3.15.95 \(\int \frac {1}{8} (x+12 x^3-20 x^4+27 x^5-84 x^6+64 x^7+(8 x-12 x^2+48 x^3-140 x^4+96 x^5) \log (2)+(16 x-48 x^2+32 x^3) \log ^2(2)) \, dx\)

Optimal. Leaf size=28 \[ (-1+x)^2 x^4 \left (\frac {1}{4} \left (1+\frac {1}{x}\right )+x+\frac {\log (2)}{x}\right )^2 \]

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Rubi [B]  time = 0.03, antiderivative size = 107, normalized size of antiderivative = 3.82, number of steps used = 4, number of rules used = 1, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {12} \begin {gather*} x^8-\frac {3 x^7}{2}+\frac {9 x^6}{16}+2 x^6 \log (2)-\frac {x^5}{2}-\frac {7}{2} x^5 \log (2)+\frac {3 x^4}{8}+x^4 \log ^2(2)+\frac {3}{2} x^4 \log (2)-2 x^3 \log ^2(2)-\frac {1}{2} x^3 \log (2)+\frac {x^2}{16}+x^2 \log ^2(2)+\frac {1}{2} x^2 \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + 12*x^3 - 20*x^4 + 27*x^5 - 84*x^6 + 64*x^7 + (8*x - 12*x^2 + 48*x^3 - 140*x^4 + 96*x^5)*Log[2] + (16*
x - 48*x^2 + 32*x^3)*Log[2]^2)/8,x]

[Out]

x^2/16 + (3*x^4)/8 - x^5/2 + (9*x^6)/16 - (3*x^7)/2 + x^8 + (x^2*Log[2])/2 - (x^3*Log[2])/2 + (3*x^4*Log[2])/2
- (7*x^5*Log[2])/2 + 2*x^6*Log[2] + x^2*Log[2]^2 - 2*x^3*Log[2]^2 + x^4*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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \left (x+12 x^3-20 x^4+27 x^5-84 x^6+64 x^7+\left (8 x-12 x^2+48 x^3-140 x^4+96 x^5\right ) \log (2)+\left (16 x-48 x^2+32 x^3\right ) \log ^2(2)\right ) \, dx\\ &=\frac {x^2}{16}+\frac {3 x^4}{8}-\frac {x^5}{2}+\frac {9 x^6}{16}-\frac {3 x^7}{2}+x^8+\frac {1}{8} \log (2) \int \left (8 x-12 x^2+48 x^3-140 x^4+96 x^5\right ) \, dx+\frac {1}{8} \log ^2(2) \int \left (16 x-48 x^2+32 x^3\right ) \, dx\\ &=\frac {x^2}{16}+\frac {3 x^4}{8}-\frac {x^5}{2}+\frac {9 x^6}{16}-\frac {3 x^7}{2}+x^8+\frac {1}{2} x^2 \log (2)-\frac {1}{2} x^3 \log (2)+\frac {3}{2} x^4 \log (2)-\frac {7}{2} x^5 \log (2)+2 x^6 \log (2)+x^2 \log ^2(2)-2 x^3 \log ^2(2)+x^4 \log ^2(2)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 82, normalized size = 2.93 \begin {gather*} \frac {1}{96} x^2 \left (-144 x^5+96 x^6+6 x^4 (9+32 \log (2))+6 (1+\log (16))^2+3 x^2 \left (12+44 \log (2)+32 \log ^2(2)+\log (16)\right )-4 x \left (32 \log ^2(2)+\log (16) (3+\log (16))\right )-48 x^3 (1+\log (128))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + 12*x^3 - 20*x^4 + 27*x^5 - 84*x^6 + 64*x^7 + (8*x - 12*x^2 + 48*x^3 - 140*x^4 + 96*x^5)*Log[2]
+ (16*x - 48*x^2 + 32*x^3)*Log[2]^2)/8,x]

[Out]

(x^2*(-144*x^5 + 96*x^6 + 6*x^4*(9 + 32*Log[2]) + 6*(1 + Log[16])^2 + 3*x^2*(12 + 44*Log[2] + 32*Log[2]^2 + Lo
g[16]) - 4*x*(32*Log[2]^2 + Log[16]*(3 + Log[16])) - 48*x^3*(1 + Log[128])))/96

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fricas [B]  time = 1.59, size = 74, normalized size = 2.64 \begin {gather*} x^{8} - \frac {3}{2} \, x^{7} + \frac {9}{16} \, x^{6} - \frac {1}{2} \, x^{5} + \frac {3}{8} \, x^{4} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + \frac {1}{16} \, x^{2} + \frac {1}{2} \, {\left (4 \, x^{6} - 7 \, x^{5} + 3 \, x^{4} - x^{3} + x^{2}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(32*x^3-48*x^2+16*x)*log(2)^2+1/8*(96*x^5-140*x^4+48*x^3-12*x^2+8*x)*log(2)+8*x^7-21/2*x^6+27/8*
x^5-5/2*x^4+3/2*x^3+1/8*x,x, algorithm="fricas")

[Out]

x^8 - 3/2*x^7 + 9/16*x^6 - 1/2*x^5 + 3/8*x^4 + (x^4 - 2*x^3 + x^2)*log(2)^2 + 1/16*x^2 + 1/2*(4*x^6 - 7*x^5 +
3*x^4 - x^3 + x^2)*log(2)

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giac [B]  time = 0.22, size = 74, normalized size = 2.64 \begin {gather*} x^{8} - \frac {3}{2} \, x^{7} + \frac {9}{16} \, x^{6} - \frac {1}{2} \, x^{5} + \frac {3}{8} \, x^{4} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + \frac {1}{16} \, x^{2} + \frac {1}{2} \, {\left (4 \, x^{6} - 7 \, x^{5} + 3 \, x^{4} - x^{3} + x^{2}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(32*x^3-48*x^2+16*x)*log(2)^2+1/8*(96*x^5-140*x^4+48*x^3-12*x^2+8*x)*log(2)+8*x^7-21/2*x^6+27/8*
x^5-5/2*x^4+3/2*x^3+1/8*x,x, algorithm="giac")

[Out]

x^8 - 3/2*x^7 + 9/16*x^6 - 1/2*x^5 + 3/8*x^4 + (x^4 - 2*x^3 + x^2)*log(2)^2 + 1/16*x^2 + 1/2*(4*x^6 - 7*x^5 +
3*x^4 - x^3 + x^2)*log(2)

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maple [A]  time = 0.04, size = 29, normalized size = 1.04

method	result	size
gosper	\(\frac {\left (4 x^{3}+4 x \ln \relax (2)-3 x^{2}-4 \ln \relax (2)-1\right )^{2} x^{2}}{16}\)	\(29\)
norman	\(x^{8}+\left (\frac {9}{16}+2 \ln \relax (2)\right ) x^{6}+\left (-2 \ln \relax (2)^{2}-\frac {\ln \relax (2)}{2}\right ) x^{3}+\left (-\frac {7 \ln \relax (2)}{2}-\frac {1}{2}\right ) x^{5}+\left (\ln \relax (2)^{2}+\frac {\ln \relax (2)}{2}+\frac {1}{16}\right ) x^{2}+\left (\ln \relax (2)^{2}+\frac {3 \ln \relax (2)}{2}+\frac {3}{8}\right ) x^{4}-\frac {3 x^{7}}{2}\)	\(73\)
default	\(\frac {\ln \relax (2)^{2} \left (8 x^{4}-16 x^{3}+8 x^{2}\right )}{8}+\frac {\ln \relax (2) \left (16 x^{6}-28 x^{5}+12 x^{4}-4 x^{3}+4 x^{2}\right )}{8}+x^{8}-\frac {3 x^{7}}{2}+\frac {9 x^{6}}{16}-\frac {x^{5}}{2}+\frac {3 x^{4}}{8}+\frac {x^{2}}{16}\)	\(82\)
risch	\(x^{4} \ln \relax (2)^{2}-2 x^{3} \ln \relax (2)^{2}+x^{2} \ln \relax (2)^{2}+2 x^{6} \ln \relax (2)-\frac {7 x^{5} \ln \relax (2)}{2}+\frac {3 x^{4} \ln \relax (2)}{2}-\frac {x^{3} \ln \relax (2)}{2}+\frac {x^{2} \ln \relax (2)}{2}+x^{8}-\frac {3 x^{7}}{2}+\frac {9 x^{6}}{16}-\frac {x^{5}}{2}+\frac {3 x^{4}}{8}+\frac {x^{2}}{16}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(32*x^3-48*x^2+16*x)*ln(2)^2+1/8*(96*x^5-140*x^4+48*x^3-12*x^2+8*x)*ln(2)+8*x^7-21/2*x^6+27/8*x^5-5/2*
x^4+3/2*x^3+1/8*x,x,method=_RETURNVERBOSE)

[Out]

1/16*(4*x^3+4*x*ln(2)-3*x^2-4*ln(2)-1)^2*x^2

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maxima [B]  time = 0.44, size = 74, normalized size = 2.64 \begin {gather*} x^{8} - \frac {3}{2} \, x^{7} + \frac {9}{16} \, x^{6} - \frac {1}{2} \, x^{5} + \frac {3}{8} \, x^{4} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + \frac {1}{16} \, x^{2} + \frac {1}{2} \, {\left (4 \, x^{6} - 7 \, x^{5} + 3 \, x^{4} - x^{3} + x^{2}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(32*x^3-48*x^2+16*x)*log(2)^2+1/8*(96*x^5-140*x^4+48*x^3-12*x^2+8*x)*log(2)+8*x^7-21/2*x^6+27/8*
x^5-5/2*x^4+3/2*x^3+1/8*x,x, algorithm="maxima")

[Out]

x^8 - 3/2*x^7 + 9/16*x^6 - 1/2*x^5 + 3/8*x^4 + (x^4 - 2*x^3 + x^2)*log(2)^2 + 1/16*x^2 + 1/2*(4*x^6 - 7*x^5 +
3*x^4 - x^3 + x^2)*log(2)

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mupad [B]  time = 1.01, size = 74, normalized size = 2.64 \begin {gather*} x^8-\frac {3\,x^7}{2}+\left (2\,\ln \relax (2)+\frac {9}{16}\right )\,x^6+\left (-\frac {7\,\ln \relax (2)}{2}-\frac {1}{2}\right )\,x^5+\left (\frac {3\,\ln \relax (2)}{2}+{\ln \relax (2)}^2+\frac {3}{8}\right )\,x^4+\left (-\frac {\ln \relax (2)}{2}-2\,{\ln \relax (2)}^2\right )\,x^3+\left (\frac {\ln \left (256\right )}{16}+{\ln \relax (2)}^2+\frac {1}{16}\right )\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/8 + (log(2)^2*(16*x - 48*x^2 + 32*x^3))/8 + (3*x^3)/2 - (5*x^4)/2 + (27*x^5)/8 - (21*x^6)/2 + 8*x^7 + (l
og(2)*(8*x - 12*x^2 + 48*x^3 - 140*x^4 + 96*x^5))/8,x)

[Out]

x^4*((3*log(2))/2 + log(2)^2 + 3/8) + x^2*(log(256)/16 + log(2)^2 + 1/16) - x^5*((7*log(2))/2 + 1/2) + x^6*(2*
log(2) + 9/16) - x^3*(log(2)/2 + 2*log(2)^2) - (3*x^7)/2 + x^8

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sympy [B]  time = 0.07, size = 88, normalized size = 3.14 \begin {gather*} x^{8} - \frac {3 x^{7}}{2} + x^{6} \left (\frac {9}{16} + 2 \log {\relax (2 )}\right ) + x^{5} \left (- \frac {7 \log {\relax (2 )}}{2} - \frac {1}{2}\right ) + x^{4} \left (\frac {3}{8} + \log {\relax (2 )}^{2} + \frac {3 \log {\relax (2 )}}{2}\right ) + x^{3} \left (- 2 \log {\relax (2 )}^{2} - \frac {\log {\relax (2 )}}{2}\right ) + x^{2} \left (\frac {1}{16} + \frac {\log {\relax (2 )}}{2} + \log {\relax (2 )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(32*x**3-48*x**2+16*x)*ln(2)**2+1/8*(96*x**5-140*x**4+48*x**3-12*x**2+8*x)*ln(2)+8*x**7-21/2*x**
6+27/8*x**5-5/2*x**4+3/2*x**3+1/8*x,x)

[Out]

x**8 - 3*x**7/2 + x**6*(9/16 + 2*log(2)) + x**5*(-7*log(2)/2 - 1/2) + x**4*(3/8 + log(2)**2 + 3*log(2)/2) + x*
*3*(-2*log(2)**2 - log(2)/2) + x**2*(1/16 + log(2)/2 + log(2)**2)

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