3.71.62 \(\int \frac {-45+182 x+3 x^2-10 x^3+(-32+8 x) \log (2)}{-45 x+31 x^2+3 x^3-x^4+(9-8 x+x^2) \log (2)} \, dx\) [7062]
Optimal. Leaf size=27 \[ \log \left (\left (-(3-x)^2+2 x\right )^4 (x (5+x)-\log (2))\right ) \]
[Out]
ln((2*x-(3-x)^2)^4*(-ln(2)+(5+x)*x))
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Rubi [A]
time = 0.06, antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps
used = 4, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2099, 642}
\begin {gather*} 4 \log \left (x^2-8 x+9\right )+\log \left (x^2+5 x-\log (2)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-45 + 182*x + 3*x^2 - 10*x^3 + (-32 + 8*x)*Log[2])/(-45*x + 31*x^2 + 3*x^3 - x^4 + (9 - 8*x + x^2)*Log[2]
),x]
[Out]
4*Log[9 - 8*x + x^2] + Log[5*x + x^2 - Log[2]]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 2099
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 {8 (-4+x)}{9-8 x+x^2}+\frac {5+2 x}{5 x+x^2-\log (2)}\right ) \, dx\\ &=8 \int \frac {-4+x}{9-8 x+x^2} \, dx+\int \frac {5+2 x}{5 x+x^2-\log (2)} \, dx\\ &=4 \log \left (9-8 x+x^2\right )+\log \left (5 x+x^2-\log (2)\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(27)=54\).
time = 0.13, size = 72, normalized size = 2.67 \begin {gather*} \frac {(\log (2) (-380+\log (16))+9 (296+\log (16))) \log \left (9-8 x+x^2\right )+\left (5 \log ^2(2)-9 (-74+\log (16))-\log (2) (50+\log (16))\right ) \log \left (5 x+x^2-\log (2)\right )}{666-86 \log (2)+\log ^2(2)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-45 + 182*x + 3*x^2 - 10*x^3 + (-32 + 8*x)*Log[2])/(-45*x + 31*x^2 + 3*x^3 - x^4 + (9 - 8*x + x^2)*
Log[2]),x]
[Out]
((Log[2]*(-380 + Log[16]) + 9*(296 + Log[16]))*Log[9 - 8*x + x^2] + (5*Log[2]^2 - 9*(-74 + Log[16]) - Log[2]*(
50 + Log[16]))*Log[5*x + x^2 - Log[2]])/(666 - 86*Log[2] + Log[2]^2)
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Maple [A]
time = 0.38, size = 25, normalized size = 0.93
| | |
| method |
result |
size |
| | |
| default |
\(4 \ln \left (x^{2}-8 x +9\right )+\ln \left (x^{2}-\ln \left (2\right )+5 x \right )\) |
\(25\) |
| norman |
\(4 \ln \left (x^{2}-8 x +9\right )+\ln \left (-x^{2}+\ln \left (2\right )-5 x \right )\) |
\(25\) |
| risch |
\(4 \ln \left (x^{2}-8 x +9\right )+\ln \left (x^{2}-\ln \left (2\right )+5 x \right )\) |
\(25\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((8*x-32)*ln(2)-10*x^3+3*x^2+182*x-45)/((x^2-8*x+9)*ln(2)-x^4+3*x^3+31*x^2-45*x),x,method=_RETURNVERBOSE)
[Out]
4*ln(x^2-8*x+9)+ln(x^2-ln(2)+5*x)
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Maxima [A]
time = 0.27, size = 24, normalized size = 0.89 \begin {gather*} \log \left (x^{2} + 5 \, x - \log \left (2\right )\right ) + 4 \, \log \left (x^{2} - 8 \, x + 9\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((8*x-32)*log(2)-10*x^3+3*x^2+182*x-45)/((x^2-8*x+9)*log(2)-x^4+3*x^3+31*x^2-45*x),x, algorithm="max
ima")
[Out]
log(x^2 + 5*x - log(2)) + 4*log(x^2 - 8*x + 9)
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Fricas [A]
time = 0.42, size = 24, normalized size = 0.89 \begin {gather*} \log \left (x^{2} + 5 \, x - \log \left (2\right )\right ) + 4 \, \log \left (x^{2} - 8 \, x + 9\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((8*x-32)*log(2)-10*x^3+3*x^2+182*x-45)/((x^2-8*x+9)*log(2)-x^4+3*x^3+31*x^2-45*x),x, algorithm="fri
cas")
[Out]
log(x^2 + 5*x - log(2)) + 4*log(x^2 - 8*x + 9)
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Sympy [A]
time = 0.78, size = 22, normalized size = 0.81 \begin {gather*} 4 \log {\left (x^{2} - 8 x + 9 \right )} + \log {\left (x^{2} + 5 x - \log {\left (2 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((8*x-32)*ln(2)-10*x**3+3*x**2+182*x-45)/((x**2-8*x+9)*ln(2)-x**4+3*x**3+31*x**2-45*x),x)
[Out]
4*log(x**2 - 8*x + 9) + log(x**2 + 5*x - log(2))
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Giac [A]
time = 0.42, size = 26, normalized size = 0.96 \begin {gather*} \log \left ({\left | x^{2} + 5 \, x - \log \left (2\right ) \right |}\right ) + 4 \, \log \left ({\left | x^{2} - 8 \, x + 9 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((8*x-32)*log(2)-10*x^3+3*x^2+182*x-45)/((x^2-8*x+9)*log(2)-x^4+3*x^3+31*x^2-45*x),x, algorithm="gia
c")
[Out]
log(abs(x^2 + 5*x - log(2))) + 4*log(abs(x^2 - 8*x + 9))
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Mupad [B]
time = 6.60, size = 2500, normalized size = 92.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((182*x + log(2)*(8*x - 32) + 3*x^2 - 10*x^3 - 45)/(31*x^2 - 45*x + 3*x^3 - x^4 + log(2)*(x^2 - 8*x + 9)),x
)
[Out]
symsum(log(653168*log(2) - 3095936*x - 1663740*root(30508128*z^4*log(2) + 6720224*z^4*log(2)^2 - 857136*z^4*lo
g(2)^3 + 18564*z^4*log(2)^4 - 112*z^4*log(2)^5 - 310489200*z^4 - 305081280*z^3*log(2) - 67202240*z^3*log(2)^2
+ 8571360*z^3*log(2)^3 - 185640*z^3*log(2)^4 + 1120*z^3*log(2)^5 + 3104892000*z^3 + 1006768224*z^2*log(2) + 22
1767392*z^2*log(2)^2 - 28285488*z^2*log(2)^3 + 612612*z^2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143600*z^2 - 268
808960*z*log(2)^2 + 34285440*z*log(2)^3 - 742560*z*log(2)^4 + 4480*z*log(2)^5 - 1220325120*z*log(2) + 12419568
000*z + 488130048*log(2) + 107523584*log(2)^2 - 13714176*log(2)^3 + 297024*log(2)^4 - 1792*log(2)^5 - 49678272
00, z, k) - 1058700*root(30508128*z^4*log(2) + 6720224*z^4*log(2)^2 - 857136*z^4*log(2)^3 + 18564*z^4*log(2)^4
- 112*z^4*log(2)^5 - 310489200*z^4 - 305081280*z^3*log(2) - 67202240*z^3*log(2)^2 + 8571360*z^3*log(2)^3 - 18
5640*z^3*log(2)^4 + 1120*z^3*log(2)^5 + 3104892000*z^3 + 1006768224*z^2*log(2) + 221767392*z^2*log(2)^2 - 2828
5488*z^2*log(2)^3 + 612612*z^2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143600*z^2 - 268808960*z*log(2)^2 + 3428544
0*z*log(2)^3 - 742560*z*log(2)^4 + 4480*z*log(2)^5 - 1220325120*z*log(2) + 12419568000*z + 488130048*log(2) +
107523584*log(2)^2 - 13714176*log(2)^3 + 297024*log(2)^4 - 1792*log(2)^5 - 4967827200, z, k)*log(2) + 5098704*
root(30508128*z^4*log(2) + 6720224*z^4*log(2)^2 - 857136*z^4*log(2)^3 + 18564*z^4*log(2)^4 - 112*z^4*log(2)^5
- 310489200*z^4 - 305081280*z^3*log(2) - 67202240*z^3*log(2)^2 + 8571360*z^3*log(2)^3 - 185640*z^3*log(2)^4 +
1120*z^3*log(2)^5 + 3104892000*z^3 + 1006768224*z^2*log(2) + 221767392*z^2*log(2)^2 - 28285488*z^2*log(2)^3 +
612612*z^2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143600*z^2 - 268808960*z*log(2)^2 + 34285440*z*log(2)^3 - 74256
0*z*log(2)^4 + 4480*z*log(2)^5 - 1220325120*z*log(2) + 12419568000*z + 488130048*log(2) + 107523584*log(2)^2 -
13714176*log(2)^3 + 297024*log(2)^4 - 1792*log(2)^5 - 4967827200, z, k)*x - 305616*x*log(2) - 21720*root(3050
8128*z^4*log(2) + 6720224*z^4*log(2)^2 - 857136*z^4*log(2)^3 + 18564*z^4*log(2)^4 - 112*z^4*log(2)^5 - 3104892
00*z^4 - 305081280*z^3*log(2) - 67202240*z^3*log(2)^2 + 8571360*z^3*log(2)^3 - 185640*z^3*log(2)^4 + 1120*z^3*
log(2)^5 + 3104892000*z^3 + 1006768224*z^2*log(2) + 221767392*z^2*log(2)^2 - 28285488*z^2*log(2)^3 + 612612*z^
2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143600*z^2 - 268808960*z*log(2)^2 + 34285440*z*log(2)^3 - 742560*z*log(2
)^4 + 4480*z*log(2)^5 - 1220325120*z*log(2) + 12419568000*z + 488130048*log(2) + 107523584*log(2)^2 - 13714176
*log(2)^3 + 297024*log(2)^4 - 1792*log(2)^5 - 4967827200, z, k)*log(2)^2 + 466092*root(30508128*z^4*log(2) + 6
720224*z^4*log(2)^2 - 857136*z^4*log(2)^3 + 18564*z^4*log(2)^4 - 112*z^4*log(2)^5 - 310489200*z^4 - 305081280*
z^3*log(2) - 67202240*z^3*log(2)^2 + 8571360*z^3*log(2)^3 - 185640*z^3*log(2)^4 + 1120*z^3*log(2)^5 + 31048920
00*z^3 + 1006768224*z^2*log(2) + 221767392*z^2*log(2)^2 - 28285488*z^2*log(2)^3 + 612612*z^2*log(2)^4 - 3696*z
^2*log(2)^5 - 10246143600*z^2 - 268808960*z*log(2)^2 + 34285440*z*log(2)^3 - 742560*z*log(2)^4 + 4480*z*log(2)
^5 - 1220325120*z*log(2) + 12419568000*z + 488130048*log(2) + 107523584*log(2)^2 - 13714176*log(2)^3 + 297024*
log(2)^4 - 1792*log(2)^5 - 4967827200, z, k)^2*log(2) - 288*root(30508128*z^4*log(2) + 6720224*z^4*log(2)^2 -
857136*z^4*log(2)^3 + 18564*z^4*log(2)^4 - 112*z^4*log(2)^5 - 310489200*z^4 - 305081280*z^3*log(2) - 67202240*
z^3*log(2)^2 + 8571360*z^3*log(2)^3 - 185640*z^3*log(2)^4 + 1120*z^3*log(2)^5 + 3104892000*z^3 + 1006768224*z^
2*log(2) + 221767392*z^2*log(2)^2 - 28285488*z^2*log(2)^3 + 612612*z^2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143
600*z^2 - 268808960*z*log(2)^2 + 34285440*z*log(2)^3 - 742560*z*log(2)^4 + 4480*z*log(2)^5 - 1220325120*z*log(
2) + 12419568000*z + 488130048*log(2) + 107523584*log(2)^2 - 13714176*log(2)^3 + 297024*log(2)^4 - 1792*log(2)
^5 - 4967827200, z, k)*log(2)^3 - 60560*root(30508128*z^4*log(2) + 6720224*z^4*log(2)^2 - 857136*z^4*log(2)^3
+ 18564*z^4*log(2)^4 - 112*z^4*log(2)^5 - 310489200*z^4 - 305081280*z^3*log(2) - 67202240*z^3*log(2)^2 + 85713
60*z^3*log(2)^3 - 185640*z^3*log(2)^4 + 1120*z^3*log(2)^5 + 3104892000*z^3 + 1006768224*z^2*log(2) + 221767392
*z^2*log(2)^2 - 28285488*z^2*log(2)^3 + 612612*z^2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143600*z^2 - 268808960*
z*log(2)^2 + 34285440*z*log(2)^3 - 742560*z*log(2)^4 + 4480*z*log(2)^5 - 1220325120*z*log(2) + 12419568000*z +
488130048*log(2) + 107523584*log(2)^2 - 13714176*log(2)^3 + 297024*log(2)^4 - 1792*log(2)^5 - 4967827200, z,
k)^3*log(2) - 2309964*root(30508128*z^4*log(2) + 6720224*z^4*log(2)^2 - 857136*z^4*log(2)^3 + 18564*z^4*log(2)
^4 - 112*z^4*log(2)^5 - 310489200*z^4 - 305081280*z^3*log(2) - 67202240*z^3*log(2)^2 + 8571360*z^3*log(2)^3 -
185640*z^3*log(2)^4 + 1120*z^3*log(2)^5 + 3104892000*z^3 + 1006768224*z^2*log(2) + 221767392*z^2*log(2)^2 - 28
285488*z^2*log(2)^3 + 612612*z^2*log(2)^4 - 3696*z^2*log(2)^5 - 10246143600*z^2 - 268808960*z*log(2)^2 + 34285
440*z*log(2)^3 - 742560*z*log(2)^4 + 4480*z*log...
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