3.9.20 \(\int \frac {347-273 x-59 x^2+x^3+(-63+64 x-x^2) \log (4-4 x)+(80-64 x-16 x^2+(-16+16 x) \log (4-4 x)) \log (-5-x+\log (4-4 x))}{320-261 x-60 x^2+x^3+(-64+65 x-x^2) \log (4-4 x)+(80-64 x-16 x^2+(-16+16 x) \log (4-4 x)) \log (-5-x+\log (4-4 x))} \, dx\)

Optimal. Leaf size=25 \[ \log \left (e^x (x-16 (4+\log (-5-x+\log (4 (1-x)))))\right ) \]

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Rubi [A]  time = 0.86, antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, integrand size = 133, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6741, 6742, 6684} \begin {gather*} x+\log (-x+16 \log (-x+\log (4-4 x)-5)+64) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(347 - 273*x - 59*x^2 + x^3 + (-63 + 64*x - x^2)*Log[4 - 4*x] + (80 - 64*x - 16*x^2 + (-16 + 16*x)*Log[4 -
4*x])*Log[-5 - x + Log[4 - 4*x]])/(320 - 261*x - 60*x^2 + x^3 + (-64 + 65*x - x^2)*Log[4 - 4*x] + (80 - 64*x
- 16*x^2 + (-16 + 16*x)*Log[4 - 4*x])*Log[-5 - x + Log[4 - 4*x]]),x]

[Out]

x + Log[64 - x + 16*Log[-5 - x + Log[4 - 4*x]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {347-273 x-59 x^2+x^3+\left (-63+64 x-x^2\right ) \log (4-4 x)+\left (80-64 x-16 x^2+(-16+16 x) \log (4-4 x)\right ) \log (-5-x+\log (4-4 x))}{(1-x) (5+x-\log (4-4 x)) (64-x+16 \log (-5-x+\log (4-4 x)))} \, dx\\ &=\int \left (1+\frac {27-12 x+x^2+\log (4-4 x)-x \log (4-4 x)}{(-1+x) (5+x-\log (4-4 x)) (-64+x-16 \log (-5-x+\log (4-4 x)))}\right ) \, dx\\ &=x+\int \frac {27-12 x+x^2+\log (4-4 x)-x \log (4-4 x)}{(-1+x) (5+x-\log (4-4 x)) (-64+x-16 \log (-5-x+\log (4-4 x)))} \, dx\\ &=x+\log (64-x+16 \log (-5-x+\log (4-4 x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 22, normalized size = 0.88 \begin {gather*} x+\log (64-x+16 \log (-5-x+\log (4-4 x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(347 - 273*x - 59*x^2 + x^3 + (-63 + 64*x - x^2)*Log[4 - 4*x] + (80 - 64*x - 16*x^2 + (-16 + 16*x)*L
og[4 - 4*x])*Log[-5 - x + Log[4 - 4*x]])/(320 - 261*x - 60*x^2 + x^3 + (-64 + 65*x - x^2)*Log[4 - 4*x] + (80 -
64*x - 16*x^2 + (-16 + 16*x)*Log[4 - 4*x])*Log[-5 - x + Log[4 - 4*x]]),x]

[Out]

x + Log[64 - x + 16*Log[-5 - x + Log[4 - 4*x]]]

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fricas [A]  time = 0.62, size = 22, normalized size = 0.88 \begin {gather*} x + \log \left (-x + 16 \, \log \left (-x + \log \left (-4 \, x + 4\right ) - 5\right ) + 64\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x-16)*log(-4*x+4)-16*x^2-64*x+80)*log(log(-4*x+4)-x-5)+(-x^2+64*x-63)*log(-4*x+4)+x^3-59*x^2-2
73*x+347)/(((16*x-16)*log(-4*x+4)-16*x^2-64*x+80)*log(log(-4*x+4)-x-5)+(-x^2+65*x-64)*log(-4*x+4)+x^3-60*x^2-2
61*x+320),x, algorithm="fricas")

[Out]

x + log(-x + 16*log(-x + log(-4*x + 4) - 5) + 64)

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giac [A]  time = 0.51, size = 20, normalized size = 0.80 \begin {gather*} x + \log \left (x - 16 \, \log \left (-x + \log \left (-4 \, x + 4\right ) - 5\right ) - 64\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x-16)*log(-4*x+4)-16*x^2-64*x+80)*log(log(-4*x+4)-x-5)+(-x^2+64*x-63)*log(-4*x+4)+x^3-59*x^2-2
73*x+347)/(((16*x-16)*log(-4*x+4)-16*x^2-64*x+80)*log(log(-4*x+4)-x-5)+(-x^2+65*x-64)*log(-4*x+4)+x^3-60*x^2-2
61*x+320),x, algorithm="giac")

[Out]

x + log(x - 16*log(-x + log(-4*x + 4) - 5) - 64)

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maple [A]  time = 0.04, size = 21, normalized size = 0.84

method	result	size
risch	\(x +\ln \left (\ln \left (\ln \left (-4 x +4\right )-x -5\right )-\frac {x}{16}+4\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((16*x-16)*ln(-4*x+4)-16*x^2-64*x+80)*ln(ln(-4*x+4)-x-5)+(-x^2+64*x-63)*ln(-4*x+4)+x^3-59*x^2-273*x+347)/
(((16*x-16)*ln(-4*x+4)-16*x^2-64*x+80)*ln(ln(-4*x+4)-x-5)+(-x^2+65*x-64)*ln(-4*x+4)+x^3-60*x^2-261*x+320),x,me
thod=_RETURNVERBOSE)

[Out]

x+ln(ln(ln(-4*x+4)-x-5)-1/16*x+4)

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maxima [C]  time = 0.61, size = 25, normalized size = 1.00 \begin {gather*} x + \log \left (-\frac {1}{16} \, x + \log \left (i \, \pi - x + 2 \, \log \relax (2) + \log \left (x - 1\right ) - 5\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x-16)*log(-4*x+4)-16*x^2-64*x+80)*log(log(-4*x+4)-x-5)+(-x^2+64*x-63)*log(-4*x+4)+x^3-59*x^2-2
73*x+347)/(((16*x-16)*log(-4*x+4)-16*x^2-64*x+80)*log(log(-4*x+4)-x-5)+(-x^2+65*x-64)*log(-4*x+4)+x^3-60*x^2-2
61*x+320),x, algorithm="maxima")

[Out]

x + log(-1/16*x + log(I*pi - x + 2*log(2) + log(x - 1) - 5) + 4)

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mupad [B]  time = 2.22, size = 20, normalized size = 0.80 \begin {gather*} x+\ln \left (\ln \left (\ln \left (4-4\,x\right )-x-5\right )-\frac {x}{16}+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((273*x + log(4 - 4*x)*(x^2 - 64*x + 63) + log(log(4 - 4*x) - x - 5)*(64*x - log(4 - 4*x)*(16*x - 16) + 16*
x^2 - 80) + 59*x^2 - x^3 - 347)/(261*x + log(4 - 4*x)*(x^2 - 65*x + 64) + log(log(4 - 4*x) - x - 5)*(64*x - lo
g(4 - 4*x)*(16*x - 16) + 16*x^2 - 80) + 60*x^2 - x^3 - 320),x)

[Out]

x + log(log(log(4 - 4*x) - x - 5) - x/16 + 4)

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sympy [A]  time = 0.72, size = 19, normalized size = 0.76 \begin {gather*} x + \log {\left (- \frac {x}{16} + \log {\left (- x + \log {\left (4 - 4 x \right )} - 5 \right )} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x-16)*ln(-4*x+4)-16*x**2-64*x+80)*ln(ln(-4*x+4)-x-5)+(-x**2+64*x-63)*ln(-4*x+4)+x**3-59*x**2-2
73*x+347)/(((16*x-16)*ln(-4*x+4)-16*x**2-64*x+80)*ln(ln(-4*x+4)-x-5)+(-x**2+65*x-64)*ln(-4*x+4)+x**3-60*x**2-2
61*x+320),x)

[Out]

x + log(-x/16 + log(-x + log(4 - 4*x) - 5) + 4)

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