3.51.93 \(\int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+(3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}) \log ^2(2)+(3 x+3 x^2-3 x^6) \log ^4(2)+\log ^6(2)} \, dx\)

Optimal. Leaf size=22 \[ \frac {3}{\left (1+\frac {1}{x}-x^4+\frac {\log ^2(2)}{x^2}\right )^2} \]

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Rubi [A]  time = 0.66, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1594, 6688, 12, 1588} \begin {gather*} \frac {3 x^4}{\left (-x^6+x^2+x+\log ^2(2)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6*x^4 + 24*x^9 + 12*x^3*Log[2]^2)/(x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*x^14 -
x^18 + (3*x^2 + 6*x^3 + 3*x^4 - 6*x^7 - 6*x^8 + 3*x^12)*Log[2]^2 + (3*x + 3*x^2 - 3*x^6)*Log[2]^4 + Log[2]^6)
,x]

[Out]

(3*x^4)/(x + x^2 - x^6 + Log[2]^2)^2

Rule 12

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

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (6 x+24 x^6+12 \log ^2(2)\right )}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx\\ &=\int \frac {6 x^3 \left (x+4 x^6+2 \log ^2(2)\right )}{\left (x+x^2-x^6+\log ^2(2)\right )^3} \, dx\\ &=6 \int \frac {x^3 \left (x+4 x^6+2 \log ^2(2)\right )}{\left (x+x^2-x^6+\log ^2(2)\right )^3} \, dx\\ &=\frac {3 x^4}{\left (x+x^2-x^6+\log ^2(2)\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 21, normalized size = 0.95 \begin {gather*} \frac {3 x^4}{\left (x+x^2-x^6+\log ^2(2)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^4 + 24*x^9 + 12*x^3*Log[2]^2)/(x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*
x^14 - x^18 + (3*x^2 + 6*x^3 + 3*x^4 - 6*x^7 - 6*x^8 + 3*x^12)*Log[2]^2 + (3*x + 3*x^2 - 3*x^6)*Log[2]^4 + Log
[2]^6),x]

[Out]

(3*x^4)/(x + x^2 - x^6 + Log[2]^2)^2

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fricas [B]  time = 1.02, size = 54, normalized size = 2.45 \begin {gather*} \frac {3 \, x^{4}}{x^{12} - 2 \, x^{8} - 2 \, x^{7} + x^{4} + \log \relax (2)^{4} + 2 \, x^{3} - 2 \, {\left (x^{6} - x^{2} - x\right )} \log \relax (2)^{2} + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log(2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3
+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3*x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x, algorithm="fricas")

[Out]

3*x^4/(x^12 - 2*x^8 - 2*x^7 + x^4 + log(2)^4 + 2*x^3 - 2*(x^6 - x^2 - x)*log(2)^2 + x^2)

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giac [A]  time = 0.51, size = 25, normalized size = 1.14 \begin {gather*} \frac {3 \, x^{4}}{{\left (x^{6} - x^{2} - \log \relax (2)^{2} - x\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log(2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3
+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3*x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x, algorithm="giac")

[Out]

3*x^4/(x^6 - x^2 - log(2)^2 - x)^2

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maple [A]  time = 0.36, size = 22, normalized size = 1.00

method	result	size
default	\(\frac {3 x^{4}}{\left (-x^{6}+\ln \relax (2)^{2}+x^{2}+x \right )^{2}}\)	\(22\)
norman	\(\frac {3 x^{4}}{\left (-x^{6}+\ln \relax (2)^{2}+x^{2}+x \right )^{2}}\)	\(22\)
gosper	\(\frac {3 x^{4}}{x^{12}-2 x^{6} \ln \relax (2)^{2}-2 x^{8}-2 x^{7}+\ln \relax (2)^{4}+2 x^{2} \ln \relax (2)^{2}+x^{4}+2 x \ln \relax (2)^{2}+2 x^{3}+x^{2}}\)	\(62\)
risch	\(\frac {3 x^{4}}{x^{12}-2 x^{6} \ln \relax (2)^{2}-2 x^{8}-2 x^{7}+\ln \relax (2)^{4}+2 x^{2} \ln \relax (2)^{2}+x^{4}+2 x \ln \relax (2)^{2}+2 x^{3}+x^{2}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^3*ln(2)^2+24*x^9+6*x^4)/(ln(2)^6+(-3*x^6+3*x^2+3*x)*ln(2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3+3*x^2)*l
n(2)^2-x^18+3*x^14+3*x^13-3*x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x,method=_RETURNVERBOSE)

[Out]

3*x^4/(-x^6+ln(2)^2+x^2+x)^2

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maxima [B]  time = 0.35, size = 61, normalized size = 2.77 \begin {gather*} \frac {3 \, x^{4}}{x^{12} - 2 \, x^{8} - 2 \, x^{6} \log \relax (2)^{2} - 2 \, x^{7} + x^{4} + \log \relax (2)^{4} + {\left (2 \, \log \relax (2)^{2} + 1\right )} x^{2} + 2 \, x^{3} + 2 \, x \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^3*log(2)^2+24*x^9+6*x^4)/(log(2)^6+(-3*x^6+3*x^2+3*x)*log(2)^4+(3*x^12-6*x^8-6*x^7+3*x^4+6*x^3
+3*x^2)*log(2)^2-x^18+3*x^14+3*x^13-3*x^10-6*x^9-3*x^8+x^6+3*x^5+3*x^4+x^3),x, algorithm="maxima")

[Out]

3*x^4/(x^12 - 2*x^8 - 2*x^6*log(2)^2 - 2*x^7 + x^4 + log(2)^4 + (2*log(2)^2 + 1)*x^2 + 2*x^3 + 2*x*log(2)^2)

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mupad [B]  time = 3.40, size = 21, normalized size = 0.95 \begin {gather*} \frac {3\,x^4}{{\left (-x^6+x^2+x+{\ln \relax (2)}^2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^3*log(2)^2 + 6*x^4 + 24*x^9)/(log(2)^6 + log(2)^4*(3*x + 3*x^2 - 3*x^6) + log(2)^2*(3*x^2 + 6*x^3 +
3*x^4 - 6*x^7 - 6*x^8 + 3*x^12) + x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*x^14 - x^18)
,x)

[Out]

(3*x^4)/(x + log(2)^2 + x^2 - x^6)^2

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sympy [B]  time = 9.61, size = 61, normalized size = 2.77 \begin {gather*} \frac {3 x^{4}}{x^{12} - 2 x^{8} - 2 x^{7} - 2 x^{6} \log {\relax (2 )}^{2} + x^{4} + 2 x^{3} + x^{2} \left (2 \log {\relax (2 )}^{2} + 1\right ) + 2 x \log {\relax (2 )}^{2} + \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**3*ln(2)**2+24*x**9+6*x**4)/(ln(2)**6+(-3*x**6+3*x**2+3*x)*ln(2)**4+(3*x**12-6*x**8-6*x**7+3*x
**4+6*x**3+3*x**2)*ln(2)**2-x**18+3*x**14+3*x**13-3*x**10-6*x**9-3*x**8+x**6+3*x**5+3*x**4+x**3),x)

[Out]

3*x**4/(x**12 - 2*x**8 - 2*x**7 - 2*x**6*log(2)**2 + x**4 + 2*x**3 + x**2*(2*log(2)**2 + 1) + 2*x*log(2)**2 +
log(2)**4)

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