqrt[17] + 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^2), x])/Sqrt[17] - (2*(17 + Sqrt[17])*Log[2]^3*Defer[In
t][1/((1 + Sqrt[17] + 4*x)*(-Log[2] + x*Log[2/(-2 + x + 2*x^2)])^2), x])/17 - 2*Log[2]^3*Defer[Int][1/(x*(-Log
[2] + x*Log[2/(-2 + x + 2*x^2)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \log ^3(2) \log \left (\frac {2}{-2+x+2 x^2}\right ) \left (x (1+4 x)-\left (-2+x+2 x^2\right ) \log \left (\frac {2}{-2+x+2 x^2}\right )\right )}{\left (2-x-2 x^2\right ) \left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx\\ &=\left (2 \log ^3(2)\right ) \int \frac {x \log \left (\frac {2}{-2+x+2 x^2}\right ) \left (x (1+4 x)-\left (-2+x+2 x^2\right ) \log \left (\frac {2}{-2+x+2 x^2}\right )\right )}{\left (2-x-2 x^2\right ) \left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx\\ &=\left (2 \log ^3(2)\right ) \int \left (\frac {\log (2) \left (4 x^3-x \log (2)+x^2 (1-\log (4))+\log (4)\right )}{x \left (2-x-2 x^2\right ) \left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}+\frac {-4 x^3-x^2 (1-4 \log (2))+x \log (4)-\log (16)}{x \left (2-x-2 x^2\right ) \left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}-\frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )}\right ) \, dx\\ &=\left (2 \log ^3(2)\right ) \int \frac {-4 x^3-x^2 (1-4 \log (2))+x \log (4)-\log (16)}{x \left (2-x-2 x^2\right ) \left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx-\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx+\left (2 \log ^4(2)\right ) \int \frac {4 x^3-x \log (2)+x^2 (1-\log (4))+\log (4)}{x \left (2-x-2 x^2\right ) \left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx\\ &=-\left (\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx\right )+\left (2 \log ^3(2)\right ) \int \left (\frac {2}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}+\frac {4-x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}-\frac {\log (16)}{2 x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}\right ) \, dx+\left (2 \log ^4(2)\right ) \int \left (-\frac {2}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}+\frac {4-x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}-\frac {\log (4)}{2 x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}\right ) \, dx\\ &=\left (2 \log ^3(2)\right ) \int \frac {4-x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx-\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx+\left (4 \log ^3(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx+\left (2 \log ^4(2)\right ) \int \frac {4-x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (4 \log ^4(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^4(2) \log (4)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^3(2) \log (16)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx\\ &=-\left (\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx\right )+\left (2 \log ^3(2)\right ) \int \left (\frac {4}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}-\frac {x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}\right ) \, dx+\left (4 \log ^3(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx+\left (2 \log ^4(2)\right ) \int \left (\frac {4}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}-\frac {x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}\right ) \, dx-\left (4 \log ^4(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^4(2) \log (4)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^3(2) \log (16)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx\\ &=-\left (\left (2 \log ^3(2)\right ) \int \frac {x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx\right )-\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx+\left (4 \log ^3(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx+\left (8 \log ^3(2)\right ) \int \frac {1}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx-\left (2 \log ^4(2)\right ) \int \frac {x}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (4 \log ^4(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx+\left (8 \log ^4(2)\right ) \int \frac {1}{\left (-2+x+2 x^2\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^4(2) \log (4)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^3(2) \log (16)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx\\ &=-\left (\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx\right )-\left (2 \log ^3(2)\right ) \int \left (\frac {1-\frac {1}{\sqrt {17}}}{\left (1-\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}+\frac {1+\frac {1}{\sqrt {17}}}{\left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}\right ) \, dx+\left (4 \log ^3(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx+\left (8 \log ^3(2)\right ) \int \left (-\frac {4}{\sqrt {17} \left (-1+\sqrt {17}-4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}-\frac {4}{\sqrt {17} \left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2}\right ) \, dx-\left (2 \log ^4(2)\right ) \int \left (\frac {1-\frac {1}{\sqrt {17}}}{\left (1-\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}+\frac {1+\frac {1}{\sqrt {17}}}{\left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}\right ) \, dx-\left (4 \log ^4(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx+\left (8 \log ^4(2)\right ) \int \left (-\frac {4}{\sqrt {17} \left (-1+\sqrt {17}-4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}-\frac {4}{\sqrt {17} \left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3}\right ) \, dx-\left (\log ^4(2) \log (4)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^3(2) \log (16)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx\\ &=-\left (\left (2 \log ^3(2)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )} \, dx\right )+\left (4 \log ^3(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx-\frac {\left (32 \log ^3(2)\right ) \int \frac {1}{\left (-1+\sqrt {17}-4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx}{\sqrt {17}}-\frac {\left (32 \log ^3(2)\right ) \int \frac {1}{\left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx}{\sqrt {17}}-\frac {1}{17} \left (2 \left (17-\sqrt {17}\right ) \log ^3(2)\right ) \int \frac {1}{\left (1-\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx-\frac {1}{17} \left (2 \left (17+\sqrt {17}\right ) \log ^3(2)\right ) \int \frac {1}{\left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx-\left (4 \log ^4(2)\right ) \int \frac {1}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\frac {\left (32 \log ^4(2)\right ) \int \frac {1}{\left (-1+\sqrt {17}-4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx}{\sqrt {17}}-\frac {\left (32 \log ^4(2)\right ) \int \frac {1}{\left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx}{\sqrt {17}}-\frac {1}{17} \left (2 \left (17-\sqrt {17}\right ) \log ^4(2)\right ) \int \frac {1}{\left (1-\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\frac {1}{17} \left (2 \left (17+\sqrt {17}\right ) \log ^4(2)\right ) \int \frac {1}{\left (1+\sqrt {17}+4 x\right ) \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^4(2) \log (4)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^3} \, dx-\left (\log ^3(2) \log (16)\right ) \int \frac {1}{x \left (-\log (2)+x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.07, size = 46, normalized size = 1.31 \begin {gather*} -\frac {\log ^3(2) \left (\log (2)-2 x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )}{\left (\log (2)-x \log \left (\frac {2}{-2+x+2 x^2}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2*x^2 + 8*x^3)*Log[2]^3*Log[2/(-2 + x + 2*x^2)] + (4*x - 2*x^2 - 4*x^3)*Log[2]^3*Log[2/(-2 + x + 2
*x^2)]^2)/((2 - x - 2*x^2)*Log[2]^3 + (-6*x + 3*x^2 + 6*x^3)*Log[2]^2*Log[2/(-2 + x + 2*x^2)] + (6*x^2 - 3*x^3
- 6*x^4)*Log[2]*Log[2/(-2 + x + 2*x^2)]^2 + (-2*x^3 + x^4 + 2*x^5)*Log[2/(-2 + x + 2*x^2)]^3),x]

[Out]

-((Log[2]^3*(Log[2] - 2*x*Log[2/(-2 + x + 2*x^2)]))/(Log[2] - x*Log[2/(-2 + x + 2*x^2)])^2)

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fricas [B]  time = 0.65, size = 72, normalized size = 2.06 \begin {gather*} \frac {2 \, x \log \relax (2)^{3} \log \left (\frac {2}{2 \, x^{2} + x - 2}\right ) - \log \relax (2)^{4}}{x^{2} \log \left (\frac {2}{2 \, x^{2} + x - 2}\right )^{2} - 2 \, x \log \relax (2) \log \left (\frac {2}{2 \, x^{2} + x - 2}\right ) + \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)*log(2)^3*log(2/(2*x^2+x-2)))/((2*x^5
+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x^4-3*x^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*l
og(2/(2*x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x, algorithm="fricas")

[Out]

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

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giac [C]  time = 0.42, size = 148, normalized size = 4.23 \begin {gather*} -\frac {4 i \, \pi x \log \relax (2)^{3} + 2 \, x \log \relax (2)^{4} - 2 \, x \log \relax (2)^{3} \log \left (2 \, x^{2} + x - 2\right ) - \log \relax (2)^{4}}{4 \, \pi ^{2} x^{2} - 4 i \, \pi x^{2} \log \relax (2) - x^{2} \log \relax (2)^{2} + 4 i \, \pi x^{2} \log \left (2 \, x^{2} + x - 2\right ) + 2 \, x^{2} \log \relax (2) \log \left (2 \, x^{2} + x - 2\right ) - x^{2} \log \left (2 \, x^{2} + x - 2\right )^{2} + 4 i \, \pi x \log \relax (2) + 2 \, x \log \relax (2)^{2} - 2 \, x \log \relax (2) \log \left (2 \, x^{2} + x - 2\right ) - \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)*log(2)^3*log(2/(2*x^2+x-2)))/((2*x^5
+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x^4-3*x^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*l
og(2/(2*x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x, algorithm="giac")

[Out]

-(4*I*pi*x*log(2)^3 + 2*x*log(2)^4 - 2*x*log(2)^3*log(2*x^2 + x - 2) - log(2)^4)/(4*pi^2*x^2 - 4*I*pi*x^2*log(
2) - x^2*log(2)^2 + 4*I*pi*x^2*log(2*x^2 + x - 2) + 2*x^2*log(2)*log(2*x^2 + x - 2) - x^2*log(2*x^2 + x - 2)^2
+ 4*I*pi*x*log(2) + 2*x*log(2)^2 - 2*x*log(2)*log(2*x^2 + x - 2) - log(2)^2)

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maple [A]  time = 0.52, size = 47, normalized size = 1.34

method	result	size
risch	\(-\frac {\left (-2 x \ln \left (\frac {2}{2 x^{2}+x -2}\right )+\ln \relax (2)\right ) \ln \relax (2)^{3}}{\left (-x \ln \left (\frac {2}{2 x^{2}+x -2}\right )+\ln \relax (2)\right )^{2}}\)	\(47\)
norman	\(\frac {2 \ln \relax (2)^{3} x \ln \left (\frac {2}{2 x^{2}+x -2}\right )-\ln \relax (2)^{4}}{\left (-x \ln \left (\frac {2}{2 x^{2}+x -2}\right )+\ln \relax (2)\right )^{2}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3-2*x^2+4*x)*ln(2)^3*ln(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)*ln(2)^3*ln(2/(2*x^2+x-2)))/((2*x^5+x^4-2*x^3
)*ln(2/(2*x^2+x-2))^3+(-6*x^4-3*x^3+6*x^2)*ln(2)*ln(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*ln(2)^2*ln(2/(2*x^2+x-2
))+(-2*x^2-x+2)*ln(2)^3),x,method=_RETURNVERBOSE)

[Out]

-(-2*x*ln(2/(2*x^2+x-2))+ln(2))*ln(2)^3/(-x*ln(2/(2*x^2+x-2))+ln(2))^2

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maxima [B]  time = 0.53, size = 91, normalized size = 2.60 \begin {gather*} \frac {2 \, x \log \relax (2)^{4} - 2 \, x \log \relax (2)^{3} \log \left (2 \, x^{2} + x - 2\right ) - \log \relax (2)^{4}}{x^{2} \log \relax (2)^{2} + x^{2} \log \left (2 \, x^{2} + x - 2\right )^{2} - 2 \, x \log \relax (2)^{2} + \log \relax (2)^{2} - 2 \, {\left (x^{2} \log \relax (2) - x \log \relax (2)\right )} \log \left (2 \, x^{2} + x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-2*x^2+4*x)*log(2)^3*log(2/(2*x^2+x-2))^2+(8*x^3+2*x^2)*log(2)^3*log(2/(2*x^2+x-2)))/((2*x^5
+x^4-2*x^3)*log(2/(2*x^2+x-2))^3+(-6*x^4-3*x^3+6*x^2)*log(2)*log(2/(2*x^2+x-2))^2+(6*x^3+3*x^2-6*x)*log(2)^2*l
og(2/(2*x^2+x-2))+(-2*x^2-x+2)*log(2)^3),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \left (\frac {2}{2\,x^2+x-2}\right )\,{\ln \relax (2)}^3\,\left (8\,x^3+2\,x^2\right )-{\ln \left (\frac {2}{2\,x^2+x-2}\right )}^2\,{\ln \relax (2)}^3\,\left (4\,x^3+2\,x^2-4\,x\right )}{\left (-2\,x^5-x^4+2\,x^3\right )\,{\ln \left (\frac {2}{2\,x^2+x-2}\right )}^3+\ln \relax (2)\,\left (6\,x^4+3\,x^3-6\,x^2\right )\,{\ln \left (\frac {2}{2\,x^2+x-2}\right )}^2-{\ln \relax (2)}^2\,\left (6\,x^3+3\,x^2-6\,x\right )\,\ln \left (\frac {2}{2\,x^2+x-2}\right )+{\ln \relax (2)}^3\,\left (2\,x^2+x-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2/(x + 2*x^2 - 2))*log(2)^3*(2*x^2 + 8*x^3) - log(2/(x + 2*x^2 - 2))^2*log(2)^3*(2*x^2 - 4*x + 4*x^3
))/(log(2)^3*(x + 2*x^2 - 2) - log(2/(x + 2*x^2 - 2))^3*(x^4 - 2*x^3 + 2*x^5) - log(2/(x + 2*x^2 - 2))*log(2)^
2*(3*x^2 - 6*x + 6*x^3) + log(2/(x + 2*x^2 - 2))^2*log(2)*(3*x^3 - 6*x^2 + 6*x^4)),x)

[Out]

-int((log(2/(x + 2*x^2 - 2))*log(2)^3*(2*x^2 + 8*x^3) - log(2/(x + 2*x^2 - 2))^2*log(2)^3*(2*x^2 - 4*x + 4*x^3
))/(log(2)^3*(x + 2*x^2 - 2) - log(2/(x + 2*x^2 - 2))^3*(x^4 - 2*x^3 + 2*x^5) - log(2/(x + 2*x^2 - 2))*log(2)^
2*(3*x^2 - 6*x + 6*x^3) + log(2/(x + 2*x^2 - 2))^2*log(2)*(3*x^3 - 6*x^2 + 6*x^4)), x)

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sympy [B]  time = 0.35, size = 65, normalized size = 1.86 \begin {gather*} \frac {2 x \log {\relax (2 )}^{3} \log {\left (\frac {2}{2 x^{2} + x - 2} \right )} - \log {\relax (2 )}^{4}}{x^{2} \log {\left (\frac {2}{2 x^{2} + x - 2} \right )}^{2} - 2 x \log {\relax (2 )} \log {\left (\frac {2}{2 x^{2} + x - 2} \right )} + \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3-2*x**2+4*x)*ln(2)**3*ln(2/(2*x**2+x-2))**2+(8*x**3+2*x**2)*ln(2)**3*ln(2/(2*x**2+x-2)))/((
2*x**5+x**4-2*x**3)*ln(2/(2*x**2+x-2))**3+(-6*x**4-3*x**3+6*x**2)*ln(2)*ln(2/(2*x**2+x-2))**2+(6*x**3+3*x**2-6
*x)*ln(2)**2*ln(2/(2*x**2+x-2))+(-2*x**2-x+2)*ln(2)**3),x)

[Out]

(2*x*log(2)**3*log(2/(2*x**2 + x - 2)) - log(2)**4)/(x**2*log(2/(2*x**2 + x - 2))**2 - 2*x*log(2)*log(2/(2*x**
2 + x - 2)) + log(2)**2)

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