\(\int \frac {-480+60 x^2+80 x^4+(-60+10 x^2+20 x^4) \log (\frac {1}{2} (6-x^2-2 x^4))}{-384+64 x^2+128 x^4+(-96+16 x^2+32 x^4) \log (\frac {1}{2} (6-x^2-2 x^4))+(-6+x^2+2 x^4) \log ^2(\frac {1}{2} (6-x^2-2 x^4))} \, dx\) [5215]

Optimal result

Integrand size = 117, antiderivative size = 22 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\frac {10 x}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \]

[Out]

10/(ln(-x^4-1/2*x^2+3)+8)*x

Rubi [F]

\[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx \]

[In]

Int[(-480 + 60*x^2 + 80*x^4 + (-60 + 10*x^2 + 20*x^4)*Log[(6 - x^2 - 2*x^4)/2])/(-384 + 64*x^2 + 128*x^4 + (-9
6 + 16*x^2 + 32*x^4)*Log[(6 - x^2 - 2*x^4)/2] + (-6 + x^2 + 2*x^4)*Log[(6 - x^2 - 2*x^4)/2]^2),x]

[Out]

-40*Defer[Int][(8 + Log[3 - x^2/2 - x^4])^(-2), x] + (10*I)*Sqrt[2]*Defer[Int][1/((I*Sqrt[2] - x)*(8 + Log[3 -
x^2/2 - x^4])^2), x] + (10*I)*Sqrt[2]*Defer[Int][1/((I*Sqrt[2] + x)*(8 + Log[3 - x^2/2 - x^4])^2), x] + 10*Sq
rt[3]*Defer[Int][1/((Sqrt[3] - Sqrt[2]*x)*(8 + Log[3 - x^2/2 - x^4])^2), x] + 10*Sqrt[3]*Defer[Int][1/((Sqrt[3
] + Sqrt[2]*x)*(8 + Log[3 - x^2/2 - x^4])^2), x] + 10*Defer[Int][(8 + Log[3 - x^2/2 - x^4])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {10 \left (48-6 x^2-8 x^4-\left (-6+x^2+2 x^4\right ) \log \left (3-\frac {x^2}{2}-x^4\right )\right )}{\left (6-x^2-2 x^4\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx \\ & = 10 \int \frac {48-6 x^2-8 x^4-\left (-6+x^2+2 x^4\right ) \log \left (3-\frac {x^2}{2}-x^4\right )}{\left (6-x^2-2 x^4\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx \\ & = 10 \int \left (\frac {2 x^2 \left (1+4 x^2\right )}{\left (6-x^2-2 x^4\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}+\frac {1}{8+\log \left (3-\frac {x^2}{2}-x^4\right )}\right ) \, dx \\ & = 10 \int \frac {1}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \, dx+20 \int \frac {x^2 \left (1+4 x^2\right )}{\left (6-x^2-2 x^4\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx \\ & = 10 \int \frac {1}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \, dx+20 \int \left (-\frac {2}{\left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}+\frac {2}{\left (2+x^2\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}-\frac {3}{\left (-3+2 x^2\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}\right ) \, dx \\ & = 10 \int \frac {1}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \, dx-40 \int \frac {1}{\left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx+40 \int \frac {1}{\left (2+x^2\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx-60 \int \frac {1}{\left (-3+2 x^2\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx \\ & = 10 \int \frac {1}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \, dx-40 \int \frac {1}{\left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx+40 \int \left (\frac {i}{2 \sqrt {2} \left (i \sqrt {2}-x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}+\frac {i}{2 \sqrt {2} \left (i \sqrt {2}+x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}\right ) \, dx-60 \int \left (-\frac {1}{2 \sqrt {3} \left (\sqrt {3}-\sqrt {2} x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}-\frac {1}{2 \sqrt {3} \left (\sqrt {3}+\sqrt {2} x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2}\right ) \, dx \\ & = 10 \int \frac {1}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \, dx-40 \int \frac {1}{\left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx+\left (10 i \sqrt {2}\right ) \int \frac {1}{\left (i \sqrt {2}-x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx+\left (10 i \sqrt {2}\right ) \int \frac {1}{\left (i \sqrt {2}+x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx+\left (10 \sqrt {3}\right ) \int \frac {1}{\left (\sqrt {3}-\sqrt {2} x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx+\left (10 \sqrt {3}\right ) \int \frac {1}{\left (\sqrt {3}+\sqrt {2} x\right ) \left (8+\log \left (3-\frac {x^2}{2}-x^4\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\frac {10 x}{8+\log \left (3-\frac {x^2}{2}-x^4\right )} \]

[In]

Integrate[(-480 + 60*x^2 + 80*x^4 + (-60 + 10*x^2 + 20*x^4)*Log[(6 - x^2 - 2*x^4)/2])/(-384 + 64*x^2 + 128*x^4
+ (-96 + 16*x^2 + 32*x^4)*Log[(6 - x^2 - 2*x^4)/2] + (-6 + x^2 + 2*x^4)*Log[(6 - x^2 - 2*x^4)/2]^2),x]

[Out]

(10*x)/(8 + Log[3 - x^2/2 - x^4])

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {10 x}{\ln \left (-x^{4}-\frac {1}{2} x^{2}+3\right )+8}\)	\(21\)
risch	\(\frac {10 x}{\ln \left (-x^{4}-\frac {1}{2} x^{2}+3\right )+8}\)	\(21\)
parallelrisch	\(\frac {10 x}{\ln \left (-x^{4}-\frac {1}{2} x^{2}+3\right )+8}\)	\(21\)

[In]

int(((20*x^4+10*x^2-60)*ln(-x^4-1/2*x^2+3)+80*x^4+60*x^2-480)/((2*x^4+x^2-6)*ln(-x^4-1/2*x^2+3)^2+(32*x^4+16*x
^2-96)*ln(-x^4-1/2*x^2+3)+128*x^4+64*x^2-384),x,method=_RETURNVERBOSE)

[Out]

10/(ln(-x^4-1/2*x^2+3)+8)*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\frac {10 \, x}{\log \left (-x^{4} - \frac {1}{2} \, x^{2} + 3\right ) + 8} \]

[In]

integrate(((20*x^4+10*x^2-60)*log(-x^4-1/2*x^2+3)+80*x^4+60*x^2-480)/((2*x^4+x^2-6)*log(-x^4-1/2*x^2+3)^2+(32*
x^4+16*x^2-96)*log(-x^4-1/2*x^2+3)+128*x^4+64*x^2-384),x, algorithm="fricas")

[Out]

10*x/(log(-x^4 - 1/2*x^2 + 3) + 8)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\frac {10 x}{\log {\left (- x^{4} - \frac {x^{2}}{2} + 3 \right )} + 8} \]

[In]

integrate(((20*x**4+10*x**2-60)*ln(-x**4-1/2*x**2+3)+80*x**4+60*x**2-480)/((2*x**4+x**2-6)*ln(-x**4-1/2*x**2+3
)**2+(32*x**4+16*x**2-96)*ln(-x**4-1/2*x**2+3)+128*x**4+64*x**2-384),x)

[Out]

10*x/(log(-x**4 - x**2/2 + 3) + 8)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=-\frac {10 \, x}{\log \left (2\right ) - \log \left (x^{2} + 2\right ) - \log \left (-2 \, x^{2} + 3\right ) - 8} \]

[In]

integrate(((20*x^4+10*x^2-60)*log(-x^4-1/2*x^2+3)+80*x^4+60*x^2-480)/((2*x^4+x^2-6)*log(-x^4-1/2*x^2+3)^2+(32*
x^4+16*x^2-96)*log(-x^4-1/2*x^2+3)+128*x^4+64*x^2-384),x, algorithm="maxima")

[Out]

-10*x/(log(2) - log(x^2 + 2) - log(-2*x^2 + 3) - 8)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\frac {10 \, x}{\log \left (-x^{4} - \frac {1}{2} \, x^{2} + 3\right ) + 8} \]

[In]

integrate(((20*x^4+10*x^2-60)*log(-x^4-1/2*x^2+3)+80*x^4+60*x^2-480)/((2*x^4+x^2-6)*log(-x^4-1/2*x^2+3)^2+(32*
x^4+16*x^2-96)*log(-x^4-1/2*x^2+3)+128*x^4+64*x^2-384),x, algorithm="giac")

[Out]

10*x/(log(-x^4 - 1/2*x^2 + 3) + 8)

Mupad [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-480+60 x^2+80 x^4+\left (-60+10 x^2+20 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )}{-384+64 x^2+128 x^4+\left (-96+16 x^2+32 x^4\right ) \log \left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )+\left (-6+x^2+2 x^4\right ) \log ^2\left (\frac {1}{2} \left (6-x^2-2 x^4\right )\right )} \, dx=\frac {10\,x}{\ln \left (-x^4-\frac {x^2}{2}+3\right )+8} \]

[In]

int((log(3 - x^4 - x^2/2)*(10*x^2 + 20*x^4 - 60) + 60*x^2 + 80*x^4 - 480)/(log(3 - x^4 - x^2/2)*(16*x^2 + 32*x
^4 - 96) + log(3 - x^4 - x^2/2)^2*(x^2 + 2*x^4 - 6) + 64*x^2 + 128*x^4 - 384),x)

[Out]

(10*x)/(log(3 - x^4 - x^2/2) + 8)