3.47.97 \(\int \frac {24-44 x+2 x^2-26 x^4+2 x^5+(20-24 x+4 x^2-5 x^4+x^5) \log (\frac {1}{4} (-4+4 x+x^4))}{-8+8 x+2 x^4} \, dx\) [4697]

Optimal. Leaf size=26 \[ 3-3 x-\frac {1}{4} (10-x) x \log \left (-1+x+\frac {x^4}{4}\right ) \]

[Out]

3-3*x-1/4*x*ln(1/4*x^4+x-1)*(10-x)

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Rubi [A]
time = 0.60, antiderivative size = 41, normalized size of antiderivative = 1.58, number of steps used = 13, number of rules used = 4, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6874, 2605, 12, 2125} \begin {gather*} \frac {1}{4} (5-x)^2 \log \left (\frac {x^4}{4}+x-1\right )-\frac {25}{4} \log \left (-x^4-4 x+4\right )-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24 - 44*x + 2*x^2 - 26*x^4 + 2*x^5 + (20 - 24*x + 4*x^2 - 5*x^4 + x^5)*Log[(-4 + 4*x + x^4)/4])/(-8 + 8*x
+ 2*x^4),x]

[Out]

-3*x - (25*Log[4 - 4*x - x^4])/4 + ((5 - x)^2*Log[-1 + x + x^4/4])/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2125

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coef
f[Qn, x, n])), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

Rule 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {12-22 x+x^2-13 x^4+x^5}{-4+4 x+x^4}+\frac {1}{2} (-5+x) \log \left (-1+x+\frac {x^4}{4}\right )\right ) \, dx\\ &=\frac {1}{2} \int (-5+x) \log \left (-1+x+\frac {x^4}{4}\right ) \, dx+\int \frac {12-22 x+x^2-13 x^4+x^5}{-4+4 x+x^4} \, dx\\ &=\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )-\frac {1}{4} \int \frac {4 (5-x)^2 \left (-1-x^3\right )}{4-4 x-x^4} \, dx+\int \left (-13+x-\frac {40-34 x+3 x^2}{-4+4 x+x^4}\right ) \, dx\\ &=-13 x+\frac {x^2}{2}+\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )-\int \frac {(5-x)^2 \left (-1-x^3\right )}{4-4 x-x^4} \, dx-\int \frac {40-34 x+3 x^2}{-4+4 x+x^4} \, dx\\ &=-13 x+\frac {x^2}{2}+\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )-\int \left (-10+x+\frac {15-34 x+3 x^2-25 x^3}{4-4 x-x^4}\right ) \, dx-\int \left (\frac {40}{-4+4 x+x^4}-\frac {34 x}{-4+4 x+x^4}+\frac {3 x^2}{-4+4 x+x^4}\right ) \, dx\\ &=-3 x+\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )-3 \int \frac {x^2}{-4+4 x+x^4} \, dx+34 \int \frac {x}{-4+4 x+x^4} \, dx-40 \int \frac {1}{-4+4 x+x^4} \, dx-\int \frac {15-34 x+3 x^2-25 x^3}{4-4 x-x^4} \, dx\\ &=-3 x-\frac {25}{4} \log \left (4-4 x-x^4\right )+\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )+\frac {1}{4} \int \frac {-160+136 x-12 x^2}{4-4 x-x^4} \, dx-3 \int \frac {x^2}{-4+4 x+x^4} \, dx+34 \int \frac {x}{-4+4 x+x^4} \, dx-40 \int \frac {1}{-4+4 x+x^4} \, dx\\ &=-3 x-\frac {25}{4} \log \left (4-4 x-x^4\right )+\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )+\frac {1}{4} \int \left (\frac {160}{-4+4 x+x^4}-\frac {136 x}{-4+4 x+x^4}+\frac {12 x^2}{-4+4 x+x^4}\right ) \, dx-3 \int \frac {x^2}{-4+4 x+x^4} \, dx+34 \int \frac {x}{-4+4 x+x^4} \, dx-40 \int \frac {1}{-4+4 x+x^4} \, dx\\ &=-3 x-\frac {25}{4} \log \left (4-4 x-x^4\right )+\frac {1}{4} (5-x)^2 \log \left (-1+x+\frac {x^4}{4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.19, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{2} \left (-6 x+\frac {1}{2} (-10+x) x \log \left (-1+x+\frac {x^4}{4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 - 44*x + 2*x^2 - 26*x^4 + 2*x^5 + (20 - 24*x + 4*x^2 - 5*x^4 + x^5)*Log[(-4 + 4*x + x^4)/4])/(-8
+ 8*x + 2*x^4),x]

[Out]

(-6*x + ((-10 + x)*x*Log[-1 + x + x^4/4])/2)/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 151, normalized size = 5.81

method	result	size
risch	\(\left (\frac {1}{4} x^{2}-\frac {5}{2} x \right ) \ln \left (\frac {1}{4} x^{4}+x -1\right )-3 x\)	\(24\)
norman	\(-3 x -\frac {5 \ln \left (\frac {1}{4} x^{4}+x -1\right ) x}{2}+\frac {\ln \left (\frac {1}{4} x^{4}+x -1\right ) x^{2}}{4}\)	\(31\)
default	\(-\frac {x^{2} \ln \left (2\right )}{2}+5 x \ln \left (2\right )+\frac {x^{2} \ln \left (x^{4}+4 x -4\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (3 \textit {\_R}^{2}-4 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{4}-\frac {5 x \ln \left (x^{4}+4 x -4\right )}{2}-3 x +\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (-3 \textit {\_R} +4\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+34 \textit {\_R} -40\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{4}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5-5*x^4+4*x^2-24*x+20)*ln(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-44*x+24)/(2*x^4+8*x-8),x,method=_RETURNVERBO
SE)

[Out]

-1/2*x^2*ln(2)+5*x*ln(2)+1/4*x^2*ln(x^4+4*x-4)+1/4*sum((3*_R^2-4*_R)/(_R^3+1)*ln(x-_R),_R=RootOf(_Z^4+4*_Z-4))
-5/2*x*ln(x^4+4*x-4)-3*x+5/2*sum((-3*_R+4)/(_R^3+1)*ln(x-_R),_R=RootOf(_Z^4+4*_Z-4))+1/4*sum((-3*_R^2+34*_R-40
)/(_R^3+1)*ln(x-_R),_R=RootOf(_Z^4+4*_Z-4))

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Maxima [A]
time = 0.52, size = 34, normalized size = 1.31 \begin {gather*} -\frac {1}{2} \, x^{2} \log \left (2\right ) + x {\left (5 \, \log \left (2\right ) - 3\right )} + \frac {1}{4} \, {\left (x^{2} - 10 \, x\right )} \log \left (x^{4} + 4 \, x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-44*x+24)/(2*x^4+8*x-8),x, algorithm="
maxima")

[Out]

-1/2*x^2*log(2) + x*(5*log(2) - 3) + 1/4*(x^2 - 10*x)*log(x^4 + 4*x - 4)

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Fricas [A]
time = 0.37, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\left (x^{2} - 10 \, x\right )} \log \left (\frac {1}{4} \, x^{4} + x - 1\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-44*x+24)/(2*x^4+8*x-8),x, algorithm="
fricas")

[Out]

1/4*(x^2 - 10*x)*log(1/4*x^4 + x - 1) - 3*x

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Sympy [A]
time = 0.07, size = 22, normalized size = 0.85 \begin {gather*} - 3 x + \left (\frac {x^{2}}{4} - \frac {5 x}{2}\right ) \log {\left (\frac {x^{4}}{4} + x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5-5*x**4+4*x**2-24*x+20)*ln(1/4*x**4+x-1)+2*x**5-26*x**4+2*x**2-44*x+24)/(2*x**4+8*x-8),x)

[Out]

-3*x + (x**2/4 - 5*x/2)*log(x**4/4 + x - 1)

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Giac [A]
time = 0.42, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\left (x^{2} - 10 \, x\right )} \log \left (\frac {1}{4} \, x^{4} + x - 1\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-44*x+24)/(2*x^4+8*x-8),x, algorithm="
giac")

[Out]

1/4*(x^2 - 10*x)*log(1/4*x^4 + x - 1) - 3*x

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Mupad [B]
time = 3.33, size = 24, normalized size = 0.92 \begin {gather*} -3\,x-\ln \left (\frac {x^4}{4}+x-1\right )\,\left (\frac {5\,x}{2}-\frac {x^2}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + x^4/4 - 1)*(4*x^2 - 24*x - 5*x^4 + x^5 + 20) - 44*x + 2*x^2 - 26*x^4 + 2*x^5 + 24)/(8*x + 2*x^4 -
8),x)

[Out]

- 3*x - log(x + x^4/4 - 1)*((5*x)/2 - x^2/4)

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