3.29.59 \(\int -\frac {2 x}{1+2 x^2+x^4+(2 x^2+2 x^4) \log (4)+x^4 \log ^2(4)} \, dx\)

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

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Rubi [A]  time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6, 12, 1989, 28, 261} \begin {gather*} \frac {1}{x^2 (1+\log (4))^2+1+\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x)/(1 + 2*x^2 + x^4 + (2*x^2 + 2*x^4)*Log[4] + x^4*Log[4]^2),x]

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1989

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] &&
TrinomialQ[u, x] &&  !TrinomialMatchQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.19 \begin {gather*} \frac {1}{(1+\log (4)) \left (1+x^2 (1+\log (4))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x)/(1 + 2*x^2 + x^4 + (2*x^2 + 2*x^4)*Log[4] + x^4*Log[4]^2),x]

[Out]

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

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fricas [B]  time = 0.84, size = 27, normalized size = 1.69 \begin {gather*} \frac {1}{4 \, x^{2} \log \relax (2)^{2} + x^{2} + 2 \, {\left (2 \, x^{2} + 1\right )} \log \relax (2) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x/(4*x^4*log(2)^2+2*(2*x^4+2*x^2)*log(2)+x^4+2*x^2+1),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.33, size = 23, normalized size = 1.44 \begin {gather*} \frac {1}{{\left (2 \, x^{2} \log \relax (2) + x^{2} + 1\right )} {\left (2 \, \log \relax (2) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x/(4*x^4*log(2)^2+2*(2*x^4+2*x^2)*log(2)+x^4+2*x^2+1),x, algorithm="giac")

[Out]

1/((2*x^2*log(2) + x^2 + 1)*(2*log(2) + 1))

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maple [A]  time = 0.06, size = 20, normalized size = 1.25

method	result	size
norman	\(-\frac {x^{2}}{2 x^{2} \ln \relax (2)+x^{2}+1}\)	\(20\)
gosper	\(\frac {1}{\left (2 x^{2} \ln \relax (2)+x^{2}+1\right ) \left (1+2 \ln \relax (2)\right )}\)	\(24\)
default	\(\frac {1}{\left (2 x^{2} \ln \relax (2)+x^{2}+1\right ) \left (1+2 \ln \relax (2)\right )}\)	\(24\)
risch	\(\frac {1}{2 \left (1+2 \ln \relax (2)\right ) \left (x^{2} \ln \relax (2)+\frac {x^{2}}{2}+\frac {1}{2}\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2*x/(4*x^4*ln(2)^2+2*(2*x^4+2*x^2)*ln(2)+x^4+2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

-x^2/(2*x^2*ln(2)+x^2+1)

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maxima [A]  time = 0.38, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{{\left (4 \, \log \relax (2)^{2} + 4 \, \log \relax (2) + 1\right )} x^{2} + 2 \, \log \relax (2) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x/(4*x^4*log(2)^2+2*(2*x^4+2*x^2)*log(2)+x^4+2*x^2+1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.01, size = 702, normalized size = 43.88 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((x^2*(16*log(16) + 64*log(2)^2*log(16) + 64*log(2)^2 + 128*log(2)^4 + 8*log(16)^2 + 8) + (32*log(16) +
x^2*(40*log(16) + 192*log(2)^2*log(16) + 128*log(2)^4*log(16) + 128*log(2)^2 + 256*log(2)^4 + 32*log(16)^2 +
8*log(16)^3 + 64*log(2)^2*log(16)^2 + 16) + 128*log(2)^2*log(16) + 128*log(2)^2 + 256*log(2)^4 + 16*log(16)^2
+ 16)/((4*log(2) + log(16))^(1/2)*(log(16) - 4*log(2))^(1/2)))*1i)/((4*log(2) + log(16))^(1/2)*(log(16) - 4*lo
g(2))^(1/2)) + ((x^2*(16*log(16) + 64*log(2)^2*log(16) + 64*log(2)^2 + 128*log(2)^4 + 8*log(16)^2 + 8) - (32*l
og(16) + x^2*(40*log(16) + 192*log(2)^2*log(16) + 128*log(2)^4*log(16) + 128*log(2)^2 + 256*log(2)^4 + 32*log(
16)^2 + 8*log(16)^3 + 64*log(2)^2*log(16)^2 + 16) + 128*log(2)^2*log(16) + 128*log(2)^2 + 256*log(2)^4 + 16*lo
g(16)^2 + 16)/((4*log(2) + log(16))^(1/2)*(log(16) - 4*log(2))^(1/2)))*1i)/((4*log(2) + log(16))^(1/2)*(log(16
) - 4*log(2))^(1/2)))/((x^2*(16*log(16) + 64*log(2)^2*log(16) + 64*log(2)^2 + 128*log(2)^4 + 8*log(16)^2 + 8)
+ (32*log(16) + x^2*(40*log(16) + 192*log(2)^2*log(16) + 128*log(2)^4*log(16) + 128*log(2)^2 + 256*log(2)^4 +
32*log(16)^2 + 8*log(16)^3 + 64*log(2)^2*log(16)^2 + 16) + 128*log(2)^2*log(16) + 128*log(2)^2 + 256*log(2)^4
+ 16*log(16)^2 + 16)/((4*log(2) + log(16))^(1/2)*(log(16) - 4*log(2))^(1/2)))/((4*log(2) + log(16))^(1/2)*(log
(16) - 4*log(2))^(1/2)) - (x^2*(16*log(16) + 64*log(2)^2*log(16) + 64*log(2)^2 + 128*log(2)^4 + 8*log(16)^2 +
8) - (32*log(16) + x^2*(40*log(16) + 192*log(2)^2*log(16) + 128*log(2)^4*log(16) + 128*log(2)^2 + 256*log(2)^4
+ 32*log(16)^2 + 8*log(16)^3 + 64*log(2)^2*log(16)^2 + 16) + 128*log(2)^2*log(16) + 128*log(2)^2 + 256*log(2)
^4 + 16*log(16)^2 + 16)/((4*log(2) + log(16))^(1/2)*(log(16) - 4*log(2))^(1/2)))/((4*log(2) + log(16))^(1/2)*(
log(16) - 4*log(2))^(1/2))))*2i)/((4*log(2) + log(16))^(1/2)*(log(16) - 4*log(2))^(1/2))

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sympy [A]  time = 0.25, size = 24, normalized size = 1.50 \begin {gather*} \frac {2}{x^{2} \left (2 + 8 \log {\relax (2 )}^{2} + 8 \log {\relax (2 )}\right ) + 2 + 4 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x/(4*x**4*ln(2)**2+2*(2*x**4+2*x**2)*ln(2)+x**4+2*x**2+1),x)

[Out]

2/(x**2*(2 + 8*log(2)**2 + 8*log(2)) + 2 + 4*log(2))

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