3.15.19 \(\int \frac {4 x^2+7 x^3+x^4-x \log (\log (2))+\log ^2(\log (2))}{4 x^3+4 x^4+x^5+(4 x^2+2 x^3) \log (\log (2))+x \log ^2(\log (2))} \, dx\)

Optimal. Leaf size=24 \[ \log (x)-\frac {5+x}{3+x+\frac {-x+\log (\log (2))}{x}} \]

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Rubi [A]  time = 0.21, antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2074, 618, 206, 638} \begin {gather*} \log (x)-\frac {3 x-\log (\log (2))}{x^2+2 x+\log (\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^2 + 7*x^3 + x^4 - x*Log[Log[2]] + Log[Log[2]]^2)/(4*x^3 + 4*x^4 + x^5 + (4*x^2 + 2*x^3)*Log[Log[2]] +
x*Log[Log[2]]^2),x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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 {3}{2 x+x^2+\log (\log (2))}+\frac {2 (-4 \log (\log (2))-x (3+\log (\log (2))))}{\left (2 x+x^2+\log (\log (2))\right )^2}\right ) \, dx\\ &=\log (x)+2 \int \frac {-4 \log (\log (2))-x (3+\log (\log (2)))}{\left (2 x+x^2+\log (\log (2))\right )^2} \, dx+3 \int \frac {1}{2 x+x^2+\log (\log (2))} \, dx\\ &=\log (x)-\frac {3 x-\log (\log (2))}{2 x+x^2+\log (\log (2))}-3 \int \frac {1}{2 x+x^2+\log (\log (2))} \, dx-6 \operatorname {Subst}\left (\int \frac {1}{-x^2+4 (1-\log (\log (2)))} \, dx,x,2+2 x\right )\\ &=\log (x)-\frac {3 \tanh ^{-1}\left (\frac {1+x}{\sqrt {1-\log (\log (2))}}\right )}{\sqrt {1-\log (\log (2))}}-\frac {3 x-\log (\log (2))}{2 x+x^2+\log (\log (2))}+6 \operatorname {Subst}\left (\int \frac {1}{-x^2+4 (1-\log (\log (2)))} \, dx,x,2+2 x\right )\\ &=\log (x)-\frac {3 x-\log (\log (2))}{2 x+x^2+\log (\log (2))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.96 \begin {gather*} \log (x)+\frac {-3 x+\log (\log (2))}{2 x+x^2+\log (\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 + 7*x^3 + x^4 - x*Log[Log[2]] + Log[Log[2]]^2)/(4*x^3 + 4*x^4 + x^5 + (4*x^2 + 2*x^3)*Log[Log
[2]] + x*Log[Log[2]]^2),x]

[Out]

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

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fricas [A]  time = 0.89, size = 35, normalized size = 1.46 \begin {gather*} \frac {{\left (x^{2} + 2 \, x\right )} \log \relax (x) + {\left (\log \relax (x) + 1\right )} \log \left (\log \relax (2)\right ) - 3 \, x}{x^{2} + 2 \, x + \log \left (\log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(log(2))^2-x*log(log(2))+x^4+7*x^3+4*x^2)/(x*log(log(2))^2+(2*x^3+4*x^2)*log(log(2))+x^5+4*x^4+4
*x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.36, size = 27, normalized size = 1.12 \begin {gather*} -\frac {3 \, x - \log \left (\log \relax (2)\right )}{x^{2} + 2 \, x + \log \left (\log \relax (2)\right )} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(log(2))^2-x*log(log(2))+x^4+7*x^3+4*x^2)/(x*log(log(2))^2+(2*x^3+4*x^2)*log(log(2))+x^5+4*x^4+4
*x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 24, normalized size = 1.00

method	result	size
norman	\(\frac {-3 x +\ln \left (\ln \relax (2)\right )}{x^{2}+\ln \left (\ln \relax (2)\right )+2 x}+\ln \relax (x )\)	\(24\)
risch	\(\frac {-3 x +\ln \left (\ln \relax (2)\right )}{x^{2}+\ln \left (\ln \relax (2)\right )+2 x}+\ln \relax (x )\)	\(24\)
default	\(-\frac {3 x -\ln \left (\ln \relax (2)\right )}{x^{2}+\ln \left (\ln \relax (2)\right )+2 x}+\ln \relax (x )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(ln(2))^2-x*ln(ln(2))+x^4+7*x^3+4*x^2)/(x*ln(ln(2))^2+(2*x^3+4*x^2)*ln(ln(2))+x^5+4*x^4+4*x^3),x,method
=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.55, size = 26, normalized size = 1.08 \begin {gather*} -\frac {3 \, x - \log \left (\log \relax (2)\right )}{x^{2} + 2 \, x + \log \left (\log \relax (2)\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(log(2))^2-x*log(log(2))+x^4+7*x^3+4*x^2)/(x*log(log(2))^2+(2*x^3+4*x^2)*log(log(2))+x^5+4*x^4+4
*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 26, normalized size = 1.08 \begin {gather*} \ln \relax (x)-\frac {3\,x-\ln \left (\ln \relax (2)\right )}{x^2+2\,x+\ln \left (\ln \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(2))^2 - x*log(log(2)) + 4*x^2 + 7*x^3 + x^4)/(log(log(2))*(4*x^2 + 2*x^3) + x*log(log(2))^2 + 4*x
^3 + 4*x^4 + x^5),x)

[Out]

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

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sympy [A]  time = 0.69, size = 22, normalized size = 0.92 \begin {gather*} \frac {- 3 x + \log {\left (\log {\relax (2 )} \right )}}{x^{2} + 2 x + \log {\left (\log {\relax (2 )} \right )}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(ln(2))**2-x*ln(ln(2))+x**4+7*x**3+4*x**2)/(x*ln(ln(2))**2+(2*x**3+4*x**2)*ln(ln(2))+x**5+4*x**4+
4*x**3),x)

[Out]

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

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