∫ 112x6+288x10−144x14+(96x10−96x14) log2(3)−16x14 log4(3)_____________________________________________ 1+12x4+32x7+54x8+192x11+108x12+256x14+288x15+81x16+(4x4+36x8+64x11+108x12+192x15+108x16) log2(3)+(6x8+36x12+32x15+54x16) log4(3)+(4x12+12x16) log6(3)+x16 log8(3) dx

3.46.42 \(\int \frac {112 x^6+288 x^{10}-144 x^{14}+(96 x^{10}-96 x^{14}) \log ^2(3)-16 x^{14} \log ^4(3)}{1+12 x^4+32 x^7+54 x^8+192 x^{11}+108 x^{12}+256 x^{14}+288 x^{15}+81 x^{16}+(4 x^4+36 x^8+64 x^{11}+108 x^{12}+192 x^{15}+108 x^{16}) \log ^2(3)+(6 x^8+36 x^{12}+32 x^{15}+54 x^{16}) \log ^4(3)+(4 x^{12}+12 x^{16}) \log ^6(3)+x^{16} \log ^8(3)} \, dx\)

Optimal. Leaf size=26 \[ -5+\frac {x}{x+\frac {1}{16} x^2 \left (3+\frac {1}{x^4}+\log ^2(3)\right )^2} \]

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Rubi [F] time = 20.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {112 x^6+288 x^{10}-144 x^{14}+\left (96 x^{10}-96 x^{14}\right ) \log ^2(3)-16 x^{14} \log ^4(3)}{1+12 x^4+32 x^7+54 x^8+192 x^{11}+108 x^{12}+256 x^{14}+288 x^{15}+81 x^{16}+\left (4 x^4+36 x^8+64 x^{11}+108 x^{12}+192 x^{15}+108 x^{16}\right ) \log ^2(3)+\left (6 x^8+36 x^{12}+32 x^{15}+54 x^{16}\right ) \log ^4(3)+\left (4 x^{12}+12 x^{16}\right ) \log ^6(3)+x^{16} \log ^8(3)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(112*x^6 + 288*x^10 - 144*x^14 + (96*x^10 - 96*x^14)*Log[3]^2 - 16*x^14*Log[3]^4)/(1 + 12*x^4 + 32*x^7 + 5

4*x^8 + 192*x^11 + 108*x^12 + 256*x^14 + 288*x^15 + 81*x^16 + (4*x^4 + 36*x^8 + 64*x^11 + 108*x^12 + 192*x^15

+ 108*x^16)*Log[3]^2 + (6*x^8 + 36*x^12 + 32*x^15 + 54*x^16)*Log[3]^4 + (4*x^12 + 12*x^16)*Log[3]^6 + x^16*Log

[3]^8),x]

[Out]

(-16384*(17459 - 35721*Log[3]^2 - 35721*Log[3]^4 - 19845*Log[3]^6 - 6615*Log[3]^8 - 1323*Log[3]^10 - 147*Log[3

]^12 - 7*Log[3]^14))/((3 + Log[3]^2)^14*(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)) + (4096*De

fer[Int][x^4/(-1 - 16*x^7 - 2*x^4*(3 + Log[3]^2) - x^8*(3 + Log[3]^2)^2), x])/(3 + Log[3]^2)^4 + 16*Defer[Int]

[x^6/(-1 - 16*x^7 - 2*x^4*(3 + Log[3]^2) - x^8*(3 + Log[3]^2)^2), x] + (16384*(9823 - 15309*Log[3]^2 - 15309*L

og[3]^4 - 8505*Log[3]^6 - 2835*Log[3]^8 - 567*Log[3]^10 - 63*Log[3]^12 - 3*Log[3]^14)*Defer[Int][(1 + 16*x^7 +

2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)^(-2), x])/(3 + Log[3]^2)^12 - (512*(21833 - 25515*Log[3]^2 - 255

15*Log[3]^4 - 14175*Log[3]^6 - 4725*Log[3]^8 - 945*Log[3]^10 - 105*Log[3]^12 - 5*Log[3]^14)*Defer[Int][x/(1 +

16*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)^2, x])/(3 + Log[3]^2)^10 + (128*(6005 - 5103*Log[3]^2 -

5103*Log[3]^4 - 2835*Log[3]^6 - 945*Log[3]^8 - 189*Log[3]^10 - 21*Log[3]^12 - Log[3]^14)*Defer[Int][x^2/(1 + 1

6*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)^2, x])/(3 + Log[3]^2)^8 - (65536*(37105 - 66339*Log[3]^2

- 66339*Log[3]^4 - 36855*Log[3]^6 - 12285*Log[3]^8 - 2457*Log[3]^10 - 273*Log[3]^12 - 13*Log[3]^14)*Defer[Int]

[x^3/(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)^2, x])/(3 + Log[3]^2)^13 + (4096*(80771 - 1173

69*Log[3]^2 - 117369*Log[3]^4 - 65205*Log[3]^6 - 21735*Log[3]^8 - 4347*Log[3]^10 - 483*Log[3]^12 - 23*Log[3]^1

4)*Defer[Int][x^4/(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)^2, x])/(3 + Log[3]^2)^11 - (256*(

89519 - 96957*Log[3]^2 - 96957*Log[3]^4 - 53865*Log[3]^6 - 17955*Log[3]^8 - 3591*Log[3]^10 - 399*Log[3]^12 - 1

9*Log[3]^14)*Defer[Int][x^5/(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)^2, x])/(3 + Log[3]^2)^9

- (128*(219243385 - 573383286*Log[3]^2 - 547342677*Log[3]^4 - 266465052*Log[3]^6 - 51207471*Log[3]^8 + 173422

62*Log[3]^10 + 17251227*Log[3]^12 + 7389048*Log[3]^14 + 2189187*Log[3]^16 + 486486*Log[3]^18 + 81081*Log[3]^20

+ 9828*Log[3]^22 + 819*Log[3]^24 + 42*Log[3]^26 + Log[3]^28)*Defer[Int][x^6/(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2

) + x^8*(3 + Log[3]^2)^2)^2, x])/(3 + Log[3]^2)^14 - (16384*(9823 - 15309*Log[3]^2 - 15309*Log[3]^4 - 8505*Log

[3]^6 - 2835*Log[3]^8 - 567*Log[3]^10 - 63*Log[3]^12 - 3*Log[3]^14)*Defer[Int][(1 + 16*x^7 + 2*x^4*(3 + Log[3]

^2) + x^8*(3 + Log[3]^2)^2)^(-1), x])/(3 + Log[3]^2)^12 + (512*(21833 - 25515*Log[3]^2 - 25515*Log[3]^4 - 1417

5*Log[3]^6 - 4725*Log[3]^8 - 945*Log[3]^10 - 105*Log[3]^12 - 5*Log[3]^14)*Defer[Int][x/(1 + 16*x^7 + 2*x^4*(3

+ Log[3]^2) + x^8*(3 + Log[3]^2)^2), x])/(3 + Log[3]^2)^10 - (128*(6005 - 5103*Log[3]^2 - 5103*Log[3]^4 - 2835

*Log[3]^6 - 945*Log[3]^8 - 189*Log[3]^10 - 21*Log[3]^12 - Log[3]^14)*Defer[Int][x^2/(1 + 16*x^7 + 2*x^4*(3 + L

og[3]^2) + x^8*(3 + Log[3]^2)^2), x])/(3 + Log[3]^2)^8 + (65536*Defer[Int][x^3/(1 + 16*x^7 + 2*x^4*(3 + Log[3]

^2) + x^8*(3 + Log[3]^2)^2), x])/(3 + Log[3]^2)^6 + (256*Defer[Int][x^5/(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2) + x

^8*(3 + Log[3]^2)^2), x])/(3 + Log[3]^2)^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {112 x^6+288 x^{10}+\left (96 x^{10}-96 x^{14}\right ) \log ^2(3)+x^{14} \left (-144-16 \log ^4(3)\right )}{1+12 x^4+32 x^7+54 x^8+192 x^{11}+108 x^{12}+256 x^{14}+288 x^{15}+81 x^{16}+\left (4 x^4+36 x^8+64 x^{11}+108 x^{12}+192 x^{15}+108 x^{16}\right ) \log ^2(3)+\left (6 x^8+36 x^{12}+32 x^{15}+54 x^{16}\right ) \log ^4(3)+\left (4 x^{12}+12 x^{16}\right ) \log ^6(3)+x^{16} \log ^8(3)} \, dx\\ &=\int \frac {112 x^6+288 x^{10}+\left (96 x^{10}-96 x^{14}\right ) \log ^2(3)+x^{14} \left (-144-16 \log ^4(3)\right )}{1+12 x^4+32 x^7+54 x^8+192 x^{11}+108 x^{12}+256 x^{14}+288 x^{15}+\left (4 x^4+36 x^8+64 x^{11}+108 x^{12}+192 x^{15}+108 x^{16}\right ) \log ^2(3)+\left (6 x^8+36 x^{12}+32 x^{15}+54 x^{16}\right ) \log ^4(3)+\left (4 x^{12}+12 x^{16}\right ) \log ^6(3)+x^{16} \left (81+\log ^8(3)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.31, size = 37, normalized size = 1.42 \begin {gather*} \frac {16 x^7}{1+16 x^7+2 x^4 \left (3+\log ^2(3)\right )+x^8 \left (3+\log ^2(3)\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(112*x^6 + 288*x^10 - 144*x^14 + (96*x^10 - 96*x^14)*Log[3]^2 - 16*x^14*Log[3]^4)/(1 + 12*x^4 + 32*x

^7 + 54*x^8 + 192*x^11 + 108*x^12 + 256*x^14 + 288*x^15 + 81*x^16 + (4*x^4 + 36*x^8 + 64*x^11 + 108*x^12 + 192

*x^15 + 108*x^16)*Log[3]^2 + (6*x^8 + 36*x^12 + 32*x^15 + 54*x^16)*Log[3]^4 + (4*x^12 + 12*x^16)*Log[3]^6 + x^

16*Log[3]^8),x]

[Out]

(16*x^7)/(1 + 16*x^7 + 2*x^4*(3 + Log[3]^2) + x^8*(3 + Log[3]^2)^2)

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fricas [A] time = 0.61, size = 47, normalized size = 1.81 \begin {gather*} \frac {16 \, x^{7}}{x^{8} \log \relax (3)^{4} + 9 \, x^{8} + 16 \, x^{7} + 6 \, x^{4} + 2 \, {\left (3 \, x^{8} + x^{4}\right )} \log \relax (3)^{2} + 1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^14*log(3)^4+(-96*x^14+96*x^10)*log(3)^2-144*x^14+288*x^10+112*x^6)/(x^16*log(3)^8+(12*x^16+4*

x^12)*log(3)^6+(54*x^16+32*x^15+36*x^12+6*x^8)*log(3)^4+(108*x^16+192*x^15+108*x^12+64*x^11+36*x^8+4*x^4)*log(

3)^2+81*x^16+288*x^15+256*x^14+108*x^12+192*x^11+54*x^8+32*x^7+12*x^4+1),x, algorithm="fricas")

[Out]

16*x^7/(x^8*log(3)^4 + 9*x^8 + 16*x^7 + 6*x^4 + 2*(3*x^8 + x^4)*log(3)^2 + 1)

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giac [A] time = 0.99, size = 50, normalized size = 1.92 \begin {gather*} \frac {16 \, x^{7}}{x^{8} \log \relax (3)^{4} + 6 \, x^{8} \log \relax (3)^{2} + 9 \, x^{8} + 16 \, x^{7} + 2 \, x^{4} \log \relax (3)^{2} + 6 \, x^{4} + 1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^14*log(3)^4+(-96*x^14+96*x^10)*log(3)^2-144*x^14+288*x^10+112*x^6)/(x^16*log(3)^8+(12*x^16+4*

x^12)*log(3)^6+(54*x^16+32*x^15+36*x^12+6*x^8)*log(3)^4+(108*x^16+192*x^15+108*x^12+64*x^11+36*x^8+4*x^4)*log(

3)^2+81*x^16+288*x^15+256*x^14+108*x^12+192*x^11+54*x^8+32*x^7+12*x^4+1),x, algorithm="giac")

[Out]

16*x^7/(x^8*log(3)^4 + 6*x^8*log(3)^2 + 9*x^8 + 16*x^7 + 2*x^4*log(3)^2 + 6*x^4 + 1)

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maple [B] time = 0.41, size = 51, normalized size = 1.96


method	result	size

gosper	\(\frac {16 x^{7}}{x^{8} \ln \relax (3)^{4}+6 x^{8} \ln \relax (3)^{2}+9 x^{8}+16 x^{7}+2 x^{4} \ln \relax (3)^{2}+6 x^{4}+1}\)	\(51\)
default	\(\frac {16 x^{7}}{x^{8} \ln \relax (3)^{4}+6 x^{8} \ln \relax (3)^{2}+9 x^{8}+16 x^{7}+2 x^{4} \ln \relax (3)^{2}+6 x^{4}+1}\)	\(51\)
norman	\(\frac {16 x^{7}}{x^{8} \ln \relax (3)^{4}+6 x^{8} \ln \relax (3)^{2}+9 x^{8}+16 x^{7}+2 x^{4} \ln \relax (3)^{2}+6 x^{4}+1}\)	\(51\)
risch	\(\frac {16 x^{7}}{x^{8} \ln \relax (3)^{4}+6 x^{8} \ln \relax (3)^{2}+9 x^{8}+16 x^{7}+2 x^{4} \ln \relax (3)^{2}+6 x^{4}+1}\)	\(51\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x^14*ln(3)^4+(-96*x^14+96*x^10)*ln(3)^2-144*x^14+288*x^10+112*x^6)/(x^16*ln(3)^8+(12*x^16+4*x^12)*ln(

3)^6+(54*x^16+32*x^15+36*x^12+6*x^8)*ln(3)^4+(108*x^16+192*x^15+108*x^12+64*x^11+36*x^8+4*x^4)*ln(3)^2+81*x^16

+288*x^15+256*x^14+108*x^12+192*x^11+54*x^8+32*x^7+12*x^4+1),x,method=_RETURNVERBOSE)

[Out]

16*x^7/(x^8*ln(3)^4+6*x^8*ln(3)^2+9*x^8+16*x^7+2*x^4*ln(3)^2+6*x^4+1)

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maxima [A] time = 0.44, size = 41, normalized size = 1.58 \begin {gather*} \frac {16 \, x^{7}}{{\left (\log \relax (3)^{4} + 6 \, \log \relax (3)^{2} + 9\right )} x^{8} + 16 \, x^{7} + 2 \, {\left (\log \relax (3)^{2} + 3\right )} x^{4} + 1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^14*log(3)^4+(-96*x^14+96*x^10)*log(3)^2-144*x^14+288*x^10+112*x^6)/(x^16*log(3)^8+(12*x^16+4*

x^12)*log(3)^6+(54*x^16+32*x^15+36*x^12+6*x^8)*log(3)^4+(108*x^16+192*x^15+108*x^12+64*x^11+36*x^8+4*x^4)*log(

3)^2+81*x^16+288*x^15+256*x^14+108*x^12+192*x^11+54*x^8+32*x^7+12*x^4+1),x, algorithm="maxima")

[Out]

16*x^7/((log(3)^4 + 6*log(3)^2 + 9)*x^8 + 16*x^7 + 2*(log(3)^2 + 3)*x^4 + 1)

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mupad [B] time = 0.58, size = 42, normalized size = 1.62 \begin {gather*} \frac {16\,x^7}{\left (6\,{\ln \relax (3)}^2+{\ln \relax (3)}^4+9\right )\,x^8+16\,x^7+\left (2\,{\ln \relax (3)}^2+6\right )\,x^4+1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((112*x^6 - 16*x^14*log(3)^4 + 288*x^10 - 144*x^14 + log(3)^2*(96*x^10 - 96*x^14))/(x^16*log(3)^8 + log(3)^

4*(6*x^8 + 36*x^12 + 32*x^15 + 54*x^16) + log(3)^2*(4*x^4 + 36*x^8 + 64*x^11 + 108*x^12 + 192*x^15 + 108*x^16)

+ 12*x^4 + 32*x^7 + 54*x^8 + 192*x^11 + 108*x^12 + 256*x^14 + 288*x^15 + 81*x^16 + log(3)^6*(4*x^12 + 12*x^16

) + 1),x)

[Out]

(16*x^7)/(x^8*(6*log(3)^2 + log(3)^4 + 9) + x^4*(2*log(3)^2 + 6) + 16*x^7 + 1)

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sympy [A] time = 136.18, size = 39, normalized size = 1.50 \begin {gather*} \frac {16 x^{7}}{x^{8} \left (\log {\relax (3 )}^{4} + 6 \log {\relax (3 )}^{2} + 9\right ) + 16 x^{7} + x^{4} \left (2 \log {\relax (3 )}^{2} + 6\right ) + 1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**14*ln(3)**4+(-96*x**14+96*x**10)*ln(3)**2-144*x**14+288*x**10+112*x**6)/(x**16*ln(3)**8+(12*

x**16+4*x**12)*ln(3)**6+(54*x**16+32*x**15+36*x**12+6*x**8)*ln(3)**4+(108*x**16+192*x**15+108*x**12+64*x**11+3

6*x**8+4*x**4)*ln(3)**2+81*x**16+288*x**15+256*x**14+108*x**12+192*x**11+54*x**8+32*x**7+12*x**4+1),x)

[Out]

16*x**7/(x**8*(log(3)**4 + 6*log(3)**2 + 9) + 16*x**7 + x**4*(2*log(3)**2 + 6) + 1)

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