3.72.63 \(\int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+(18+e^2 (18-9 x)+3 x-4 x^2-x^3) \log (\frac {1}{9} (9+9 e^2+6 x+x^2))+(-48-28 x+2 x^2+2 x^3+(-24-2 x+2 x^2) \log (16-8 x+x^2)) \log (2+x+\log (16-8 x+x^2))}{(-72-66 x-11 x^2+4 x^3+x^4+e^2 (-72-18 x+9 x^2)+(-36-15 x+2 x^2+x^3+e^2 (-36+9 x)) \log (16-8 x+x^2)) \log ^2(2+x+\log (16-8 x+x^2))} \, dx\)

Optimal. Leaf size=29 \[ \frac {-4+\log \left (e^2+\left (1+\frac {x}{3}\right )^2\right )}{\log \left (2+x+\log \left ((-4+x)^2\right )\right )} \]

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Rubi [F]  time = 5.64, 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 {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-72 - 12*x + 16*x^2 + 4*x^3 + E^2*(-72 + 36*x) + (18 + E^2*(18 - 9*x) + 3*x - 4*x^2 - x^3)*Log[(9 + 9*E^2
+ 6*x + x^2)/9] + (-48 - 28*x + 2*x^2 + 2*x^3 + (-24 - 2*x + 2*x^2)*Log[16 - 8*x + x^2])*Log[2 + x + Log[16 -
8*x + x^2]])/((-72 - 66*x - 11*x^2 + 4*x^3 + x^4 + E^2*(-72 - 18*x + 9*x^2) + (-36 - 15*x + 2*x^2 + x^3 + E^2
*(-36 + 9*x))*Log[16 - 8*x + x^2])*Log[2 + x + Log[16 - 8*x + x^2]]^2),x]

[Out]

4*Defer[Int][1/((2 + x + Log[(-4 + x)^2])*Log[2 + x + Log[(-4 + x)^2]]^2), x] + 8*Defer[Int][1/((-4 + x)*(2 +
x + Log[(-4 + x)^2])*Log[2 + x + Log[(-4 + x)^2]]^2), x] - Defer[Int][Log[1 + E^2 + (2*x)/3 + x^2/9]/((2 + x +
Log[(-4 + x)^2])*Log[2 + x + Log[(-4 + x)^2]]^2), x] - 2*Defer[Int][Log[1 + E^2 + (2*x)/3 + x^2/9]/((-4 + x)*
(2 + x + Log[(-4 + x)^2])*Log[2 + x + Log[(-4 + x)^2]]^2), x] + ((2*I)*Defer[Int][1/((-6 + (6*I)*E - 2*x)*Log[
2 + x + Log[(-4 + x)^2]]), x])/E + (2*(I + E)*Defer[Int][1/((6 - (6*I)*E + 2*x)*Log[2 + x + Log[(-4 + x)^2]]),
x])/E + ((2*I)*Defer[Int][1/((6 + (6*I)*E + 2*x)*Log[2 + x + Log[(-4 + x)^2]]), x])/E - (2*(I - E)*Defer[Int]
[1/((6 + (6*I)*E + 2*x)*Log[2 + x + Log[(-4 + x)^2]]), x])/E

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 (-2+x) \left (9 e^2+(3+x)^2\right )+(-2+x) \left (9 e^2+(3+x)^2\right ) \log \left (e^2+\frac {1}{9} (3+x)^2\right )-2 \left (-12-x+x^2\right ) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}{(4-x) \left (9 \left (1+e^2\right )+6 x+x^2\right ) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx\\ &=\int \left (-\frac {(-2+x) \left (-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )\right )}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {2 (3+x)}{\left (9 \left (1+e^2\right )+6 x+x^2\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx\\ &=2 \int \frac {3+x}{\left (9 \left (1+e^2\right )+6 x+x^2\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx-\int \frac {(-2+x) \left (-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )\right )}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx\\ &=2 \int \left (\frac {3}{\left (9 \left (1+e^2\right )+6 x+x^2\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {x}{\left (9 \left (1+e^2\right )+6 x+x^2\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx-\int \left (\frac {-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )}{\left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {2 \left (-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )\right )}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx\right )+2 \int \frac {x}{\left (9 \left (1+e^2\right )+6 x+x^2\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx+6 \int \frac {1}{\left (9 \left (1+e^2\right )+6 x+x^2\right ) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx-\int \frac {-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )}{\left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {4}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {\log \left (1+e^2+\frac {2 x}{3}+\frac {x^2}{9}\right )}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx\right )+2 \int \left (\frac {1+\frac {i}{e}}{(6-6 i e+2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {1-\frac {i}{e}}{(6+6 i e+2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx+6 \int \left (\frac {i}{3 e (-6+6 i e-2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {i}{3 e (6+6 i e+2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx-\int \left (-\frac {4}{\left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}+\frac {\log \left (1+e^2+\frac {2 x}{3}+\frac {x^2}{9}\right )}{\left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\log \left (1+e^2+\frac {2 x}{3}+\frac {x^2}{9}\right )}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx\right )+4 \int \frac {1}{\left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx+8 \int \frac {1}{(-4+x) \left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx+\frac {(2 i) \int \frac {1}{(-6+6 i e-2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx}{e}+\frac {(2 i) \int \frac {1}{(6+6 i e+2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx}{e}-\frac {(2 (i-e)) \int \frac {1}{(6+6 i e+2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx}{e}+\frac {(2 (i+e)) \int \frac {1}{(6-6 i e+2 x) \log \left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx}{e}-\int \frac {\log \left (1+e^2+\frac {2 x}{3}+\frac {x^2}{9}\right )}{\left (2+x+\log \left ((-4+x)^2\right )\right ) \log ^2\left (2+x+\log \left ((-4+x)^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 29, normalized size = 1.00 \begin {gather*} \frac {-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )}{\log \left (2+x+\log \left ((-4+x)^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 - 12*x + 16*x^2 + 4*x^3 + E^2*(-72 + 36*x) + (18 + E^2*(18 - 9*x) + 3*x - 4*x^2 - x^3)*Log[(9 +
9*E^2 + 6*x + x^2)/9] + (-48 - 28*x + 2*x^2 + 2*x^3 + (-24 - 2*x + 2*x^2)*Log[16 - 8*x + x^2])*Log[2 + x + Lo
g[16 - 8*x + x^2]])/((-72 - 66*x - 11*x^2 + 4*x^3 + x^4 + E^2*(-72 - 18*x + 9*x^2) + (-36 - 15*x + 2*x^2 + x^3
+ E^2*(-36 + 9*x))*Log[16 - 8*x + x^2])*Log[2 + x + Log[16 - 8*x + x^2]]^2),x]

[Out]

(-4 + Log[E^2 + (3 + x)^2/9])/Log[2 + x + Log[(-4 + x)^2]]

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fricas [A]  time = 0.66, size = 31, normalized size = 1.07 \begin {gather*} \frac {\log \left (\frac {1}{9} \, x^{2} + \frac {2}{3} \, x + e^{2} + 1\right ) - 4}{\log \left (x + \log \left (x^{2} - 8 \, x + 16\right ) + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3
-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3*x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-
15*x-36)*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/log(log(x^2-8*x+16)+2+x)^2,x, algori
thm="fricas")

[Out]

(log(1/9*x^2 + 2/3*x + e^2 + 1) - 4)/log(x + log(x^2 - 8*x + 16) + 2)

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giac [A]  time = 0.93, size = 38, normalized size = 1.31 \begin {gather*} -\frac {2 \, \log \relax (3) - \log \left (x^{2} + 6 \, x + 9 \, e^{2} + 9\right ) + 4}{\log \left (x + \log \left (x^{2} - 8 \, x + 16\right ) + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3
-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3*x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-
15*x-36)*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/log(log(x^2-8*x+16)+2+x)^2,x, algori
thm="giac")

[Out]

-(2*log(3) - log(x^2 + 6*x + 9*e^2 + 9) + 4)/log(x + log(x^2 - 8*x + 16) + 2)

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maple [C]  time = 0.12, size = 63, normalized size = 2.17

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{2}+\frac {x^{2}}{9}+\frac {2 x}{3}+1\right )-4}{\ln \left (2 \ln \left (x -4\right )-\frac {i \pi \,\mathrm {csgn}\left (i \left (x -4\right )^{2}\right ) \left (-\mathrm {csgn}\left (i \left (x -4\right )^{2}\right )+\mathrm {csgn}\left (i \left (x -4\right )\right )\right )^{2}}{2}+2+x \right )}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-2*x-24)*ln(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*ln(ln(x^2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3-4*x^2+3*
x+18)*ln(exp(2)+1/9*x^2+2/3*x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-15*x-36)*l
n(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/ln(ln(x^2-8*x+16)+2+x)^2,x,method=_RETURNVERBOS
E)

[Out]

(ln(exp(2)+1/9*x^2+2/3*x+1)-4)/ln(2*ln(x-4)-1/2*I*Pi*csgn(I*(x-4)^2)*(-csgn(I*(x-4)^2)+csgn(I*(x-4)))^2+2+x)

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maxima [A]  time = 0.51, size = 35, normalized size = 1.21 \begin {gather*} -\frac {2 \, \log \relax (3) - \log \left (x^{2} + 6 \, x + 9 \, e^{2} + 9\right ) + 4}{\log \left (x + 2 \, \log \left (x - 4\right ) + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3
-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3*x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-
15*x-36)*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/log(log(x^2-8*x+16)+2+x)^2,x, algori
thm="maxima")

[Out]

-(2*log(3) - log(x^2 + 6*x + 9*e^2 + 9) + 4)/log(x + 2*log(x - 4) + 2)

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mupad [B]  time = 5.55, size = 49, normalized size = 1.69 \begin {gather*} -\frac {2\,\ln \left (x+\ln \left (x^2-8\,x+16\right )+2\right )-\ln \left (\frac {x^2}{9}+\frac {2\,x}{3}+{\mathrm {e}}^2+1\right )+4}{\ln \left (x+\ln \left (x^2-8\,x+16\right )+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + log(x + log(x^2 - 8*x + 16) + 2)*(28*x + log(x^2 - 8*x + 16)*(2*x - 2*x^2 + 24) - 2*x^2 - 2*x^3 +
48) - 16*x^2 - 4*x^3 + log((2*x)/3 + exp(2) + x^2/9 + 1)*(4*x^2 - 3*x + x^3 + exp(2)*(9*x - 18) - 18) - exp(2)
*(36*x - 72) + 72)/(log(x + log(x^2 - 8*x + 16) + 2)^2*(66*x - log(x^2 - 8*x + 16)*(2*x^2 - 15*x + x^3 + exp(2
)*(9*x - 36) - 36) + exp(2)*(18*x - 9*x^2 + 72) + 11*x^2 - 4*x^3 - x^4 + 72)),x)

[Out]

-(2*log(x + log(x^2 - 8*x + 16) + 2) - log((2*x)/3 + exp(2) + x^2/9 + 1) + 4)/log(x + log(x^2 - 8*x + 16) + 2)

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sympy [A]  time = 2.46, size = 32, normalized size = 1.10 \begin {gather*} \frac {\log {\left (\frac {x^{2}}{9} + \frac {2 x}{3} + 1 + e^{2} \right )} - 4}{\log {\left (x + \log {\left (x^{2} - 8 x + 16 \right )} + 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-2*x-24)*ln(x**2-8*x+16)+2*x**3+2*x**2-28*x-48)*ln(ln(x**2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x
**3-4*x**2+3*x+18)*ln(exp(2)+1/9*x**2+2/3*x+1)+(36*x-72)*exp(2)+4*x**3+16*x**2-12*x-72)/(((9*x-36)*exp(2)+x**3
+2*x**2-15*x-36)*ln(x**2-8*x+16)+(9*x**2-18*x-72)*exp(2)+x**4+4*x**3-11*x**2-66*x-72)/ln(ln(x**2-8*x+16)+2+x)*
*2,x)

[Out]

(log(x**2/9 + 2*x/3 + 1 + exp(2)) - 4)/log(x + log(x**2 - 8*x + 16) + 2)

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