3.88.25 \(\int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+(108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6) \log (3)+(108 x^3+1728 x^4+9216 x^5+16384 x^6) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx\)

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

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Rubi [B]  time = 0.18, antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps used = 2, number of rules used = 1, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2074} \begin {gather*} x^4 (1-\log (3))^2-2 x^3 (1-\log (3))-\frac {1}{8} x^2 (7-15 \log (3))+\frac {9}{256 (16 x+3)^2}+\frac {3}{128} x (79+\log (3))+\frac {9 (61+\log (27))}{2048 (16 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 + 684*x + 3348*x^2 + 1068*x^3 - 19264*x^4 - 15360*x^5 + 16384*x^6 + (108*x + 1836*x^2 + 11112*x^3 + 25
728*x^4 + 6144*x^5 - 32768*x^6)*Log[3] + (108*x^3 + 1728*x^4 + 9216*x^5 + 16384*x^6)*Log[3]^2)/(27 + 432*x + 2
304*x^2 + 4096*x^3),x]

[Out]

9/(256*(3 + 16*x)^2) - (x^2*(7 - 15*Log[3]))/8 - 2*x^3*(1 - Log[3]) + x^4*(1 - Log[3])^2 + (3*x*(79 + Log[3]))
/128 + (9*(61 + Log[27]))/(2048*(3 + 16*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 (-\frac {9}{8 (3+16 x)^3}+6 x^2 (-1+\log (3))+4 x^3 (-1+\log (3))^2+\frac {3}{128} (79+\log (3))+\frac {1}{4} x (-7+15 \log (3))-\frac {9 (61+\log (27))}{128 (3+16 x)^2}\right ) \, dx\\ &=\frac {9}{256 (3+16 x)^2}-\frac {1}{8} x^2 (7-15 \log (3))-2 x^3 (1-\log (3))+x^4 (1-\log (3))^2+\frac {3}{128} x (79+\log (3))+\frac {9 (61+\log (27))}{2048 (3+16 x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 140, normalized size = 5.38 \begin {gather*} \frac {16777216 x^6 (-1+\log (3))^2-6912 x^2 \left (-2493+370 \log (3)+3 \log ^2(3)-220 \log (9)\right )+65536 x^4 \left (-407+714 \log (3)+9 \log ^2(3)-30 \log (9)\right )+98304 x^3 (248+106 \log (3)+15 \log (9))-288 x \left (-12709+1170 \log (3)+27 \log ^2(3)-900 \log (9)+512 \log (27)\right )-9 \left (-29935+270 \log (3)+81 \log ^2(3)-1080 \log (9)+1536 \log (27)\right )+2097152 x^5 \left (-13+3 \log ^2(3)+\log (59049)\right )}{65536 (3+16 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 + 684*x + 3348*x^2 + 1068*x^3 - 19264*x^4 - 15360*x^5 + 16384*x^6 + (108*x + 1836*x^2 + 11112*x^
3 + 25728*x^4 + 6144*x^5 - 32768*x^6)*Log[3] + (108*x^3 + 1728*x^4 + 9216*x^5 + 16384*x^6)*Log[3]^2)/(27 + 432
*x + 2304*x^2 + 4096*x^3),x]

[Out]

(16777216*x^6*(-1 + Log[3])^2 - 6912*x^2*(-2493 + 370*Log[3] + 3*Log[3]^2 - 220*Log[9]) + 65536*x^4*(-407 + 71
4*Log[3] + 9*Log[3]^2 - 30*Log[9]) + 98304*x^3*(248 + 106*Log[3] + 15*Log[9]) - 288*x*(-12709 + 1170*Log[3] +
27*Log[3]^2 - 900*Log[9] + 512*Log[27]) - 9*(-29935 + 270*Log[3] + 81*Log[3]^2 - 1080*Log[9] + 1536*Log[27]) +
2097152*x^5*(-13 + 3*Log[3]^2 + Log[59049]))/(65536*(3 + 16*x)^2)

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fricas [B]  time = 0.59, size = 100, normalized size = 3.85 \begin {gather*} \frac {524288 \, x^{6} - 851968 \, x^{5} - 833536 \, x^{4} + 761856 \, x^{3} + 2048 \, {\left (256 \, x^{6} + 96 \, x^{5} + 9 \, x^{4}\right )} \log \relax (3)^{2} + 347904 \, x^{2} - {\left (1048576 \, x^{6} - 655360 \, x^{5} - 1339392 \, x^{4} - 417792 \, x^{3} - 39168 \, x^{2} - 864 \, x - 81\right )} \log \relax (3) + 42912 \, x + 1719}{2048 \, {\left (256 \, x^{2} + 96 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*log(3)^2+(-32768*x^6+6144*x^5+25728*x^4+11112*x^3+1836*x^2+10
8*x)*log(3)+16384*x^6-15360*x^5-19264*x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x, algorith
m="fricas")

[Out]

1/2048*(524288*x^6 - 851968*x^5 - 833536*x^4 + 761856*x^3 + 2048*(256*x^6 + 96*x^5 + 9*x^4)*log(3)^2 + 347904*
x^2 - (1048576*x^6 - 655360*x^5 - 1339392*x^4 - 417792*x^3 - 39168*x^2 - 864*x - 81)*log(3) + 42912*x + 1719)/
(256*x^2 + 96*x + 9)

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giac [B]  time = 0.16, size = 74, normalized size = 2.85 \begin {gather*} x^{4} \log \relax (3)^{2} - 2 \, x^{4} \log \relax (3) + x^{4} + 2 \, x^{3} \log \relax (3) - 2 \, x^{3} + \frac {15}{8} \, x^{2} \log \relax (3) - \frac {7}{8} \, x^{2} + \frac {3}{128} \, x \log \relax (3) + \frac {237}{128} \, x + \frac {9 \, {\left (48 \, x \log \relax (3) + 976 \, x + 9 \, \log \relax (3) + 191\right )}}{2048 \, {\left (16 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*log(3)^2+(-32768*x^6+6144*x^5+25728*x^4+11112*x^3+1836*x^2+10
8*x)*log(3)+16384*x^6-15360*x^5-19264*x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x, algorith
m="giac")

[Out]

x^4*log(3)^2 - 2*x^4*log(3) + x^4 + 2*x^3*log(3) - 2*x^3 + 15/8*x^2*log(3) - 7/8*x^2 + 3/128*x*log(3) + 237/12
8*x + 9/2048*(48*x*log(3) + 976*x + 9*log(3) + 191)/(16*x + 3)^2

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maple [B]  time = 0.09, size = 76, normalized size = 2.92

method	result	size
default	\(x^{4} \ln \relax (3)^{2}-2 x^{4} \ln \relax (3)+x^{4}+2 x^{3} \ln \relax (3)-2 x^{3}+\frac {15 x^{2} \ln \relax (3)}{8}-\frac {7 x^{2}}{8}+\frac {3 x \ln \relax (3)}{128}+\frac {237 x}{128}+\frac {9}{256 \left (16 x +3\right )^{2}}-\frac {4 \left (-\frac {27 \ln \relax (3)}{8192}-\frac {549}{8192}\right )}{16 x +3}\)	\(76\)
risch	\(x^{4} \ln \relax (3)^{2}-2 x^{4} \ln \relax (3)+2 x^{3} \ln \relax (3)+x^{4}+\frac {15 x^{2} \ln \relax (3)}{8}-2 x^{3}+\frac {3 x \ln \relax (3)}{128}-\frac {7 x^{2}}{8}+\frac {237 x}{128}+\frac {\frac {\left (\frac {549}{256}+\frac {27 \ln \relax (3)}{256}\right ) x}{128}+\frac {1719}{524288}+\frac {81 \ln \relax (3)}{524288}}{x^{2}+\frac {3}{8} x +\frac {9}{256}}\)	\(78\)
norman	\(\frac {\left (372+204 \ln \relax (3)\right ) x^{3}+12 x +\left (-416+96 \ln \relax (3)^{2}+320 \ln \relax (3)\right ) x^{5}+\left (-407+654 \ln \relax (3)+9 \ln \relax (3)^{2}\right ) x^{4}+\left (146+18 \ln \relax (3)\right ) x^{2}+\left (256 \ln \relax (3)^{2}-512 \ln \relax (3)+256\right ) x^{6}}{\left (16 x +3\right )^{2}}\)	\(81\)
gosper	\(\frac {x \left (256 x^{5} \ln \relax (3)^{2}+96 x^{4} \ln \relax (3)^{2}-512 x^{5} \ln \relax (3)+9 x^{3} \ln \relax (3)^{2}+320 x^{4} \ln \relax (3)+256 x^{5}+654 x^{3} \ln \relax (3)-416 x^{4}+204 x^{2} \ln \relax (3)-407 x^{3}+18 x \ln \relax (3)+372 x^{2}+146 x +12\right )}{256 x^{2}+96 x +9}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*ln(3)^2+(-32768*x^6+6144*x^5+25728*x^4+11112*x^3+1836*x^2+108*x)*ln
(3)+16384*x^6-15360*x^5-19264*x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x,method=_RETURNVER
BOSE)

[Out]

x^4*ln(3)^2-2*x^4*ln(3)+x^4+2*x^3*ln(3)-2*x^3+15/8*x^2*ln(3)-7/8*x^2+3/128*x*ln(3)+237/128*x+9/256/(16*x+3)^2-
4*(-27/8192*ln(3)-549/8192)/(16*x+3)

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maxima [B]  time = 0.34, size = 71, normalized size = 2.73 \begin {gather*} {\left (\log \relax (3)^{2} - 2 \, \log \relax (3) + 1\right )} x^{4} + 2 \, x^{3} {\left (\log \relax (3) - 1\right )} + \frac {1}{8} \, x^{2} {\left (15 \, \log \relax (3) - 7\right )} + \frac {3}{128} \, x {\left (\log \relax (3) + 79\right )} + \frac {9 \, {\left (16 \, x {\left (3 \, \log \relax (3) + 61\right )} + 9 \, \log \relax (3) + 191\right )}}{2048 \, {\left (256 \, x^{2} + 96 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*log(3)^2+(-32768*x^6+6144*x^5+25728*x^4+11112*x^3+1836*x^2+10
8*x)*log(3)+16384*x^6-15360*x^5-19264*x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x, algorith
m="maxima")

[Out]

(log(3)^2 - 2*log(3) + 1)*x^4 + 2*x^3*(log(3) - 1) + 1/8*x^2*(15*log(3) - 7) + 3/128*x*(log(3) + 79) + 9/2048*
(16*x*(3*log(3) + 61) + 9*log(3) + 191)/(256*x^2 + 96*x + 9)

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mupad [B]  time = 5.65, size = 109, normalized size = 4.19 \begin {gather*} x^3\,\left (\frac {\ln \relax (3)}{2}-\frac {3\,{\left (\ln \relax (3)-1\right )}^2}{4}+\frac {3\,{\ln \relax (3)}^2}{4}-\frac {5}{4}\right )+x^2\,\left (\frac {87\,\ln \relax (3)}{32}+\frac {27\,{\left (\ln \relax (3)-1\right )}^2}{64}-\frac {27\,{\ln \relax (3)}^2}{64}-\frac {83}{64}\right )+\frac {\frac {81\,\ln \relax (3)}{16}+x\,\left (27\,\ln \relax (3)+549\right )+\frac {1719}{16}}{32768\,x^2+12288\,x+1152}+x^4\,{\left (\ln \relax (3)-1\right )}^2-x\,\left (\frac {129\,\ln \relax (3)}{256}+\frac {135\,{\left (\ln \relax (3)-1\right )}^2}{512}-\frac {135\,{\ln \relax (3)}^2}{512}-\frac {1083}{512}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((684*x + log(3)*(108*x + 1836*x^2 + 11112*x^3 + 25728*x^4 + 6144*x^5 - 32768*x^6) + log(3)^2*(108*x^3 + 17
28*x^4 + 9216*x^5 + 16384*x^6) + 3348*x^2 + 1068*x^3 - 19264*x^4 - 15360*x^5 + 16384*x^6 + 36)/(432*x + 2304*x
^2 + 4096*x^3 + 27),x)

[Out]

x^3*(log(3)/2 - (3*(log(3) - 1)^2)/4 + (3*log(3)^2)/4 - 5/4) + x^2*((87*log(3))/32 + (27*(log(3) - 1)^2)/64 -
(27*log(3)^2)/64 - 83/64) + ((81*log(3))/16 + x*(27*log(3) + 549) + 1719/16)/(12288*x + 32768*x^2 + 1152) + x^
4*(log(3) - 1)^2 - x*((129*log(3))/256 + (135*(log(3) - 1)^2)/512 - (135*log(3)^2)/512 - 1083/512)

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sympy [B]  time = 0.37, size = 75, normalized size = 2.88 \begin {gather*} x^{4} \left (- 2 \log {\relax (3 )} + 1 + \log {\relax (3 )}^{2}\right ) + x^{3} \left (-2 + 2 \log {\relax (3 )}\right ) + x^{2} \left (- \frac {7}{8} + \frac {15 \log {\relax (3 )}}{8}\right ) + x \left (\frac {3 \log {\relax (3 )}}{128} + \frac {237}{128}\right ) + \frac {x \left (432 \log {\relax (3 )} + 8784\right ) + 81 \log {\relax (3 )} + 1719}{524288 x^{2} + 196608 x + 18432} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16384*x**6+9216*x**5+1728*x**4+108*x**3)*ln(3)**2+(-32768*x**6+6144*x**5+25728*x**4+11112*x**3+183
6*x**2+108*x)*ln(3)+16384*x**6-15360*x**5-19264*x**4+1068*x**3+3348*x**2+684*x+36)/(4096*x**3+2304*x**2+432*x+
27),x)

[Out]

x**4*(-2*log(3) + 1 + log(3)**2) + x**3*(-2 + 2*log(3)) + x**2*(-7/8 + 15*log(3)/8) + x*(3*log(3)/128 + 237/12
8) + (x*(432*log(3) + 8784) + 81*log(3) + 1719)/(524288*x**2 + 196608*x + 18432)

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