3.82.95 \(\int \frac {-12+e^x (-3-3 x-x^2)+(e^{3 x} (6 x+2 x^2)+e^{2 x} (24 x+8 x^2)) \log (\frac {16+4 e^x}{3+x})+(e^{3 x} (6 x+2 x^2)+e^{2 x} (24 x+8 x^2)) \log (\frac {x}{\log (3)})}{(12 x+4 x^2+e^x (3 x+x^2)) \log (\frac {16+4 e^x}{3+x})+(12 x+4 x^2+e^x (3 x+x^2)) \log (\frac {x}{\log (3)})} \, dx\)

Optimal. Leaf size=30 \[ e^{2 x}-\log \left (\log \left (\frac {4 \left (4+e^x\right )}{3+x}\right )+\log \left (\frac {x}{\log (3)}\right )\right ) \]

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Rubi [F]  time = 5.98, 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 {-12+e^x \left (-3-3 x-x^2\right )+\left (e^{3 x} \left (6 x+2 x^2\right )+e^{2 x} \left (24 x+8 x^2\right )\right ) \log \left (\frac {16+4 e^x}{3+x}\right )+\left (e^{3 x} \left (6 x+2 x^2\right )+e^{2 x} \left (24 x+8 x^2\right )\right ) \log \left (\frac {x}{\log (3)}\right )}{\left (12 x+4 x^2+e^x \left (3 x+x^2\right )\right ) \log \left (\frac {16+4 e^x}{3+x}\right )+\left (12 x+4 x^2+e^x \left (3 x+x^2\right )\right ) \log \left (\frac {x}{\log (3)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12 + E^x*(-3 - 3*x - x^2) + (E^(3*x)*(6*x + 2*x^2) + E^(2*x)*(24*x + 8*x^2))*Log[(16 + 4*E^x)/(3 + x)] +
(E^(3*x)*(6*x + 2*x^2) + E^(2*x)*(24*x + 8*x^2))*Log[x/Log[3]])/((12*x + 4*x^2 + E^x*(3*x + x^2))*Log[(16 + 4
*E^x)/(3 + x)] + (12*x + 4*x^2 + E^x*(3*x + x^2))*Log[x/Log[3]]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.21, size = 30, normalized size = 1.00 \begin {gather*} e^{2 x}-\log \left (\log \left (\frac {4+e^x}{3+x}\right )+\log \left (\frac {4 x}{\log (3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + E^x*(-3 - 3*x - x^2) + (E^(3*x)*(6*x + 2*x^2) + E^(2*x)*(24*x + 8*x^2))*Log[(16 + 4*E^x)/(3 +
x)] + (E^(3*x)*(6*x + 2*x^2) + E^(2*x)*(24*x + 8*x^2))*Log[x/Log[3]])/((12*x + 4*x^2 + E^x*(3*x + x^2))*Log[(
16 + 4*E^x)/(3 + x)] + (12*x + 4*x^2 + E^x*(3*x + x^2))*Log[x/Log[3]]),x]

[Out]

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

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fricas [A]  time = 0.65, size = 28, normalized size = 0.93 \begin {gather*} e^{\left (2 \, x\right )} - \log \left (\log \left (\frac {4 \, {\left (e^{x} + 4\right )}}{x + 3}\right ) + \log \left (\frac {x}{\log \relax (3)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+6*x)*exp(x)^3+(8*x^2+24*x)*exp(x)^2)*log((4*exp(x)+16)/(3+x))+((2*x^2+6*x)*exp(x)^3+(8*x^2+
24*x)*exp(x)^2)*log(x/log(3))+(-x^2-3*x-3)*exp(x)-12)/(((x^2+3*x)*exp(x)+4*x^2+12*x)*log((4*exp(x)+16)/(3+x))+
((x^2+3*x)*exp(x)+4*x^2+12*x)*log(x/log(3))),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.33, size = 31, normalized size = 1.03 \begin {gather*} e^{\left (2 \, x\right )} - \log \left (2 \, \log \relax (2) - \log \left (x + 3\right ) + \log \relax (x) + \log \left (e^{x} + 4\right ) - \log \left (\log \relax (3)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+6*x)*exp(x)^3+(8*x^2+24*x)*exp(x)^2)*log((4*exp(x)+16)/(3+x))+((2*x^2+6*x)*exp(x)^3+(8*x^2+
24*x)*exp(x)^2)*log(x/log(3))+(-x^2-3*x-3)*exp(x)-12)/(((x^2+3*x)*exp(x)+4*x^2+12*x)*log((4*exp(x)+16)/(3+x))+
((x^2+3*x)*exp(x)+4*x^2+12*x)*log(x/log(3))),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.15, size = 143, normalized size = 4.77

method	result	size
risch	\({\mathrm e}^{2 x}-\ln \left (\ln \left ({\mathrm e}^{x}+4\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+4\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+4\right )}{3+x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+4\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+4\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i}{3+x}\right )-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+4\right )}{3+x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+4\right )}{3+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{3+x}\right )-4 i \ln \relax (2)-2 i \ln \left (\frac {x}{\ln \relax (3)}\right )+2 i \ln \left (3+x \right )\right )}{2}\right )\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2+6*x)*exp(x)^3+(8*x^2+24*x)*exp(x)^2)*ln((4*exp(x)+16)/(3+x))+((2*x^2+6*x)*exp(x)^3+(8*x^2+24*x)*e
xp(x)^2)*ln(x/ln(3))+(-x^2-3*x-3)*exp(x)-12)/(((x^2+3*x)*exp(x)+4*x^2+12*x)*ln((4*exp(x)+16)/(3+x))+((x^2+3*x)
*exp(x)+4*x^2+12*x)*ln(x/ln(3))),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.55, size = 31, normalized size = 1.03 \begin {gather*} e^{\left (2 \, x\right )} - \log \left (2 \, \log \relax (2) - \log \left (x + 3\right ) + \log \relax (x) + \log \left (e^{x} + 4\right ) - \log \left (\log \relax (3)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+6*x)*exp(x)^3+(8*x^2+24*x)*exp(x)^2)*log((4*exp(x)+16)/(3+x))+((2*x^2+6*x)*exp(x)^3+(8*x^2+
24*x)*exp(x)^2)*log(x/log(3))+(-x^2-3*x-3)*exp(x)-12)/(((x^2+3*x)*exp(x)+4*x^2+12*x)*log((4*exp(x)+16)/(3+x))+
((x^2+3*x)*exp(x)+4*x^2+12*x)*log(x/log(3))),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.28, size = 28, normalized size = 0.93 \begin {gather*} {\mathrm {e}}^{2\,x}-\ln \left (\ln \left (\frac {4\,{\mathrm {e}}^x+16}{\ln \relax (3)\,\left (x+3\right )}\right )+\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(3*x + x^2 + 3) - log((4*exp(x) + 16)/(x + 3))*(exp(3*x)*(6*x + 2*x^2) + exp(2*x)*(24*x + 8*x^2))
- log(x/log(3))*(exp(3*x)*(6*x + 2*x^2) + exp(2*x)*(24*x + 8*x^2)) + 12)/(log(x/log(3))*(12*x + exp(x)*(3*x +
x^2) + 4*x^2) + log((4*exp(x) + 16)/(x + 3))*(12*x + exp(x)*(3*x + x^2) + 4*x^2)),x)

[Out]

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

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sympy [A]  time = 0.89, size = 24, normalized size = 0.80 \begin {gather*} e^{2 x} - \log {\left (\log {\left (\frac {x}{\log {\relax (3 )}} \right )} + \log {\left (\frac {4 e^{x} + 16}{x + 3} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2+6*x)*exp(x)**3+(8*x**2+24*x)*exp(x)**2)*ln((4*exp(x)+16)/(3+x))+((2*x**2+6*x)*exp(x)**3+(8
*x**2+24*x)*exp(x)**2)*ln(x/ln(3))+(-x**2-3*x-3)*exp(x)-12)/(((x**2+3*x)*exp(x)+4*x**2+12*x)*ln((4*exp(x)+16)/
(3+x))+((x**2+3*x)*exp(x)+4*x**2+12*x)*ln(x/ln(3))),x)

[Out]

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

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