3.94.44 \(\int \frac {-768+448 x-64 x^2-4 x^3+x^4+(-192+112 x-20 x^2+x^3) \log (2)+e^{-\frac {4 x}{-4+x}} (64-32 x+20 x^2+(16-8 x+x^2) \log (2))+(-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} (32 x-16 x^2+2 x^3)) \log (-12+e^{-\frac {4 x}{-4+x}}+x)}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} (16-8 x+x^2)} \, dx\)

Optimal. Leaf size=27 \[ x (4+\log (2))+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right ) \]

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Rubi [A]  time = 7.97, antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 5, integrand size = 160, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6742, 6688, 698, 2551, 43} \begin {gather*} x^2 \log \left (x+e^{\frac {4 x}{4-x}}-12\right )-12 x+x (16+\log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-768 + 448*x - 64*x^2 - 4*x^3 + x^4 + (-192 + 112*x - 20*x^2 + x^3)*Log[2] + (64 - 32*x + 20*x^2 + (16 -
8*x + x^2)*Log[2])/E^((4*x)/(-4 + x)) + (-384*x + 224*x^2 - 40*x^3 + 2*x^4 + (32*x - 16*x^2 + 2*x^3)/E^((4*x)/
(-4 + x)))*Log[-12 + E^((-4*x)/(-4 + x)) + x])/(-192 + 112*x - 20*x^2 + x^3 + (16 - 8*x + x^2)/E^((4*x)/(-4 +
x))),x]

[Out]

-12*x + x*(16 + Log[2]) + x^2*Log[-12 + E^((4*x)/(4 - x)) + x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

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 \left (-\frac {x^2 \left (208-24 x+x^2\right )}{(-12+x) (-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}+\frac {-x^2+48 \left (1+\frac {\log (2)}{4}\right )-4 x \left (1+\frac {\log (2)}{4}\right )+24 x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )-2 x^2 \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{12-x}\right ) \, dx\\ &=-\int \frac {x^2 \left (208-24 x+x^2\right )}{(-12+x) (-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+\int \frac {-x^2+48 \left (1+\frac {\log (2)}{4}\right )-4 x \left (1+\frac {\log (2)}{4}\right )+24 x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )-2 x^2 \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{12-x} \, dx\\ &=-\int \left (-\frac {4}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x}+\frac {144}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}-\frac {256}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}-\frac {128}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}+\frac {x}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x}\right ) \, dx+\int \frac {-x^2+12 (4+\log (2))-x (4+\log (2))-2 (-12+x) x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{12-x} \, dx\\ &=4 \int \frac {1}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x} \, dx+128 \int \frac {1}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-144 \int \frac {1}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+256 \int \frac {1}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-\int \frac {x}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x} \, dx+\int \left (\frac {-x^2+12 (4+\log (2))-x (4+\log (2))}{12-x}+2 x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )\right ) \, dx\\ &=2 \int x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right ) \, dx+4 \int \frac {1}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+128 \int \frac {1}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-144 \int \frac {1}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+256 \int \frac {1}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-\int \frac {x}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+\int \frac {-x^2+12 (4+\log (2))-x (4+\log (2))}{12-x} \, dx\\ &=x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right )+4 \int \frac {1}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+128 \int \frac {1}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-144 \int \frac {1}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+256 \int \frac {1}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-\int \frac {x}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx-\int \frac {\left (16+e^{\frac {4 x}{-4+x}} (-4+x)^2\right ) x^2}{\left (1+e^{\frac {4 x}{-4+x}} (-12+x)\right ) (4-x)^2} \, dx+\int \left (16-\frac {144}{12-x}+x+\log (2)\right ) \, dx\\ &=\frac {x^2}{2}+x (16+\log (2))+144 \log (12-x)+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right )+4 \int \frac {1}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+128 \int \frac {1}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-144 \int \frac {1}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+256 \int \frac {1}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-\int \frac {x}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx-\int \left (\frac {x^2}{-12+x}-\frac {x^2 \left (208-24 x+x^2\right )}{(-12+x) (-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}\right ) \, dx\\ &=\frac {x^2}{2}+x (16+\log (2))+144 \log (12-x)+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right )+4 \int \frac {1}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+128 \int \frac {1}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-144 \int \frac {1}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+256 \int \frac {1}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-\int \frac {x}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx-\int \frac {x^2}{-12+x} \, dx+\int \frac {x^2 \left (208-24 x+x^2\right )}{(-12+x) (-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx\\ &=\frac {x^2}{2}+x (16+\log (2))+144 \log (12-x)+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right )+4 \int \frac {1}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+128 \int \frac {1}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-144 \int \frac {1}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx+256 \int \frac {1}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )} \, dx-\int \frac {x}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx-\int \left (12+\frac {144}{-12+x}+x\right ) \, dx+\int \left (-\frac {4}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x}+\frac {144}{(-12+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}-\frac {256}{(-4+x)^2 \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}-\frac {128}{(-4+x) \left (1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x\right )}+\frac {x}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x}\right ) \, dx\\ &=-12 x+x (16+\log (2))+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right )+4 \int \frac {1}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx-4 \int \frac {1}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x} \, dx-\int \frac {x}{1+e^{\frac {4 x}{-4+x}} (-12+x)} \, dx+\int \frac {x}{1-12 e^{\frac {4 x}{-4+x}}+e^{\frac {4 x}{-4+x}} x} \, dx\\ &=-12 x+x (16+\log (2))+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 22, normalized size = 0.81 \begin {gather*} x \left (4+\log (2)+x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-768 + 448*x - 64*x^2 - 4*x^3 + x^4 + (-192 + 112*x - 20*x^2 + x^3)*Log[2] + (64 - 32*x + 20*x^2 +
(16 - 8*x + x^2)*Log[2])/E^((4*x)/(-4 + x)) + (-384*x + 224*x^2 - 40*x^3 + 2*x^4 + (32*x - 16*x^2 + 2*x^3)/E^(
(4*x)/(-4 + x)))*Log[-12 + E^((-4*x)/(-4 + x)) + x])/(-192 + 112*x - 20*x^2 + x^3 + (16 - 8*x + x^2)/E^((4*x)/
(-4 + x))),x]

[Out]

x*(4 + Log[2] + x*Log[-12 + E^((-4*x)/(-4 + x)) + x])

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fricas [A]  time = 0.82, size = 25, normalized size = 0.93 \begin {gather*} x^{2} \log \left (x + e^{\left (-\frac {4 \, x}{x - 4}\right )} - 12\right ) + x \log \relax (2) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-16*x^2+32*x)*exp(-4*x/(x-4))+2*x^4-40*x^3+224*x^2-384*x)*log(exp(-4*x/(x-4))+x-12)+((x^2-8*
x+16)*log(2)+20*x^2-32*x+64)*exp(-4*x/(x-4))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2-8
*x+16)*exp(-4*x/(x-4))+x^3-20*x^2+112*x-192),x, algorithm="fricas")

[Out]

x^2*log(x + e^(-4*x/(x - 4)) - 12) + x*log(2) + 4*x

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giac [A]  time = 0.68, size = 25, normalized size = 0.93 \begin {gather*} x^{2} \log \left (x + e^{\left (-\frac {4 \, x}{x - 4}\right )} - 12\right ) + x \log \relax (2) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-16*x^2+32*x)*exp(-4*x/(x-4))+2*x^4-40*x^3+224*x^2-384*x)*log(exp(-4*x/(x-4))+x-12)+((x^2-8*
x+16)*log(2)+20*x^2-32*x+64)*exp(-4*x/(x-4))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2-8
*x+16)*exp(-4*x/(x-4))+x^3-20*x^2+112*x-192),x, algorithm="giac")

[Out]

x^2*log(x + e^(-4*x/(x - 4)) - 12) + x*log(2) + 4*x

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maple [A]  time = 0.22, size = 26, normalized size = 0.96

method	result	size
risch	\(\ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{2}+4 x +x \ln \relax (2)\)	\(26\)
norman	\(\frac {\left (4+\ln \relax (2)\right ) x^{2}+\ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{3}-64-4 \ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{2}-16 \ln \relax (2)}{x -4}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3-16*x^2+32*x)*exp(-4*x/(x-4))+2*x^4-40*x^3+224*x^2-384*x)*ln(exp(-4*x/(x-4))+x-12)+((x^2-8*x+16)*l
n(2)+20*x^2-32*x+64)*exp(-4*x/(x-4))+(x^3-20*x^2+112*x-192)*ln(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2-8*x+16)*ex
p(-4*x/(x-4))+x^3-20*x^2+112*x-192),x,method=_RETURNVERBOSE)

[Out]

ln(exp(-4*x/(x-4))+x-12)*x^2+4*x+x*ln(2)

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maxima [B]  time = 0.54, size = 62, normalized size = 2.30 \begin {gather*} -\frac {4 \, x^{3} - x^{2} {\left (\log \relax (2) + 4\right )} + 4 \, x {\left (\log \relax (2) - 12\right )} - {\left (x^{3} - 4 \, x^{2}\right )} \log \left ({\left (x e^{4} - 12 \, e^{4}\right )} e^{\left (\frac {16}{x - 4}\right )} + 1\right ) + 256}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-16*x^2+32*x)*exp(-4*x/(x-4))+2*x^4-40*x^3+224*x^2-384*x)*log(exp(-4*x/(x-4))+x-12)+((x^2-8*
x+16)*log(2)+20*x^2-32*x+64)*exp(-4*x/(x-4))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2-8
*x+16)*exp(-4*x/(x-4))+x^3-20*x^2+112*x-192),x, algorithm="maxima")

[Out]

-(4*x^3 - x^2*(log(2) + 4) + 4*x*(log(2) - 12) - (x^3 - 4*x^2)*log((x*e^4 - 12*e^4)*e^(16/(x - 4)) + 1) + 256)
/(x - 4)

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mupad [B]  time = 8.26, size = 38, normalized size = 1.41 \begin {gather*} x\,\left (\ln \relax (2)+4\right )-\frac {\ln \left (x+{\mathrm {e}}^{-\frac {4\,x}{x-4}}-12\right )\,\left (4\,x^2-x^3\right )}{x-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((448*x + log(2)*(112*x - 20*x^2 + x^3 - 192) + exp(-(4*x)/(x - 4))*(20*x^2 - 32*x + log(2)*(x^2 - 8*x + 16
) + 64) + log(x + exp(-(4*x)/(x - 4)) - 12)*(exp(-(4*x)/(x - 4))*(32*x - 16*x^2 + 2*x^3) - 384*x + 224*x^2 - 4
0*x^3 + 2*x^4) - 64*x^2 - 4*x^3 + x^4 - 768)/(112*x - 20*x^2 + x^3 + exp(-(4*x)/(x - 4))*(x^2 - 8*x + 16) - 19
2),x)

[Out]

x*(log(2) + 4) - (log(x + exp(-(4*x)/(x - 4)) - 12)*(4*x^2 - x^3))/(x - 4)

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sympy [A]  time = 0.65, size = 24, normalized size = 0.89 \begin {gather*} x^{2} \log {\left (x - 12 + e^{- \frac {4 x}{x - 4}} \right )} + x \left (\log {\relax (2 )} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3-16*x**2+32*x)*exp(-4*x/(x-4))+2*x**4-40*x**3+224*x**2-384*x)*ln(exp(-4*x/(x-4))+x-12)+((x*
*2-8*x+16)*ln(2)+20*x**2-32*x+64)*exp(-4*x/(x-4))+(x**3-20*x**2+112*x-192)*ln(2)+x**4-4*x**3-64*x**2+448*x-768
)/((x**2-8*x+16)*exp(-4*x/(x-4))+x**3-20*x**2+112*x-192),x)

[Out]

x**2*log(x - 12 + exp(-4*x/(x - 4))) + x*(log(2) + 4)

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