3.83.53 \(\int \frac {(16-40 x-4 x^2) \log (16)}{(-5-6 x+4 x^2+x^3) \log (4) \log ^3(\frac {1+2 x-x^2-2 x^3+x^4}{50+20 x+2 x^2})} \, dx\)

Optimal. Leaf size=29 \[ \frac {\log (16)}{\log (4) \log ^2\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \]

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Rubi [F]  time = 0.79, 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 (16-40 x-4 x^2\right ) \log (16)}{\left (-5-6 x+4 x^2+x^3\right ) \log (4) \log ^3\left (\frac {1+2 x-x^2-2 x^3+x^4}{50+20 x+2 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((16 - 40*x - 4*x^2)*Log[16])/((-5 - 6*x + 4*x^2 + x^3)*Log[4]*Log[(1 + 2*x - x^2 - 2*x^3 + x^4)/(50 + 20*
x + 2*x^2)]^3),x]

[Out]

(-8*Log[16]*Defer[Int][1/((1 + Sqrt[5] - 2*x)*Log[(1 + x - x^2)^2/(2*(5 + x)^2)]^3), x])/(Sqrt[5]*Log[4]) + (4
*Log[16]*Defer[Int][1/((5 + x)*Log[(1 + x - x^2)^2/(2*(5 + x)^2)]^3), x])/Log[4] - (8*(5 + Sqrt[5])*Log[16]*De
fer[Int][1/((-1 - Sqrt[5] + 2*x)*Log[(1 + x - x^2)^2/(2*(5 + x)^2)]^3), x])/(5*Log[4]) - (8*Log[16]*Defer[Int]
[1/((-1 + Sqrt[5] + 2*x)*Log[(1 + x - x^2)^2/(2*(5 + x)^2)]^3), x])/(Sqrt[5]*Log[4]) - (8*(5 - Sqrt[5])*Log[16
]*Defer[Int][1/((-1 + Sqrt[5] + 2*x)*Log[(1 + x - x^2)^2/(2*(5 + x)^2)]^3), x])/(5*Log[4])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\log (16) \int \frac {16-40 x-4 x^2}{\left (-5-6 x+4 x^2+x^3\right ) \log ^3\left (\frac {1+2 x-x^2-2 x^3+x^4}{50+20 x+2 x^2}\right )} \, dx}{\log (4)}\\ &=\frac {\log (16) \int \left (\frac {4}{(5+x) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}-\frac {4 (-1+2 x)}{\left (-1-x+x^2\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}\right ) \, dx}{\log (4)}\\ &=\frac {(4 \log (16)) \int \frac {1}{(5+x) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}-\frac {(4 \log (16)) \int \frac {-1+2 x}{\left (-1-x+x^2\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}\\ &=-\frac {(4 \log (16)) \int \left (-\frac {1}{\left (-1-x+x^2\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}+\frac {2 x}{\left (-1-x+x^2\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}\right ) \, dx}{\log (4)}+\frac {(4 \log (16)) \int \frac {1}{(5+x) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}\\ &=\frac {(4 \log (16)) \int \frac {1}{(5+x) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}+\frac {(4 \log (16)) \int \frac {1}{\left (-1-x+x^2\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}-\frac {(8 \log (16)) \int \frac {x}{\left (-1-x+x^2\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}\\ &=\frac {(4 \log (16)) \int \left (-\frac {2}{\sqrt {5} \left (1+\sqrt {5}-2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}-\frac {2}{\sqrt {5} \left (-1+\sqrt {5}+2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}\right ) \, dx}{\log (4)}+\frac {(4 \log (16)) \int \frac {1}{(5+x) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}-\frac {(8 \log (16)) \int \left (\frac {1+\frac {1}{\sqrt {5}}}{\left (-1-\sqrt {5}+2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}+\frac {1-\frac {1}{\sqrt {5}}}{\left (-1+\sqrt {5}+2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )}\right ) \, dx}{\log (4)}\\ &=\frac {(4 \log (16)) \int \frac {1}{(5+x) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\log (4)}-\frac {(8 \log (16)) \int \frac {1}{\left (1+\sqrt {5}-2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\sqrt {5} \log (4)}-\frac {(8 \log (16)) \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{\sqrt {5} \log (4)}-\frac {\left (8 \left (5-\sqrt {5}\right ) \log (16)\right ) \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{5 \log (4)}-\frac {\left (8 \left (5+\sqrt {5}\right ) \log (16)\right ) \int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \log ^3\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \, dx}{5 \log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 29, normalized size = 1.00 \begin {gather*} \frac {\log (16)}{\log (4) \log ^2\left (\frac {\left (1+x-x^2\right )^2}{2 (5+x)^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((16 - 40*x - 4*x^2)*Log[16])/((-5 - 6*x + 4*x^2 + x^3)*Log[4]*Log[(1 + 2*x - x^2 - 2*x^3 + x^4)/(50
+ 20*x + 2*x^2)]^3),x]

[Out]

Log[16]/(Log[4]*Log[(1 + x - x^2)^2/(2*(5 + x)^2)]^2)

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fricas [A]  time = 0.60, size = 35, normalized size = 1.21 \begin {gather*} \frac {2}{\log \left (\frac {x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}{2 \, {\left (x^{2} + 10 \, x + 25\right )}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(-4*x^2-40*x+16)/(x^3+4*x^2-6*x-5)/log((x^4-2*x^3-x^2+2*x+1)/(2*x^2+20*x+50))^3,x, algorithm="fric
as")

[Out]

2/log(1/2*(x^4 - 2*x^3 - x^2 + 2*x + 1)/(x^2 + 10*x + 25))^2

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giac [B]  time = 0.26, size = 121, normalized size = 4.17 \begin {gather*} \frac {2 \, {\left (x^{2} + 10 \, x - 4\right )}}{x^{2} \log \left (\frac {x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}{2 \, {\left (x^{2} + 10 \, x + 25\right )}}\right )^{2} + 10 \, x \log \left (\frac {x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}{2 \, {\left (x^{2} + 10 \, x + 25\right )}}\right )^{2} - 4 \, \log \left (\frac {x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}{2 \, {\left (x^{2} + 10 \, x + 25\right )}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(-4*x^2-40*x+16)/(x^3+4*x^2-6*x-5)/log((x^4-2*x^3-x^2+2*x+1)/(2*x^2+20*x+50))^3,x, algorithm="giac
")

[Out]

2*(x^2 + 10*x - 4)/(x^2*log(1/2*(x^4 - 2*x^3 - x^2 + 2*x + 1)/(x^2 + 10*x + 25))^2 + 10*x*log(1/2*(x^4 - 2*x^3
- x^2 + 2*x + 1)/(x^2 + 10*x + 25))^2 - 4*log(1/2*(x^4 - 2*x^3 - x^2 + 2*x + 1)/(x^2 + 10*x + 25))^2)

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maple [A]  time = 0.06, size = 37, normalized size = 1.28

method	result	size
norman	\(\frac {2}{\ln \left (\frac {x^{4}-2 x^{3}-x^{2}+2 x +1}{2 x^{2}+20 x +50}\right )^{2}}\)	\(37\)
risch	\(\frac {2}{\ln \left (\frac {x^{4}-2 x^{3}-x^{2}+2 x +1}{2 x^{2}+20 x +50}\right )^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*(-4*x^2-40*x+16)/(x^3+4*x^2-6*x-5)/ln((x^4-2*x^3-x^2+2*x+1)/(2*x^2+20*x+50))^3,x,method=_RETURNVERBOSE)

[Out]

2/ln((x^4-2*x^3-x^2+2*x+1)/(2*x^2+20*x+50))^2

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maxima [B]  time = 0.49, size = 58, normalized size = 2.00 \begin {gather*} \frac {2}{\log \relax (2)^{2} - 4 \, {\left (\log \relax (2) + 2 \, \log \left (x + 5\right )\right )} \log \left (x^{2} - x - 1\right ) + 4 \, \log \left (x^{2} - x - 1\right )^{2} + 4 \, \log \relax (2) \log \left (x + 5\right ) + 4 \, \log \left (x + 5\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(-4*x^2-40*x+16)/(x^3+4*x^2-6*x-5)/log((x^4-2*x^3-x^2+2*x+1)/(2*x^2+20*x+50))^3,x, algorithm="maxi
ma")

[Out]

2/(log(2)^2 - 4*(log(2) + 2*log(x + 5))*log(x^2 - x - 1) + 4*log(x^2 - x - 1)^2 + 4*log(2)*log(x + 5) + 4*log(
x + 5)^2)

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mupad [B]  time = 0.68, size = 36, normalized size = 1.24 \begin {gather*} \frac {2}{{\ln \left (\frac {x^4-2\,x^3-x^2+2\,x+1}{2\,x^2+20\,x+50}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((80*x + 8*x^2 - 32)/(log((2*x - x^2 - 2*x^3 + x^4 + 1)/(20*x + 2*x^2 + 50))^3*(6*x - 4*x^2 - x^3 + 5)),x)

[Out]

2/log((2*x - x^2 - 2*x^3 + x^4 + 1)/(20*x + 2*x^2 + 50))^2

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sympy [A]  time = 0.21, size = 31, normalized size = 1.07 \begin {gather*} \frac {2}{\log {\left (\frac {x^{4} - 2 x^{3} - x^{2} + 2 x + 1}{2 x^{2} + 20 x + 50} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(-4*x**2-40*x+16)/(x**3+4*x**2-6*x-5)/ln((x**4-2*x**3-x**2+2*x+1)/(2*x**2+20*x+50))**3,x)

[Out]

2/log((x**4 - 2*x**3 - x**2 + 2*x + 1)/(2*x**2 + 20*x + 50))**2

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