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

Optimal. Leaf size=18 \[ \log \left (30 x \left (x^2+\frac {2}{-x+\log (4)}\right )\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 3, number of rules used = 2, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {2074, 1587} \begin {gather*} \log \left (-x^3+x^2 \log (4)+2\right )+\log (x)-\log (x-\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - Log[x - Log[4]] + Log[2 - x^3 + x^2*Log[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 {1}{x}+\frac {1}{-x+\log (4)}+\frac {x (3 x-2 \log (4))}{-2+x^3-x^2 \log (4)}\right ) \, dx\\ &=\log (x)-\log (x-\log (4))+\int \frac {x (3 x-2 \log (4))}{-2+x^3-x^2 \log (4)} \, dx\\ &=\log (x)-\log (x-\log (4))+\log \left (2-x^3+x^2 \log (4)\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.05, size = 109, normalized size = 6.06 \begin {gather*} \log (x)+\text {RootSum}\left [\log (16)-2 \text {$\#$1}+\log ^2(4) \text {$\#$1}^2-2 \log (4) \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})+\log ^2(4) \log (x-\text {$\#$1}) \text {$\#$1}-2 \log (4) \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-1+\log ^2(4) \text {$\#$1}-3 \log (4) \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + RootSum[Log[16] - 2*#1 + Log[4]^2*#1^2 - 2*Log[4]*#1^3 + #1^4 & , (Log[x - #1] + Log[4]^2*Log[x - #1]
*#1 - 2*Log[4]*Log[x - #1]*#1^2 + Log[x - #1]*#1^3)/(-1 + Log[4]^2*#1 - 3*Log[4]*#1^2 + 2*#1^3) & ]

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fricas [A]  time = 0.72, size = 25, normalized size = 1.39 \begin {gather*} \log \left (x^{4} - 2 \, x^{3} \log \relax (2) - 2 \, x\right ) - \log \left (x - 2 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(-2*x^4+2*x)*log(2)+x^5-2*x^2),x, algo
rithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 28, normalized size = 1.56 \begin {gather*} \log \left ({\left | x^{3} - 2 \, x^{2} \log \relax (2) - 2 \right |}\right ) - \log \left ({\left | x - 2 \, \log \relax (2) \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(-2*x^4+2*x)*log(2)+x^5-2*x^2),x, algo
rithm="giac")

[Out]

log(abs(x^3 - 2*x^2*log(2) - 2)) - log(abs(x - 2*log(2))) + log(abs(x))

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maple [A]  time = 0.10, size = 26, normalized size = 1.44

method	result	size
default	\(\ln \left (-2 x^{2} \ln \relax (2)+x^{3}-2\right )-\ln \left (x -2 \ln \relax (2)\right )+\ln \relax (x )\)	\(26\)
risch	\(-\ln \left (x -2 \ln \relax (2)\right )+\ln \left (-2 x^{3} \ln \relax (2)+x^{4}-2 x \right )\)	\(26\)
norman	\(-\ln \left (2 \ln \relax (2)-x \right )+\ln \relax (x )+\ln \left (2 x^{2} \ln \relax (2)-x^{3}+2\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^2*ln(2)^2+2*(-6*x^3+2)*ln(2)+3*x^4)/(4*x^3*ln(2)^2+2*(-2*x^4+2*x)*ln(2)+x^5-2*x^2),x,method=_RETURNV
ERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 25, normalized size = 1.39 \begin {gather*} \log \left (x^{3} - 2 \, x^{2} \log \relax (2) - 2\right ) - \log \left (x - 2 \, \log \relax (2)\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(-2*x^4+2*x)*log(2)+x^5-2*x^2),x, algo
rithm="maxima")

[Out]

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

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mupad [B]  time = 0.20, size = 25, normalized size = 1.39 \begin {gather*} \ln \left (x^4-2\,\ln \relax (2)\,x^3-2\,x\right )-\ln \left (x-\ln \relax (4)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^2*log(2)^2 - 2*log(2)*(6*x^3 - 2) + 3*x^4)/(4*x^3*log(2)^2 + 2*log(2)*(2*x - 2*x^4) - 2*x^2 + x^5),x
)

[Out]

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

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sympy [A]  time = 0.54, size = 24, normalized size = 1.33 \begin {gather*} - \log {\left (x - 2 \log {\relax (2 )} \right )} + \log {\left (x^{4} - 2 x^{3} \log {\relax (2 )} - 2 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**2*ln(2)**2+2*(-6*x**3+2)*ln(2)+3*x**4)/(4*x**3*ln(2)**2+2*(-2*x**4+2*x)*ln(2)+x**5-2*x**2),x)

[Out]

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

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