3.13.93 \(\int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+(6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}) \log (2)+(1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx\)

Optimal. Leaf size=26 \[ \left (2 x+\frac {1}{\left (x-x^2\right )^2}\right )^2 \left (x+\frac {\log (2)}{4}\right )^2 \]

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Rubi [B]  time = 0.53, antiderivative size = 163, normalized size of antiderivative = 6.27, number of steps used = 2, number of rules used = 1, integrand size = 195, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2074} \begin {gather*} 4 x^4+\frac {\log ^2(2)}{16 x^4}+\frac {\log ^2(2)+\log (4)}{4 x^3}+\frac {1}{3} x^3 \log (64)+\frac {1}{4} x^2 \log ^2(2)+\frac {8+5 \log ^2(2)+16 \log (2)}{8 x^2}+\frac {8+\log ^2(2)+\log (64)}{4 (1-x)^3}+\frac {56+7 \log ^2(2)+40 \log (2)}{8 (1-x)^2}+\frac {8+3 \log ^2(2)+\log (1024)}{2 x}+\frac {\log (2) (10+\log (8))}{2 (1-x)}+\frac {(4+\log (2))^2}{16 (1-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*x^2 - 24*x^3 - 16*x^5 + 16*x^6 + 16*x^7 - 80*x^8 + 320*x^9 - 640*x^10 + 640*x^11 - 320*x^12 + 64*x^13 +
(6*x - 14*x^2 - 16*x^5 + 32*x^6 - 40*x^7 + 120*x^8 - 240*x^9 + 240*x^10 - 120*x^11 + 24*x^12)*Log[2] + (1 - 2
*x + x^3 - 5*x^4 + 7*x^5 - 5*x^6 + 10*x^7 - 20*x^8 + 20*x^9 - 10*x^10 + 2*x^11)*Log[2]^2)/(-4*x^5 + 20*x^6 - 4
0*x^7 + 40*x^8 - 20*x^9 + 4*x^10),x]

[Out]

4*x^4 + Log[2]^2/(16*x^4) + (x^2*Log[2]^2)/4 + (4 + Log[2])^2/(16*(1 - x)^4) + (8 + 16*Log[2] + 5*Log[2]^2)/(8
*x^2) + (56 + 40*Log[2] + 7*Log[2]^2)/(8*(1 - x)^2) + (Log[2]^2 + Log[4])/(4*x^3) + (Log[2]*(10 + Log[8]))/(2*
(1 - x)) + (x^3*Log[64])/3 + (8 + Log[2]^2 + Log[64])/(4*(1 - x)^3) + (8 + 3*Log[2]^2 + Log[1024])/(2*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 (16 x^3-\frac {\log ^2(2)}{4 x^5}+\frac {1}{2} x \log ^2(2)-\frac {(4+\log (2))^2}{4 (-1+x)^5}+\frac {-56-40 \log (2)-7 \log ^2(2)}{4 (-1+x)^3}+\frac {-8-16 \log (2)-5 \log ^2(2)}{4 x^3}-\frac {3 \left (\log ^2(2)+\log (4)\right )}{4 x^4}+\frac {\log (2) (10+\log (8))}{2 (-1+x)^2}+x^2 \log (64)+\frac {3 \left (8+\log ^2(2)+\log (64)\right )}{4 (-1+x)^4}+\frac {-8-3 \log ^2(2)-\log (1024)}{2 x^2}\right ) \, dx\\ &=4 x^4+\frac {\log ^2(2)}{16 x^4}+\frac {1}{4} x^2 \log ^2(2)+\frac {(4+\log (2))^2}{16 (1-x)^4}+\frac {8+16 \log (2)+5 \log ^2(2)}{8 x^2}+\frac {56+40 \log (2)+7 \log ^2(2)}{8 (1-x)^2}+\frac {\log ^2(2)+\log (4)}{4 x^3}+\frac {\log (2) (10+\log (8))}{2 (1-x)}+\frac {1}{3} x^3 \log (64)+\frac {8+\log ^2(2)+\log (64)}{4 (1-x)^3}+\frac {8+3 \log ^2(2)+\log (1024)}{2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 39, normalized size = 1.50 \begin {gather*} \frac {\left (1+2 x^3-4 x^4+2 x^5\right )^2 (4 x+\log (2))^2}{16 (-1+x)^4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^2 - 24*x^3 - 16*x^5 + 16*x^6 + 16*x^7 - 80*x^8 + 320*x^9 - 640*x^10 + 640*x^11 - 320*x^12 + 64*
x^13 + (6*x - 14*x^2 - 16*x^5 + 32*x^6 - 40*x^7 + 120*x^8 - 240*x^9 + 240*x^10 - 120*x^11 + 24*x^12)*Log[2] +
(1 - 2*x + x^3 - 5*x^4 + 7*x^5 - 5*x^6 + 10*x^7 - 20*x^8 + 20*x^9 - 10*x^10 + 2*x^11)*Log[2]^2)/(-4*x^5 + 20*x
^6 - 40*x^7 + 40*x^8 - 20*x^9 + 4*x^10),x]

[Out]

((1 + 2*x^3 - 4*x^4 + 2*x^5)^2*(4*x + Log[2])^2)/(16*(-1 + x)^4*x^4)

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fricas [B]  time = 0.93, size = 165, normalized size = 6.35 \begin {gather*} \frac {64 \, x^{12} - 256 \, x^{11} + 384 \, x^{10} - 256 \, x^{9} + 64 \, x^{8} + 64 \, x^{7} - 128 \, x^{6} + 64 \, x^{5} + {\left (4 \, x^{10} - 16 \, x^{9} + 24 \, x^{8} - 16 \, x^{7} + 4 \, x^{6} + 4 \, x^{5} - 8 \, x^{4} + 4 \, x^{3} + 1\right )} \log \relax (2)^{2} + 16 \, x^{2} + 8 \, {\left (4 \, x^{11} - 16 \, x^{10} + 24 \, x^{9} - 16 \, x^{8} + 4 \, x^{7} + 4 \, x^{6} - 8 \, x^{5} + 4 \, x^{4} + x\right )} \log \relax (2)}{16 \, {\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^11-10*x^10+20*x^9-20*x^8+10*x^7-5*x^6+7*x^5-5*x^4+x^3-2*x+1)*log(2)^2+(24*x^12-120*x^11+240*x^
10-240*x^9+120*x^8-40*x^7+32*x^6-16*x^5-14*x^2+6*x)*log(2)+64*x^13-320*x^12+640*x^11-640*x^10+320*x^9-80*x^8+1
6*x^7+16*x^6-16*x^5-24*x^3+8*x^2)/(4*x^10-20*x^9+40*x^8-40*x^7+20*x^6-4*x^5),x, algorithm="fricas")

[Out]

1/16*(64*x^12 - 256*x^11 + 384*x^10 - 256*x^9 + 64*x^8 + 64*x^7 - 128*x^6 + 64*x^5 + (4*x^10 - 16*x^9 + 24*x^8
- 16*x^7 + 4*x^6 + 4*x^5 - 8*x^4 + 4*x^3 + 1)*log(2)^2 + 16*x^2 + 8*(4*x^11 - 16*x^10 + 24*x^9 - 16*x^8 + 4*x
^7 + 4*x^6 - 8*x^5 + 4*x^4 + x)*log(2))/(x^8 - 4*x^7 + 6*x^6 - 4*x^5 + x^4)

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giac [B]  time = 0.27, size = 111, normalized size = 4.27 \begin {gather*} 4 \, x^{4} + 2 \, x^{3} \log \relax (2) + \frac {1}{4} \, x^{2} \log \relax (2)^{2} + \frac {64 \, x^{7} + 32 \, x^{6} \log \relax (2) + 4 \, x^{5} \log \relax (2)^{2} - 128 \, x^{6} - 64 \, x^{5} \log \relax (2) - 8 \, x^{4} \log \relax (2)^{2} + 64 \, x^{5} + 32 \, x^{4} \log \relax (2) + 4 \, x^{3} \log \relax (2)^{2} + 16 \, x^{2} + 8 \, x \log \relax (2) + \log \relax (2)^{2}}{16 \, {\left (x^{2} - x\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^11-10*x^10+20*x^9-20*x^8+10*x^7-5*x^6+7*x^5-5*x^4+x^3-2*x+1)*log(2)^2+(24*x^12-120*x^11+240*x^
10-240*x^9+120*x^8-40*x^7+32*x^6-16*x^5-14*x^2+6*x)*log(2)+64*x^13-320*x^12+640*x^11-640*x^10+320*x^9-80*x^8+1
6*x^7+16*x^6-16*x^5-24*x^3+8*x^2)/(4*x^10-20*x^9+40*x^8-40*x^7+20*x^6-4*x^5),x, algorithm="giac")

[Out]

4*x^4 + 2*x^3*log(2) + 1/4*x^2*log(2)^2 + 1/16*(64*x^7 + 32*x^6*log(2) + 4*x^5*log(2)^2 - 128*x^6 - 64*x^5*log
(2) - 8*x^4*log(2)^2 + 64*x^5 + 32*x^4*log(2) + 4*x^3*log(2)^2 + 16*x^2 + 8*x*log(2) + log(2)^2)/(x^2 - x)^4

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maple [B]  time = 0.13, size = 120, normalized size = 4.62

method	result	size
risch	\(\frac {x^{2} \ln \relax (2)^{2}}{4}+2 x^{3} \ln \relax (2)+4 x^{4}+\frac {4 x^{7}+\frac {\left (-16+4 \ln \relax (2)\right ) x^{6}}{2}+\frac {\left (\frac {\ln \relax (2)^{2}}{2}-8 \ln \relax (2)+8\right ) x^{5}}{2}+\frac {\left (-\ln \relax (2)^{2}+4 \ln \relax (2)\right ) x^{4}}{2}+\frac {x^{3} \ln \relax (2)^{2}}{4}+x^{2}+\frac {x \ln \relax (2)}{2}+\frac {\ln \relax (2)^{2}}{16}}{x^{4} \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )}\)	\(120\)
norman	\(\frac {x^{2}+\left (-16+2 \ln \relax (2)\right ) x^{11}+\left (24-8 \ln \relax (2)+\frac {\ln \relax (2)^{2}}{4}\right ) x^{10}+\left (-\ln \relax (2)^{2}+12 \ln \relax (2)-16\right ) x^{9}+\left (-2 \ln \relax (2)^{2}+10 \ln \relax (2)-4\right ) x^{4}+\left (5 \ln \relax (2)^{2}-30 \ln \relax (2)+20\right ) x^{7}+\left (-\frac {35 \ln \relax (2)^{2}}{4}+50 \ln \relax (2)-32\right ) x^{6}+\left (\frac {25 \ln \relax (2)^{2}}{4}+20-36 \ln \relax (2)\right ) x^{5}+4 x^{12}+\frac {\ln \relax (2)^{2}}{16}+\frac {x \ln \relax (2)}{2}+\frac {x^{3} \ln \relax (2)^{2}}{4}}{x^{4} \left (x -1\right )^{4}}\)	\(145\)
default	\(4 x^{4}+2 x^{3} \ln \relax (2)+\frac {x^{2} \ln \relax (2)^{2}}{4}-\frac {-6 \ln \relax (2)^{2}-20 \ln \relax (2)-16}{4 x}-\frac {-5 \ln \relax (2)^{2}-16 \ln \relax (2)-8}{8 x^{2}}+\frac {\ln \relax (2)^{2}}{16 x^{4}}+\frac {\ln \relax (2) \left (\ln \relax (2)+2\right )}{4 x^{3}}-\frac {-7 \ln \relax (2)^{2}-40 \ln \relax (2)-56}{8 \left (x -1\right )^{2}}-\frac {-\ln \relax (2)^{2}-8 \ln \relax (2)-16}{16 \left (x -1\right )^{4}}-\frac {3 \ln \relax (2)^{2}+18 \ln \relax (2)+24}{12 \left (x -1\right )^{3}}-\frac {\ln \relax (2) \left (3 \ln \relax (2)+10\right )}{2 \left (x -1\right )}\)	\(149\)
gosper	\(\frac {-140 x^{6} \ln \relax (2)^{2}+64 x^{12}-256 x^{11}+384 x^{10}-256 x^{9}+\ln \relax (2)^{2}+320 x^{7}-64 x^{4}+16 x^{2}-512 x^{6}+320 x^{5}+100 x^{5} \ln \relax (2)^{2}+80 x^{7} \ln \relax (2)^{2}-480 x^{7} \ln \relax (2)-32 x^{4} \ln \relax (2)^{2}+4 x^{3} \ln \relax (2)^{2}+160 x^{4} \ln \relax (2)-576 x^{5} \ln \relax (2)+800 x^{6} \ln \relax (2)+8 x \ln \relax (2)+4 \ln \relax (2)^{2} x^{10}+32 \ln \relax (2) x^{11}-16 \ln \relax (2)^{2} x^{9}-128 \ln \relax (2) x^{10}+192 \ln \relax (2) x^{9}}{16 x^{4} \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )}\)	\(193\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^11-10*x^10+20*x^9-20*x^8+10*x^7-5*x^6+7*x^5-5*x^4+x^3-2*x+1)*ln(2)^2+(24*x^12-120*x^11+240*x^10-240*
x^9+120*x^8-40*x^7+32*x^6-16*x^5-14*x^2+6*x)*ln(2)+64*x^13-320*x^12+640*x^11-640*x^10+320*x^9-80*x^8+16*x^7+16
*x^6-16*x^5-24*x^3+8*x^2)/(4*x^10-20*x^9+40*x^8-40*x^7+20*x^6-4*x^5),x,method=_RETURNVERBOSE)

[Out]

1/4*x^2*ln(2)^2+2*x^3*ln(2)+4*x^4+(4*x^7+1/2*(-16+4*ln(2))*x^6+1/2*(1/2*ln(2)^2-8*ln(2)+8)*x^5+1/2*(-ln(2)^2+4
*ln(2))*x^4+1/4*x^3*ln(2)^2+x^2+1/2*x*ln(2)+1/16*ln(2)^2)/x^4/(x^4-4*x^3+6*x^2-4*x+1)

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maxima [B]  time = 0.34, size = 115, normalized size = 4.42 \begin {gather*} 4 \, x^{4} + 2 \, x^{3} \log \relax (2) + \frac {1}{4} \, x^{2} \log \relax (2)^{2} + \frac {64 \, x^{7} + 32 \, x^{6} {\left (\log \relax (2) - 4\right )} + 4 \, {\left (\log \relax (2)^{2} - 16 \, \log \relax (2) + 16\right )} x^{5} - 8 \, {\left (\log \relax (2)^{2} - 4 \, \log \relax (2)\right )} x^{4} + 4 \, x^{3} \log \relax (2)^{2} + 16 \, x^{2} + 8 \, x \log \relax (2) + \log \relax (2)^{2}}{16 \, {\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^11-10*x^10+20*x^9-20*x^8+10*x^7-5*x^6+7*x^5-5*x^4+x^3-2*x+1)*log(2)^2+(24*x^12-120*x^11+240*x^
10-240*x^9+120*x^8-40*x^7+32*x^6-16*x^5-14*x^2+6*x)*log(2)+64*x^13-320*x^12+640*x^11-640*x^10+320*x^9-80*x^8+1
6*x^7+16*x^6-16*x^5-24*x^3+8*x^2)/(4*x^10-20*x^9+40*x^8-40*x^7+20*x^6-4*x^5),x, algorithm="maxima")

[Out]

4*x^4 + 2*x^3*log(2) + 1/4*x^2*log(2)^2 + 1/16*(64*x^7 + 32*x^6*(log(2) - 4) + 4*(log(2)^2 - 16*log(2) + 16)*x
^5 - 8*(log(2)^2 - 4*log(2))*x^4 + 4*x^3*log(2)^2 + 16*x^2 + 8*x*log(2) + log(2)^2)/(x^8 - 4*x^7 + 6*x^6 - 4*x
^5 + x^4)

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mupad [B]  time = 0.21, size = 131, normalized size = 5.04 \begin {gather*} \frac {x^2\,{\ln \relax (2)}^2}{4}+\frac {16\,x^7+\left (2\,\ln \left (16\right )-32\right )\,x^6+\left ({\ln \relax (2)}^2-\frac {4\,\ln \left (256\right )}{3}-\frac {4\,\ln \left (16\right )}{3}+16\right )\,x^5+\left (\frac {4\,\ln \left (16\right )}{3}+\frac {\ln \left (256\right )}{3}-2\,{\ln \relax (2)}^2\right )\,x^4+{\ln \relax (2)}^2\,x^3+4\,x^2+2\,\ln \relax (2)\,x+\frac {{\ln \relax (2)}^2}{4}}{4\,x^8-16\,x^7+24\,x^6-16\,x^5+4\,x^4}+2\,x^3\,\ln \relax (2)+4\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^2 - 24*x^3 - 16*x^5 + 16*x^6 + 16*x^7 - 80*x^8 + 320*x^9 - 640*x^10 + 640*x^11 - 320*x^12 + 64*x^13
+ log(2)*(6*x - 14*x^2 - 16*x^5 + 32*x^6 - 40*x^7 + 120*x^8 - 240*x^9 + 240*x^10 - 120*x^11 + 24*x^12) + log(2
)^2*(x^3 - 2*x - 5*x^4 + 7*x^5 - 5*x^6 + 10*x^7 - 20*x^8 + 20*x^9 - 10*x^10 + 2*x^11 + 1))/(4*x^5 - 20*x^6 + 4
0*x^7 - 40*x^8 + 20*x^9 - 4*x^10),x)

[Out]

(x^2*log(2)^2)/4 + (x^3*log(2)^2 + 2*x*log(2) + x^4*((4*log(16))/3 + log(256)/3 - 2*log(2)^2) - x^5*((4*log(16
))/3 + (4*log(256))/3 - log(2)^2 - 16) + x^6*(2*log(16) - 32) + log(2)^2/4 + 4*x^2 + 16*x^7)/(4*x^4 - 16*x^5 +
24*x^6 - 16*x^7 + 4*x^8) + 2*x^3*log(2) + 4*x^4

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sympy [B]  time = 9.25, size = 122, normalized size = 4.69 \begin {gather*} 4 x^{4} + 2 x^{3} \log {\relax (2 )} + \frac {x^{2} \log {\relax (2 )}^{2}}{4} + \frac {64 x^{7} + x^{6} \left (-128 + 32 \log {\relax (2 )}\right ) + x^{5} \left (- 64 \log {\relax (2 )} + 4 \log {\relax (2 )}^{2} + 64\right ) + x^{4} \left (- 8 \log {\relax (2 )}^{2} + 32 \log {\relax (2 )}\right ) + 4 x^{3} \log {\relax (2 )}^{2} + 16 x^{2} + 8 x \log {\relax (2 )} + \log {\relax (2 )}^{2}}{16 x^{8} - 64 x^{7} + 96 x^{6} - 64 x^{5} + 16 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**11-10*x**10+20*x**9-20*x**8+10*x**7-5*x**6+7*x**5-5*x**4+x**3-2*x+1)*ln(2)**2+(24*x**12-120*x
**11+240*x**10-240*x**9+120*x**8-40*x**7+32*x**6-16*x**5-14*x**2+6*x)*ln(2)+64*x**13-320*x**12+640*x**11-640*x
**10+320*x**9-80*x**8+16*x**7+16*x**6-16*x**5-24*x**3+8*x**2)/(4*x**10-20*x**9+40*x**8-40*x**7+20*x**6-4*x**5)
,x)

[Out]

4*x**4 + 2*x**3*log(2) + x**2*log(2)**2/4 + (64*x**7 + x**6*(-128 + 32*log(2)) + x**5*(-64*log(2) + 4*log(2)**
2 + 64) + x**4*(-8*log(2)**2 + 32*log(2)) + 4*x**3*log(2)**2 + 16*x**2 + 8*x*log(2) + log(2)**2)/(16*x**8 - 64
*x**7 + 96*x**6 - 64*x**5 + 16*x**4)

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