3.70.20 \(\int \frac {-72+e^{e^x} (e^{2+x} (468+444 x+144 x^2+16 x^3)+e^x (468+444 x+144 x^2+16 x^3) \log (\frac {39+24 x+4 x^2}{9+6 x+x^2}))}{e^2 (117+111 x+36 x^2+4 x^3)+(117+111 x+36 x^2+4 x^3) \log (\frac {39+24 x+4 x^2}{9+6 x+x^2})} \, dx\)

Optimal. Leaf size=25 \[ 4 \left (e^{e^x}+3 \log \left (e^2+\log \left (4+\frac {3}{(3+x)^2}\right )\right )\right ) \]

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Rubi [F]  time = 2.62, 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+e^{e^x} \left (e^{2+x} \left (468+444 x+144 x^2+16 x^3\right )+e^x \left (468+444 x+144 x^2+16 x^3\right ) \log \left (\frac {39+24 x+4 x^2}{9+6 x+x^2}\right )\right )}{e^2 \left (117+111 x+36 x^2+4 x^3\right )+\left (117+111 x+36 x^2+4 x^3\right ) \log \left (\frac {39+24 x+4 x^2}{9+6 x+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-72 + E^E^x*(E^(2 + x)*(468 + 444*x + 144*x^2 + 16*x^3) + E^x*(468 + 444*x + 144*x^2 + 16*x^3)*Log[(39 +
24*x + 4*x^2)/(9 + 6*x + x^2)]))/(E^2*(117 + 111*x + 36*x^2 + 4*x^3) + (117 + 111*x + 36*x^2 + 4*x^3)*Log[(39
+ 24*x + 4*x^2)/(9 + 6*x + x^2)]),x]

[Out]

4*E^E^x + (192*I)*Sqrt[3]*Defer[Int][1/((-24 + (4*I)*Sqrt[3] - 8*x)*(E^2 + Log[(39 + 24*x + 4*x^2)/(3 + x)^2])
), x] - 24*Defer[Int][1/((3 + x)*(E^2 + Log[(39 + 24*x + 4*x^2)/(3 + x)^2])), x] + 96*(1 + (2*I)*Sqrt[3])*Defe
r[Int][1/((24 - (4*I)*Sqrt[3] + 8*x)*(E^2 + Log[(39 + 24*x + 4*x^2)/(3 + x)^2])), x] + (192*I)*Sqrt[3]*Defer[I
nt][1/((24 + (4*I)*Sqrt[3] + 8*x)*(E^2 + Log[(39 + 24*x + 4*x^2)/(3 + x)^2])), x] + 96*(1 - (2*I)*Sqrt[3])*Def
er[Int][1/((24 + (4*I)*Sqrt[3] + 8*x)*(E^2 + Log[(39 + 24*x + 4*x^2)/(3 + x)^2])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-72+e^{e^x} \left (e^{2+x} \left (468+444 x+144 x^2+16 x^3\right )+e^x \left (468+444 x+144 x^2+16 x^3\right ) \log \left (\frac {39+24 x+4 x^2}{9+6 x+x^2}\right )\right )}{\left (117+111 x+36 x^2+4 x^3\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx\\ &=\int \left (4 e^{e^x+x}-\frac {72}{(3+x) \left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}\right ) \, dx\\ &=4 \int e^{e^x+x} \, dx-72 \int \frac {1}{(3+x) \left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx\\ &=4 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )-72 \int \left (\frac {1}{3 (3+x) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}-\frac {4 (3+x)}{3 \left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}\right ) \, dx\\ &=4 e^{e^x}-24 \int \frac {1}{(3+x) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+96 \int \frac {3+x}{\left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx\\ &=4 e^{e^x}-24 \int \frac {1}{(3+x) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+96 \int \left (\frac {3}{\left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}+\frac {x}{\left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}\right ) \, dx\\ &=4 e^{e^x}-24 \int \frac {1}{(3+x) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+96 \int \frac {x}{\left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+288 \int \frac {1}{\left (39+24 x+4 x^2\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx\\ &=4 e^{e^x}-24 \int \frac {1}{(3+x) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+96 \int \left (\frac {1+2 i \sqrt {3}}{\left (24-4 i \sqrt {3}+8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}+\frac {1-2 i \sqrt {3}}{\left (24+4 i \sqrt {3}+8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}\right ) \, dx+288 \int \left (\frac {2 i}{\sqrt {3} \left (-24+4 i \sqrt {3}-8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}+\frac {2 i}{\sqrt {3} \left (24+4 i \sqrt {3}+8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )}\right ) \, dx\\ &=4 e^{e^x}-24 \int \frac {1}{(3+x) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+\left (192 i \sqrt {3}\right ) \int \frac {1}{\left (-24+4 i \sqrt {3}-8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+\left (192 i \sqrt {3}\right ) \int \frac {1}{\left (24+4 i \sqrt {3}+8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+\left (96 \left (1-2 i \sqrt {3}\right )\right ) \int \frac {1}{\left (24+4 i \sqrt {3}+8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx+\left (96 \left (1+2 i \sqrt {3}\right )\right ) \int \frac {1}{\left (24-4 i \sqrt {3}+8 x\right ) \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 32, normalized size = 1.28 \begin {gather*} 4 \left (e^{e^x}+3 \log \left (e^2+\log \left (\frac {39+24 x+4 x^2}{(3+x)^2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 + E^E^x*(E^(2 + x)*(468 + 444*x + 144*x^2 + 16*x^3) + E^x*(468 + 444*x + 144*x^2 + 16*x^3)*Log[
(39 + 24*x + 4*x^2)/(9 + 6*x + x^2)]))/(E^2*(117 + 111*x + 36*x^2 + 4*x^3) + (117 + 111*x + 36*x^2 + 4*x^3)*Lo
g[(39 + 24*x + 4*x^2)/(9 + 6*x + x^2)]),x]

[Out]

4*(E^E^x + 3*Log[E^2 + Log[(39 + 24*x + 4*x^2)/(3 + x)^2]])

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fricas [A]  time = 0.54, size = 34, normalized size = 1.36 \begin {gather*} 4 \, e^{\left (e^{x}\right )} + 12 \, \log \left (e^{2} + \log \left (\frac {4 \, x^{2} + 24 \, x + 39}{x^{2} + 6 \, x + 9}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^3+144*x^2+444*x+468)*exp(x)*log((4*x^2+24*x+39)/(x^2+6*x+9))+(16*x^3+144*x^2+444*x+468)*exp(
2)*exp(x))*exp(exp(x))-72)/((4*x^3+36*x^2+111*x+117)*log((4*x^2+24*x+39)/(x^2+6*x+9))+(4*x^3+36*x^2+111*x+117)
*exp(2)),x, algorithm="fricas")

[Out]

4*e^(e^x) + 12*log(e^2 + log((4*x^2 + 24*x + 39)/(x^2 + 6*x + 9)))

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giac [A]  time = 0.55, size = 42, normalized size = 1.68 \begin {gather*} 4 \, {\left (3 \, e^{x} \log \left (e^{2} + \log \left (\frac {4 \, x^{2} + 24 \, x + 39}{x^{2} + 6 \, x + 9}\right )\right ) + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^3+144*x^2+444*x+468)*exp(x)*log((4*x^2+24*x+39)/(x^2+6*x+9))+(16*x^3+144*x^2+444*x+468)*exp(
2)*exp(x))*exp(exp(x))-72)/((4*x^3+36*x^2+111*x+117)*log((4*x^2+24*x+39)/(x^2+6*x+9))+(4*x^3+36*x^2+111*x+117)
*exp(2)),x, algorithm="giac")

[Out]

4*(3*e^x*log(e^2 + log((4*x^2 + 24*x + 39)/(x^2 + 6*x + 9))) + e^(x + e^x))*e^(-x)

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maple [C]  time = 0.19, size = 223, normalized size = 8.92

method	result	size
risch	\(12 \ln \left (\ln \left (x^{2}+6 x +\frac {39}{4}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+6 x +\frac {39}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+6 x +\frac {39}{4}\right )}{\left (3+x \right )^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+6 x +\frac {39}{4}\right )}{\left (3+x \right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (3+x \right )\right ) \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (x^{2}+6 x +\frac {39}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+6 x +\frac {39}{4}\right )}{\left (3+x \right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{2}+6 x +\frac {39}{4}\right )}{\left (3+x \right )^{2}}\right )^{3}+2 i {\mathrm e}^{2}+4 i \ln \relax (2)-4 i \ln \left (3+x \right )\right )}{2}\right )+4 \,{\mathrm e}^{{\mathrm e}^{x}}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((16*x^3+144*x^2+444*x+468)*exp(x)*ln((4*x^2+24*x+39)/(x^2+6*x+9))+(16*x^3+144*x^2+444*x+468)*exp(2)*exp(
x))*exp(exp(x))-72)/((4*x^3+36*x^2+111*x+117)*ln((4*x^2+24*x+39)/(x^2+6*x+9))+(4*x^3+36*x^2+111*x+117)*exp(2))
,x,method=_RETURNVERBOSE)

[Out]

12*ln(ln(x^2+6*x+39/4)-1/2*I*(Pi*csgn(I/(3+x)^2)*csgn(I*(x^2+6*x+39/4))*csgn(I/(3+x)^2*(x^2+6*x+39/4))-Pi*csgn
(I/(3+x)^2)*csgn(I/(3+x)^2*(x^2+6*x+39/4))^2-Pi*csgn(I*(3+x))^2*csgn(I*(3+x)^2)+2*Pi*csgn(I*(3+x))*csgn(I*(3+x
)^2)^2-Pi*csgn(I*(3+x)^2)^3-Pi*csgn(I*(x^2+6*x+39/4))*csgn(I/(3+x)^2*(x^2+6*x+39/4))^2+Pi*csgn(I/(3+x)^2*(x^2+
6*x+39/4))^3+2*I*exp(2)+4*I*ln(2)-4*I*ln(3+x)))+4*exp(exp(x))

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maxima [A]  time = 0.55, size = 29, normalized size = 1.16 \begin {gather*} 4 \, e^{\left (e^{x}\right )} + 12 \, \log \left (e^{2} + \log \left (4 \, x^{2} + 24 \, x + 39\right ) - 2 \, \log \left (x + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^3+144*x^2+444*x+468)*exp(x)*log((4*x^2+24*x+39)/(x^2+6*x+9))+(16*x^3+144*x^2+444*x+468)*exp(
2)*exp(x))*exp(exp(x))-72)/((4*x^3+36*x^2+111*x+117)*log((4*x^2+24*x+39)/(x^2+6*x+9))+(4*x^3+36*x^2+111*x+117)
*exp(2)),x, algorithm="maxima")

[Out]

4*e^(e^x) + 12*log(e^2 + log(4*x^2 + 24*x + 39) - 2*log(x + 3))

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mupad [B]  time = 4.45, size = 34, normalized size = 1.36 \begin {gather*} 4\,{\mathrm {e}}^{{\mathrm {e}}^x}+12\,\ln \left ({\mathrm {e}}^2+\ln \left (\frac {4\,x^2+24\,x+39}{x^2+6\,x+9}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*(exp(2)*exp(x)*(444*x + 144*x^2 + 16*x^3 + 468) + exp(x)*log((24*x + 4*x^2 + 39)/(6*x + x^2 +
9))*(444*x + 144*x^2 + 16*x^3 + 468)) - 72)/(exp(2)*(111*x + 36*x^2 + 4*x^3 + 117) + log((24*x + 4*x^2 + 39)/
(6*x + x^2 + 9))*(111*x + 36*x^2 + 4*x^3 + 117)),x)

[Out]

4*exp(exp(x)) + 12*log(exp(2) + log((24*x + 4*x^2 + 39)/(6*x + x^2 + 9)))

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sympy [A]  time = 0.73, size = 32, normalized size = 1.28 \begin {gather*} 4 e^{e^{x}} + 12 \log {\left (\log {\left (\frac {4 x^{2} + 24 x + 39}{x^{2} + 6 x + 9} \right )} + e^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x**3+144*x**2+444*x+468)*exp(x)*ln((4*x**2+24*x+39)/(x**2+6*x+9))+(16*x**3+144*x**2+444*x+468)
*exp(2)*exp(x))*exp(exp(x))-72)/((4*x**3+36*x**2+111*x+117)*ln((4*x**2+24*x+39)/(x**2+6*x+9))+(4*x**3+36*x**2+
111*x+117)*exp(2)),x)

[Out]

4*exp(exp(x)) + 12*log(log((4*x**2 + 24*x + 39)/(x**2 + 6*x + 9)) + exp(2))

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