\(\int \frac {-4050+(-180-90 x) \log ^2(2)+(48+88 x+60 x^2+18 x^3+2 x^4) \log ^4(2)}{(8+12 x+6 x^2+x^3) \log ^4(2)} \, dx\) [3236]

Optimal result

Integrand size = 57, antiderivative size = 18 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=-3+\left (3+x+\frac {45}{(2+x) \log ^2(2)}\right )^2 \]

[Out]

(3+45/(2+x)/ln(2)^2+x)^2-3

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {12, 2099} \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=x^2+6 x+\frac {2025}{(x+2)^2 \log ^4(2)}+\frac {90}{(x+2) \log ^2(2)} \]

[In]

Int[(-4050 + (-180 - 90*x)*Log[2]^2 + (48 + 88*x + 60*x^2 + 18*x^3 + 2*x^4)*Log[2]^4)/((8 + 12*x + 6*x^2 + x^3
)*Log[2]^4),x]

[Out]

6*x + x^2 + 2025/((2 + x)^2*Log[2]^4) + 90/((2 + x)*Log[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 2099

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{align*} \text {integral}& = \frac {\int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{8+12 x+6 x^2+x^3} \, dx}{\log ^4(2)} \\ & = \frac {\int \left (-\frac {4050}{(2+x)^3}-\frac {90 \log ^2(2)}{(2+x)^2}+6 \log ^4(2)+2 x \log ^4(2)\right ) \, dx}{\log ^4(2)} \\ & = 6 x+x^2+\frac {2025}{(2+x)^2 \log ^4(2)}+\frac {90}{(2+x) \log ^2(2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=6 x+x^2+\frac {2025}{(2+x)^2 \log ^4(2)}+\frac {90}{(2+x) \log ^2(2)} \]

[In]

Integrate[(-4050 + (-180 - 90*x)*Log[2]^2 + (48 + 88*x + 60*x^2 + 18*x^3 + 2*x^4)*Log[2]^4)/((8 + 12*x + 6*x^2
+ x^3)*Log[2]^4),x]

[Out]

6*x + x^2 + 2025/((2 + x)^2*Log[2]^4) + 90/((2 + x)*Log[2]^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).

Time = 0.50 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11

method	result	size
risch	\(x^{2}+6 x +\frac {90 x \ln \left (2\right )^{2}+180 \ln \left (2\right )^{2}+2025}{\ln \left (2\right )^{4} \left (x^{2}+4 x +4\right )}\)	\(38\)
default	\(\frac {x^{2} \ln \left (2\right )^{4}+6 x \ln \left (2\right )^{4}+\frac {90 \ln \left (2\right )^{2}}{2+x}+\frac {2025}{\left (2+x \right )^{2}}}{\ln \left (2\right )^{4}}\)	\(40\)
gosper	\(\frac {\ln \left (2\right )^{4} x^{4}+10 x^{3} \ln \left (2\right )^{4}-88 x \ln \left (2\right )^{4}-112 \ln \left (2\right )^{4}+90 x \ln \left (2\right )^{2}+180 \ln \left (2\right )^{2}+2025}{\ln \left (2\right )^{4} \left (x^{2}+4 x +4\right )}\)	\(61\)
parallelrisch	\(\frac {\ln \left (2\right )^{4} x^{4}+10 x^{3} \ln \left (2\right )^{4}-88 x \ln \left (2\right )^{4}-112 \ln \left (2\right )^{4}+90 x \ln \left (2\right )^{2}+180 \ln \left (2\right )^{2}+2025}{\ln \left (2\right )^{4} \left (x^{2}+4 x +4\right )}\)	\(61\)
norman	\(\frac {x^{4} \ln \left (2\right )^{3}-2 \ln \left (2\right ) \left (44 \ln \left (2\right )^{2}-45\right ) x +10 x^{3} \ln \left (2\right )^{3}-\frac {112 \ln \left (2\right )^{4}-180 \ln \left (2\right )^{2}-2025}{\ln \left (2\right )}}{\left (2+x \right )^{2} \ln \left (2\right )^{3}}\)	\(62\)

[In]

int(((2*x^4+18*x^3+60*x^2+88*x+48)*ln(2)^4+(-90*x-180)*ln(2)^2-4050)/(x^3+6*x^2+12*x+8)/ln(2)^4,x,method=_RETU
RNVERBOSE)

[Out]

x^2+6*x+1/ln(2)^4*(90*x*ln(2)^2+180*ln(2)^2+2025)/(x^2+4*x+4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.67 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=\frac {{\left (x^{4} + 10 \, x^{3} + 28 \, x^{2} + 24 \, x\right )} \log \left (2\right )^{4} + 90 \, {\left (x + 2\right )} \log \left (2\right )^{2} + 2025}{{\left (x^{2} + 4 \, x + 4\right )} \log \left (2\right )^{4}} \]

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="fricas")

[Out]

((x^4 + 10*x^3 + 28*x^2 + 24*x)*log(2)^4 + 90*(x + 2)*log(2)^2 + 2025)/((x^2 + 4*x + 4)*log(2)^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).

Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=x^{2} + 6 x + \frac {90 x \log {\left (2 \right )}^{2} + 180 \log {\left (2 \right )}^{2} + 2025}{x^{2} \log {\left (2 \right )}^{4} + 4 x \log {\left (2 \right )}^{4} + 4 \log {\left (2 \right )}^{4}} \]

[In]

integrate(((2*x**4+18*x**3+60*x**2+88*x+48)*ln(2)**4+(-90*x-180)*ln(2)**2-4050)/(x**3+6*x**2+12*x+8)/ln(2)**4,
x)

[Out]

x**2 + 6*x + (90*x*log(2)**2 + 180*log(2)**2 + 2025)/(x**2*log(2)**4 + 4*x*log(2)**4 + 4*log(2)**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.67 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=\frac {x^{2} \log \left (2\right )^{4} + 6 \, x \log \left (2\right )^{4} + \frac {45 \, {\left (2 \, x \log \left (2\right )^{2} + 4 \, \log \left (2\right )^{2} + 45\right )}}{x^{2} + 4 \, x + 4}}{\log \left (2\right )^{4}} \]

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="maxima")

[Out]

(x^2*log(2)^4 + 6*x*log(2)^4 + 45*(2*x*log(2)^2 + 4*log(2)^2 + 45)/(x^2 + 4*x + 4))/log(2)^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (18) = 36\).

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=\frac {x^{2} \log \left (2\right )^{4} + 6 \, x \log \left (2\right )^{4} + \frac {45 \, {\left (2 \, x \log \left (2\right )^{2} + 4 \, \log \left (2\right )^{2} + 45\right )}}{{\left (x + 2\right )}^{2}}}{\log \left (2\right )^{4}} \]

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="giac")

[Out]

(x^2*log(2)^4 + 6*x*log(2)^4 + 45*(2*x*log(2)^2 + 4*log(2)^2 + 45)/(x + 2)^2)/log(2)^4

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {-4050+(-180-90 x) \log ^2(2)+\left (48+88 x+60 x^2+18 x^3+2 x^4\right ) \log ^4(2)}{\left (8+12 x+6 x^2+x^3\right ) \log ^4(2)} \, dx=\frac {28\,x^2\,{\ln \left (2\right )}^4+10\,x^3\,{\ln \left (2\right )}^4+x^4\,{\ln \left (2\right )}^4+90\,x\,{\ln \left (2\right )}^2+24\,x\,{\ln \left (2\right )}^4+180\,{\ln \left (2\right )}^2+2025}{{\ln \left (2\right )}^4\,{\left (x+2\right )}^2} \]

[In]

int(-(log(2)^2*(90*x + 180) - log(2)^4*(88*x + 60*x^2 + 18*x^3 + 2*x^4 + 48) + 4050)/(log(2)^4*(12*x + 6*x^2 +
x^3 + 8)),x)

[Out]

(28*x^2*log(2)^4 + 10*x^3*log(2)^4 + x^4*log(2)^4 + 90*x*log(2)^2 + 24*x*log(2)^4 + 180*log(2)^2 + 2025)/(log(
2)^4*(x + 2)^2)