3.99.9 \(\int \frac {-8+16 x-8 x^2+(-12-14 x+34 x^2-6 x^3-2 x^4) \log (6+x)+(-24+20 x+4 x^2) \log (6+x) \log (\log ^2(6+x))}{(6 x^3+19 x^4+21 x^5+9 x^6+x^7) \log (6+x)+(36 x^2+78 x^3+48 x^4+6 x^5) \log (6+x) \log (\log ^2(6+x))+(72 x+84 x^2+12 x^3) \log (6+x) \log ^2(\log ^2(6+x))+(48+8 x) \log (6+x) \log ^3(\log ^2(6+x))} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{\left (-x+\frac {2 \left (x+\log \left (\log ^2(6+x)\right )\right )}{1-x}\right )^2} \]

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Rubi [F]  time = 3.43, 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 {-8+16 x-8 x^2+\left (-12-14 x+34 x^2-6 x^3-2 x^4\right ) \log (6+x)+\left (-24+20 x+4 x^2\right ) \log (6+x) \log \left (\log ^2(6+x)\right )}{\left (6 x^3+19 x^4+21 x^5+9 x^6+x^7\right ) \log (6+x)+\left (36 x^2+78 x^3+48 x^4+6 x^5\right ) \log (6+x) \log \left (\log ^2(6+x)\right )+\left (72 x+84 x^2+12 x^3\right ) \log (6+x) \log ^2\left (\log ^2(6+x)\right )+(48+8 x) \log (6+x) \log ^3\left (\log ^2(6+x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 + 16*x - 8*x^2 + (-12 - 14*x + 34*x^2 - 6*x^3 - 2*x^4)*Log[6 + x] + (-24 + 20*x + 4*x^2)*Log[6 + x]*Lo
g[Log[6 + x]^2])/((6*x^3 + 19*x^4 + 21*x^5 + 9*x^6 + x^7)*Log[6 + x] + (36*x^2 + 78*x^3 + 48*x^4 + 6*x^5)*Log[
6 + x]*Log[Log[6 + x]^2] + (72*x + 84*x^2 + 12*x^3)*Log[6 + x]*Log[Log[6 + x]^2]^2 + (48 + 8*x)*Log[6 + x]*Log
[Log[6 + x]^2]^3),x]

[Out]

49/(x + x^2 + 2*Log[Log[6 + x]^2])^2 + 96*Defer[Int][(x + x^2 + 2*Log[Log[6 + x]^2])^(-3), x] + 196*Defer[Int]
[x/(x + x^2 + 2*Log[Log[6 + x]^2])^3, x] + 6*Defer[Int][x^2/(x + x^2 + 2*Log[Log[6 + x]^2])^3, x] - 4*Defer[In
t][x^3/(x + x^2 + 2*Log[Log[6 + x]^2])^3, x] + 64*Defer[Int][1/(Log[6 + x]*(x + x^2 + 2*Log[Log[6 + x]^2])^3),
x] - 8*Defer[Int][x/(Log[6 + x]*(x + x^2 + 2*Log[Log[6 + x]^2])^3), x] - 2*Defer[Int][(x + x^2 + 2*Log[Log[6
+ x]^2])^(-2), x] + 2*Defer[Int][x/(x + x^2 + 2*Log[Log[6 + x]^2])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (1-x) \left (4 (-1+x)+(6+x) \log (6+x) \left (-1-2 x+x^2-2 \log \left (\log ^2(6+x)\right )\right )\right )}{(6+x) \log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\\ &=2 \int \frac {(1-x) \left (4 (-1+x)+(6+x) \log (6+x) \left (-1-2 x+x^2-2 \log \left (\log ^2(6+x)\right )\right )\right )}{(6+x) \log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\\ &=2 \int \left (-\frac {(-1+x)^2 \left (4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)\right )}{(6+x) \log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {-1+x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {(-1+x)^2 \left (4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)\right )}{(6+x) \log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\right )+2 \int \frac {-1+x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx\\ &=-\left (2 \int \left (-\frac {8 \left (4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)\right )}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {x \left (4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)\right )}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {49 \left (4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)\right )}{(6+x) \log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}\right ) \, dx\right )+2 \int \left (-\frac {1}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2}+\frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)\right )}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\right )-2 \int \frac {1}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx+2 \int \frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx+16 \int \frac {4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx-98 \int \frac {4+6 \log (6+x)+13 x \log (6+x)+2 x^2 \log (6+x)}{(6+x) \log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\\ &=\frac {49}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2}-2 \int \frac {x \left (4+\left (6+13 x+2 x^2\right ) \log (6+x)\right )}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx-2 \int \frac {1}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx+2 \int \frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx+16 \int \frac {4+\left (6+13 x+2 x^2\right ) \log (6+x)}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\\ &=\frac {49}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2}-2 \int \frac {1}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx+2 \int \frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx-2 \int \left (\frac {6 x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {13 x^2}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {2 x^3}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {4 x}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}\right ) \, dx+16 \int \left (\frac {6}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {13 x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {2 x^2}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}+\frac {4}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3}\right ) \, dx\\ &=\frac {49}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2}-2 \int \frac {1}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx+2 \int \frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \, dx-4 \int \frac {x^3}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx-8 \int \frac {x}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx-12 \int \frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx-26 \int \frac {x^2}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx+32 \int \frac {x^2}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx+64 \int \frac {1}{\log (6+x) \left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx+96 \int \frac {1}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx+208 \int \frac {x}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.90, size = 22, normalized size = 0.92 \begin {gather*} \frac {(-1+x)^2}{\left (x+x^2+2 \log \left (\log ^2(6+x)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 16*x - 8*x^2 + (-12 - 14*x + 34*x^2 - 6*x^3 - 2*x^4)*Log[6 + x] + (-24 + 20*x + 4*x^2)*Log[6 +
x]*Log[Log[6 + x]^2])/((6*x^3 + 19*x^4 + 21*x^5 + 9*x^6 + x^7)*Log[6 + x] + (36*x^2 + 78*x^3 + 48*x^4 + 6*x^5
)*Log[6 + x]*Log[Log[6 + x]^2] + (72*x + 84*x^2 + 12*x^3)*Log[6 + x]*Log[Log[6 + x]^2]^2 + (48 + 8*x)*Log[6 +
x]*Log[Log[6 + x]^2]^3),x]

[Out]

(-1 + x)^2/(x + x^2 + 2*Log[Log[6 + x]^2])^2

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fricas [B]  time = 0.75, size = 48, normalized size = 2.00 \begin {gather*} \frac {x^{2} - 2 \, x + 1}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, {\left (x^{2} + x\right )} \log \left (\log \left (x + 6\right )^{2}\right ) + 4 \, \log \left (\log \left (x + 6\right )^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+20*x-24)*log(x+6)*log(log(x+6)^2)+(-2*x^4-6*x^3+34*x^2-14*x-12)*log(x+6)-8*x^2+16*x-8)/((8*x
+48)*log(x+6)*log(log(x+6)^2)^3+(12*x^3+84*x^2+72*x)*log(x+6)*log(log(x+6)^2)^2+(6*x^5+48*x^4+78*x^3+36*x^2)*l
og(x+6)*log(log(x+6)^2)+(x^7+9*x^6+21*x^5+19*x^4+6*x^3)*log(x+6)),x, algorithm="fricas")

[Out]

(x^2 - 2*x + 1)/(x^4 + 2*x^3 + x^2 + 4*(x^2 + x)*log(log(x + 6)^2) + 4*log(log(x + 6)^2)^2)

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giac [B]  time = 1.27, size = 257, normalized size = 10.71 \begin {gather*} \frac {2 \, x^{4} \log \left (x + 6\right ) + 9 \, x^{3} \log \left (x + 6\right ) - 18 \, x^{2} \log \left (x + 6\right ) + 4 \, x^{2} + x \log \left (x + 6\right ) - 8 \, x + 6 \, \log \left (x + 6\right ) + 4}{2 \, x^{6} \log \left (x + 6\right ) + 17 \, x^{5} \log \left (x + 6\right ) + 8 \, x^{4} \log \left (\log \left (x + 6\right )^{2}\right ) \log \left (x + 6\right ) + 34 \, x^{4} \log \left (x + 6\right ) + 60 \, x^{3} \log \left (\log \left (x + 6\right )^{2}\right ) \log \left (x + 6\right ) + 8 \, x^{2} \log \left (\log \left (x + 6\right )^{2}\right )^{2} \log \left (x + 6\right ) + 4 \, x^{4} + 25 \, x^{3} \log \left (x + 6\right ) + 76 \, x^{2} \log \left (\log \left (x + 6\right )^{2}\right ) \log \left (x + 6\right ) + 52 \, x \log \left (\log \left (x + 6\right )^{2}\right )^{2} \log \left (x + 6\right ) + 8 \, x^{3} + 16 \, x^{2} \log \left (\log \left (x + 6\right )^{2}\right ) + 6 \, x^{2} \log \left (x + 6\right ) + 24 \, x \log \left (\log \left (x + 6\right )^{2}\right ) \log \left (x + 6\right ) + 24 \, \log \left (\log \left (x + 6\right )^{2}\right )^{2} \log \left (x + 6\right ) + 4 \, x^{2} + 16 \, x \log \left (\log \left (x + 6\right )^{2}\right ) + 16 \, \log \left (\log \left (x + 6\right )^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+20*x-24)*log(x+6)*log(log(x+6)^2)+(-2*x^4-6*x^3+34*x^2-14*x-12)*log(x+6)-8*x^2+16*x-8)/((8*x
+48)*log(x+6)*log(log(x+6)^2)^3+(12*x^3+84*x^2+72*x)*log(x+6)*log(log(x+6)^2)^2+(6*x^5+48*x^4+78*x^3+36*x^2)*l
og(x+6)*log(log(x+6)^2)+(x^7+9*x^6+21*x^5+19*x^4+6*x^3)*log(x+6)),x, algorithm="giac")

[Out]

(2*x^4*log(x + 6) + 9*x^3*log(x + 6) - 18*x^2*log(x + 6) + 4*x^2 + x*log(x + 6) - 8*x + 6*log(x + 6) + 4)/(2*x
^6*log(x + 6) + 17*x^5*log(x + 6) + 8*x^4*log(log(x + 6)^2)*log(x + 6) + 34*x^4*log(x + 6) + 60*x^3*log(log(x
+ 6)^2)*log(x + 6) + 8*x^2*log(log(x + 6)^2)^2*log(x + 6) + 4*x^4 + 25*x^3*log(x + 6) + 76*x^2*log(log(x + 6)^
2)*log(x + 6) + 52*x*log(log(x + 6)^2)^2*log(x + 6) + 8*x^3 + 16*x^2*log(log(x + 6)^2) + 6*x^2*log(x + 6) + 24
*x*log(log(x + 6)^2)*log(x + 6) + 24*log(log(x + 6)^2)^2*log(x + 6) + 4*x^2 + 16*x*log(log(x + 6)^2) + 16*log(
log(x + 6)^2)^2)

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maple [C]  time = 0.34, size = 88, normalized size = 3.67

method	result	size
risch	\(\frac {x^{2}-2 x +1}{\left (4 \ln \left (\ln \left (x +6\right )\right )-i \pi \mathrm {csgn}\left (i \ln \left (x +6\right )^{2}\right )^{3}-i \pi \mathrm {csgn}\left (i \ln \left (x +6\right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x +6\right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \ln \left (x +6\right )\right ) \mathrm {csgn}\left (i \ln \left (x +6\right )^{2}\right )^{2}+x^{2}+x \right )^{2}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+20*x-24)*ln(x+6)*ln(ln(x+6)^2)+(-2*x^4-6*x^3+34*x^2-14*x-12)*ln(x+6)-8*x^2+16*x-8)/((8*x+48)*ln(x+
6)*ln(ln(x+6)^2)^3+(12*x^3+84*x^2+72*x)*ln(x+6)*ln(ln(x+6)^2)^2+(6*x^5+48*x^4+78*x^3+36*x^2)*ln(x+6)*ln(ln(x+6
)^2)+(x^7+9*x^6+21*x^5+19*x^4+6*x^3)*ln(x+6)),x,method=_RETURNVERBOSE)

[Out]

(x^2-2*x+1)/(4*ln(ln(x+6))-I*Pi*csgn(I*ln(x+6)^2)^3-I*Pi*csgn(I*ln(x+6))^2*csgn(I*ln(x+6)^2)+2*I*Pi*csgn(I*ln(
x+6))*csgn(I*ln(x+6)^2)^2+x^2+x)^2

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maxima [B]  time = 0.44, size = 44, normalized size = 1.83 \begin {gather*} \frac {x^{2} - 2 \, x + 1}{x^{4} + 2 \, x^{3} + x^{2} + 8 \, {\left (x^{2} + x\right )} \log \left (\log \left (x + 6\right )\right ) + 16 \, \log \left (\log \left (x + 6\right )\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+20*x-24)*log(x+6)*log(log(x+6)^2)+(-2*x^4-6*x^3+34*x^2-14*x-12)*log(x+6)-8*x^2+16*x-8)/((8*x
+48)*log(x+6)*log(log(x+6)^2)^3+(12*x^3+84*x^2+72*x)*log(x+6)*log(log(x+6)^2)^2+(6*x^5+48*x^4+78*x^3+36*x^2)*l
og(x+6)*log(log(x+6)^2)+(x^7+9*x^6+21*x^5+19*x^4+6*x^3)*log(x+6)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (x+6\right )\,\left (2\,x^4+6\,x^3-34\,x^2+14\,x+12\right )-16\,x+8\,x^2-\ln \left ({\ln \left (x+6\right )}^2\right )\,\ln \left (x+6\right )\,\left (4\,x^2+20\,x-24\right )+8}{\ln \left (x+6\right )\,\left (8\,x+48\right )\,{\ln \left ({\ln \left (x+6\right )}^2\right )}^3+\ln \left (x+6\right )\,\left (12\,x^3+84\,x^2+72\,x\right )\,{\ln \left ({\ln \left (x+6\right )}^2\right )}^2+\ln \left (x+6\right )\,\left (6\,x^5+48\,x^4+78\,x^3+36\,x^2\right )\,\ln \left ({\ln \left (x+6\right )}^2\right )+\ln \left (x+6\right )\,\left (x^7+9\,x^6+21\,x^5+19\,x^4+6\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + 6)*(14*x - 34*x^2 + 6*x^3 + 2*x^4 + 12) - 16*x + 8*x^2 - log(log(x + 6)^2)*log(x + 6)*(20*x + 4*
x^2 - 24) + 8)/(log(x + 6)*(6*x^3 + 19*x^4 + 21*x^5 + 9*x^6 + x^7) + log(log(x + 6)^2)^3*log(x + 6)*(8*x + 48)
+ log(log(x + 6)^2)^2*log(x + 6)*(72*x + 84*x^2 + 12*x^3) + log(log(x + 6)^2)*log(x + 6)*(36*x^2 + 78*x^3 + 4
8*x^4 + 6*x^5)),x)

[Out]

int(-(log(x + 6)*(14*x - 34*x^2 + 6*x^3 + 2*x^4 + 12) - 16*x + 8*x^2 - log(log(x + 6)^2)*log(x + 6)*(20*x + 4*
x^2 - 24) + 8)/(log(x + 6)*(6*x^3 + 19*x^4 + 21*x^5 + 9*x^6 + x^7) + log(log(x + 6)^2)^3*log(x + 6)*(8*x + 48)
+ log(log(x + 6)^2)^2*log(x + 6)*(72*x + 84*x^2 + 12*x^3) + log(log(x + 6)^2)*log(x + 6)*(36*x^2 + 78*x^3 + 4
8*x^4 + 6*x^5)), x)

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sympy [B]  time = 0.41, size = 48, normalized size = 2.00 \begin {gather*} \frac {x^{2} - 2 x + 1}{x^{4} + 2 x^{3} + x^{2} + \left (4 x^{2} + 4 x\right ) \log {\left (\log {\left (x + 6 \right )}^{2} \right )} + 4 \log {\left (\log {\left (x + 6 \right )}^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+20*x-24)*ln(x+6)*ln(ln(x+6)**2)+(-2*x**4-6*x**3+34*x**2-14*x-12)*ln(x+6)-8*x**2+16*x-8)/((8
*x+48)*ln(x+6)*ln(ln(x+6)**2)**3+(12*x**3+84*x**2+72*x)*ln(x+6)*ln(ln(x+6)**2)**2+(6*x**5+48*x**4+78*x**3+36*x
**2)*ln(x+6)*ln(ln(x+6)**2)+(x**7+9*x**6+21*x**5+19*x**4+6*x**3)*ln(x+6)),x)

[Out]

(x**2 - 2*x + 1)/(x**4 + 2*x**3 + x**2 + (4*x**2 + 4*x)*log(log(x + 6)**2) + 4*log(log(x + 6)**2)**2)

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