3.92.27 \(\int \frac {-1+(16+12 x-4 x^2+e^4 (8+6 x-2 x^2)) \log (4-x)+(-8-6 x+2 x^2) \log (4-x) \log (\log (4-x))}{(8+e^4 (4-x)-2 x) \log (4-x)+(-4+x) \log (4-x) \log (\log (4-x))} \, dx\)

Optimal. Leaf size=25 \[ -5+2 x+x^2-\log \left (2+e^4-\log (\log (4-x))\right ) \]

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Rubi [A]  time = 0.30, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6688, 6742, 6684} \begin {gather*} (x+1)^2-\log \left (-\log (\log (4-x))+e^4+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + (16 + 12*x - 4*x^2 + E^4*(8 + 6*x - 2*x^2))*Log[4 - x] + (-8 - 6*x + 2*x^2)*Log[4 - x]*Log[Log[4 - x
]])/((8 + E^4*(4 - x) - 2*x)*Log[4 - x] + (-4 + x)*Log[4 - x]*Log[Log[4 - x]]),x]

[Out]

(1 + x)^2 - Log[2 + E^4 - Log[Log[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 6688

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

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 {8+6 x-2 x^2-\frac {1}{\log (4-x) \left (2+e^4-\log (\log (4-x))\right )}}{4-x} \, dx\\ &=\int \left (2 (1+x)+\frac {1}{(-4+x) \log (4-x) \left (2 \left (1+\frac {e^4}{2}\right )-\log (\log (4-x))\right )}\right ) \, dx\\ &=(1+x)^2+\int \frac {1}{(-4+x) \log (4-x) \left (2 \left (1+\frac {e^4}{2}\right )-\log (\log (4-x))\right )} \, dx\\ &=(1+x)^2-\log \left (2+e^4-\log (\log (4-x))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 24, normalized size = 0.96 \begin {gather*} 2 x+x^2-\log \left (2+e^4-\log (\log (4-x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + (16 + 12*x - 4*x^2 + E^4*(8 + 6*x - 2*x^2))*Log[4 - x] + (-8 - 6*x + 2*x^2)*Log[4 - x]*Log[Log
[4 - x]])/((8 + E^4*(4 - x) - 2*x)*Log[4 - x] + (-4 + x)*Log[4 - x]*Log[Log[4 - x]]),x]

[Out]

2*x + x^2 - Log[2 + E^4 - Log[Log[4 - x]]]

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fricas [A]  time = 0.71, size = 23, normalized size = 0.92 \begin {gather*} x^{2} + 2 \, x - \log \left (-e^{4} + \log \left (\log \left (-x + 4\right )\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-6*x-8)*log(-x+4)*log(log(-x+4))+((-2*x^2+6*x+8)*exp(4)-4*x^2+12*x+16)*log(-x+4)-1)/((x-4)*lo
g(-x+4)*log(log(-x+4))+((-x+4)*exp(4)-2*x+8)*log(-x+4)),x, algorithm="fricas")

[Out]

x^2 + 2*x - log(-e^4 + log(log(-x + 4)) - 2)

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giac [A]  time = 0.16, size = 23, normalized size = 0.92 \begin {gather*} x^{2} + 2 \, x - \log \left (-e^{4} + \log \left (\log \left (-x + 4\right )\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-6*x-8)*log(-x+4)*log(log(-x+4))+((-2*x^2+6*x+8)*exp(4)-4*x^2+12*x+16)*log(-x+4)-1)/((x-4)*lo
g(-x+4)*log(log(-x+4))+((-x+4)*exp(4)-2*x+8)*log(-x+4)),x, algorithm="giac")

[Out]

x^2 + 2*x - log(-e^4 + log(log(-x + 4)) - 2)

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maple [A]  time = 0.32, size = 24, normalized size = 0.96

method	result	size
default	\(2 x -\ln \left (2-\ln \left (\ln \left (-x +4\right )\right )+{\mathrm e}^{4}\right )+x^{2}\)	\(24\)
norman	\(2 x -\ln \left (2-\ln \left (\ln \left (-x +4\right )\right )+{\mathrm e}^{4}\right )+x^{2}\)	\(24\)
risch	\(x^{2}+2 x -\ln \left (-{\mathrm e}^{4}+\ln \left (\ln \left (-x +4\right )\right )-2\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-6*x-8)*ln(-x+4)*ln(ln(-x+4))+((-2*x^2+6*x+8)*exp(4)-4*x^2+12*x+16)*ln(-x+4)-1)/((x-4)*ln(-x+4)*ln(
ln(-x+4))+((-x+4)*exp(4)-2*x+8)*ln(-x+4)),x,method=_RETURNVERBOSE)

[Out]

2*x-ln(2-ln(ln(-x+4))+exp(4))+x^2

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maxima [C]  time = 0.45, size = 173, normalized size = 6.92 \begin {gather*} 8 \, e^{4} \log \left (-x + 4\right ) \log \left (-e^{4} + \log \left (\log \left (-x + 4\right )\right ) - 2\right ) + 8 \, e^{\left (e^{4} + 2\right )} E_{1}\left (e^{4} - \log \left (\log \left (-x + 4\right )\right ) + 2\right ) \log \left (\log \left (-x + 4\right )\right ) + x^{2} - 8 \, {\left (e^{\left (e^{4} + 2\right )} E_{1}\left (e^{4} - \log \left (\log \left (-x + 4\right )\right ) + 2\right ) + \log \left (-x + 4\right ) \log \left (-e^{4} + \log \left (\log \left (-x + 4\right )\right ) - 2\right )\right )} e^{4} - 8 \, e^{\left (e^{4} + 2\right )} E_{2}\left (e^{4} - \log \left (\log \left (-x + 4\right )\right ) + 2\right ) - 16 \, e^{\left (e^{4} + 2\right )} E_{1}\left (e^{4} - \log \left (\log \left (-x + 4\right )\right ) + 2\right ) + 2 \, x + 8 \, \log \left (x - 4\right ) - \log \left (-e^{4} + \log \left (\log \left (-x + 4\right )\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-6*x-8)*log(-x+4)*log(log(-x+4))+((-2*x^2+6*x+8)*exp(4)-4*x^2+12*x+16)*log(-x+4)-1)/((x-4)*lo
g(-x+4)*log(log(-x+4))+((-x+4)*exp(4)-2*x+8)*log(-x+4)),x, algorithm="maxima")

[Out]

8*e^4*log(-x + 4)*log(-e^4 + log(log(-x + 4)) - 2) + 8*e^(e^4 + 2)*exp_integral_e(1, e^4 - log(log(-x + 4)) +
2)*log(log(-x + 4)) + x^2 - 8*(e^(e^4 + 2)*exp_integral_e(1, e^4 - log(log(-x + 4)) + 2) + log(-x + 4)*log(-e^
4 + log(log(-x + 4)) - 2))*e^4 - 8*e^(e^4 + 2)*exp_integral_e(2, e^4 - log(log(-x + 4)) + 2) - 16*e^(e^4 + 2)*
exp_integral_e(1, e^4 - log(log(-x + 4)) + 2) + 2*x + 8*log(x - 4) - log(-e^4 + log(log(-x + 4)) - 2)

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mupad [B]  time = 9.66, size = 23, normalized size = 0.92 \begin {gather*} 2\,x-\ln \left (\ln \left (\ln \left (4-x\right )\right )-{\mathrm {e}}^4-2\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(4 - x))*log(4 - x)*(6*x - 2*x^2 + 8) - log(4 - x)*(12*x + exp(4)*(6*x - 2*x^2 + 8) - 4*x^2 + 16)
+ 1)/(log(4 - x)*(2*x + exp(4)*(x - 4) - 8) - log(log(4 - x))*log(4 - x)*(x - 4)),x)

[Out]

2*x - log(log(log(4 - x)) - exp(4) - 2) + x^2

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sympy [A]  time = 0.40, size = 19, normalized size = 0.76 \begin {gather*} x^{2} + 2 x - \log {\left (\log {\left (\log {\left (4 - x \right )} \right )} - e^{4} - 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-6*x-8)*ln(-x+4)*ln(ln(-x+4))+((-2*x**2+6*x+8)*exp(4)-4*x**2+12*x+16)*ln(-x+4)-1)/((x-4)*ln(
-x+4)*ln(ln(-x+4))+((-x+4)*exp(4)-2*x+8)*ln(-x+4)),x)

[Out]

x**2 + 2*x - log(log(log(4 - x)) - exp(4) - 2)

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