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

Optimal. Leaf size=23 \[ (2-\log (4)) \log \left (1+x+\frac {\frac {1}{x}+x}{4 x^2}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 34, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2074, 1587} \begin {gather*} (2-\log (4)) \log \left (4 x^4+4 x^3+x^2+1\right )-3 (2-\log (4)) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-3*(2 - Log[4])*Log[x] + (2 - Log[4])*Log[1 + x^2 + 4*x^3 + 4*x^4]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.17 \begin {gather*} -\left ((-2+\log (4)) \left (-3 \log (x)+\log \left (1+x^2+4 x^3+4 x^4\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-2 + Log[4])*(-3*Log[x] + Log[1 + x^2 + 4*x^3 + 4*x^4]))

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fricas [A]  time = 0.67, size = 31, normalized size = 1.35 \begin {gather*} -2 \, {\left (\log \relax (2) - 1\right )} \log \left (4 \, x^{4} + 4 \, x^{3} + x^{2} + 1\right ) + 6 \, {\left (\log \relax (2) - 1\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(log(2) - 1)*log(4*x^4 + 4*x^3 + x^2 + 1) + 6*(log(2) - 1)*log(x)

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giac [A]  time = 0.20, size = 32, normalized size = 1.39 \begin {gather*} -2 \, {\left (\log \relax (2) - 1\right )} \log \left (4 \, x^{4} + 4 \, x^{3} + x^{2} + 1\right ) + 6 \, {\left (\log \relax (2) - 1\right )} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(log(2) - 1)*log(4*x^4 + 4*x^3 + x^2 + 1) + 6*(log(2) - 1)*log(abs(x))

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maple [A]  time = 0.04, size = 30, normalized size = 1.30

method result size
default \(2 \left (\ln \relax (2)-1\right ) \left (3 \ln \relax (x )-\ln \left (4 x^{4}+4 x^{3}+x^{2}+1\right )\right )\) \(30\)
norman \(\left (2-2 \ln \relax (2)\right ) \ln \left (4 x^{4}+4 x^{3}+x^{2}+1\right )+\left (6 \ln \relax (2)-6\right ) \ln \relax (x )\) \(34\)
risch \(-2 \ln \left (4 x^{4}+4 x^{3}+x^{2}+1\right ) \ln \relax (2)+2 \ln \left (4 x^{4}+4 x^{3}+x^{2}+1\right )+6 \ln \relax (2) \ln \relax (x )-6 \ln \relax (x )\) \(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.39, size = 31, normalized size = 1.35 \begin {gather*} -2 \, {\left (\log \relax (2) - 1\right )} \log \left (4 \, x^{4} + 4 \, x^{3} + x^{2} + 1\right ) + 6 \, {\left (\log \relax (2) - 1\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(log(2) - 1)*log(4*x^4 + 4*x^3 + x^2 + 1) + 6*(log(2) - 1)*log(x)

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mupad [B]  time = 4.41, size = 1785, normalized size = 77.61 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(253952*x - 262144*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z
*log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z,
k) - 786432*log(2) + 786432*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*
log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k
)*log(2) - 1048576*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2
+ 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)*x - 101
5808*x*log(2) - 786432*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2
)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)*log
(2)^2 - 196608*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32
*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^2*log(2) +
262144*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*log(2
)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)*log(2)^3 + 1191936*
root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*log(2)^3 + 9
6*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^2*x - 532480*root(z^4 + 8*
z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) -
32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^3*x + 83456*root(z^4 + 8*z^3*log(2) - 8
*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(
2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^4*x + 1523712*x*log(2)^2 - 1015808*x*log(2)^3 + 25395
2*x*log(2)^4 + 98304*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^
2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^2 - 4
096*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*log(2)^3
 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^4 + 1179648*log(2)^2 -
 786432*log(2)^3 + 196608*log(2)^4 + 98304*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 +
 24*z^2 - 96*z*log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(
2)^4 + 16, z, k)^2*log(2)^2 + 1191936*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z
^2 - 96*z*log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4
+ 16, z, k)^2*x*log(2)^2 + 3145728*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2
- 96*z*log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 1
6, z, k)*x*log(2) - 3145728*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*
log(2)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k
)*x*log(2)^2 - 2383872*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2
)^2 + 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^2*x
*log(2) + 1048576*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 +
 32*z*log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)*x*log(2)
^3 + 532480*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*
log(2)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k)^3*x*log(2) + 1
96608)*root(z^4 + 8*z^3*log(2) - 8*z^3 - 48*z^2*log(2) + 24*z^2*log(2)^2 + 24*z^2 - 96*z*log(2)^2 + 32*z*log(2
)^3 + 96*z*log(2) - 32*z - 64*log(2) + 96*log(2)^2 - 64*log(2)^3 + 16*log(2)^4 + 16, z, k), k, 1, 4) - 6*log(x
) + 6*log(2)*log(x)

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sympy [A]  time = 1.01, size = 32, normalized size = 1.39 \begin {gather*} 6 \left (-1 + \log {\relax (2 )}\right ) \log {\relax (x )} - 2 \left (-1 + \log {\relax (2 )}\right ) \log {\left (4 x^{4} + 4 x^{3} + x^{2} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*(-1 + log(2))*log(x) - 2*(-1 + log(2))*log(4*x**4 + 4*x**3 + x**2 + 1)

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