3.81.28 \(\int \frac {(12 x+12 e^x x) \log (48+3 e^{2 x}+24 x+3 x^2+e^x (24+6 x))+(-12-3 e^x-3 x) \log ^2(48+3 e^{2 x}+24 x+3 x^2+e^x (24+6 x))}{16 x^2+4 e^x x^2+4 x^3} \, dx\)

Optimal. Leaf size=20 \[ \frac {3 \log ^2\left (3 \left (4+e^x+x\right )^2\right )}{4 x} \]

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Rubi [F]  time = 3.08, 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 {\left (12 x+12 e^x x\right ) \log \left (48+3 e^{2 x}+24 x+3 x^2+e^x (24+6 x)\right )+\left (-12-3 e^x-3 x\right ) \log ^2\left (48+3 e^{2 x}+24 x+3 x^2+e^x (24+6 x)\right )}{16 x^2+4 e^x x^2+4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((12*x + 12*E^x*x)*Log[48 + 3*E^(2*x) + 24*x + 3*x^2 + E^x*(24 + 6*x)] + (-12 - 3*E^x - 3*x)*Log[48 + 3*E^
(2*x) + 24*x + 3*x^2 + E^x*(24 + 6*x)]^2)/(16*x^2 + 4*E^x*x^2 + 4*x^3),x]

[Out]

-3*Log[3*(4 + E^x + x)^2]*Defer[Int][(4 + E^x + x)^(-1), x] - 9*Log[3*(4 + E^x + x)^2]*Defer[Int][1/(x*(4 + E^
x + x)), x] + 3*Defer[Int][Log[3*(4 + E^x + x)^2]/x, x] - (3*Defer[Int][Log[3*(4 + E^x + x)^2]^2/x^2, x])/4 +
6*Defer[Int][Defer[Int][(4 + E^x + x)^(-1), x], x] - 18*Defer[Int][Defer[Int][(4 + E^x + x)^(-1), x]/(4 + E^x
+ x), x] - 6*Defer[Int][(x*Defer[Int][(4 + E^x + x)^(-1), x])/(4 + E^x + x), x] + 18*Defer[Int][Defer[Int][1/(
x*(4 + E^x + x)), x], x] - 54*Defer[Int][Defer[Int][1/(x*(4 + E^x + x)), x]/(4 + E^x + x), x] - 18*Defer[Int][
(x*Defer[Int][1/(x*(4 + E^x + x)), x])/(4 + E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (\frac {4 \left (1+e^x\right ) x}{4+e^x+x}-\log \left (3 \left (4+e^x+x\right )^2\right )\right ) \log \left (3 \left (4+e^x+x\right )^2\right )}{4 x^2} \, dx\\ &=\frac {3}{4} \int \frac {\left (\frac {4 \left (1+e^x\right ) x}{4+e^x+x}-\log \left (3 \left (4+e^x+x\right )^2\right )\right ) \log \left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\\ &=\frac {3}{4} \int \left (-\frac {4 (3+x) \log \left (3 \left (4+e^x+x\right )^2\right )}{x \left (4+e^x+x\right )}+\frac {\left (4 x-\log \left (3 \left (4+e^x+x\right )^2\right )\right ) \log \left (3 \left (4+e^x+x\right )^2\right )}{x^2}\right ) \, dx\\ &=\frac {3}{4} \int \frac {\left (4 x-\log \left (3 \left (4+e^x+x\right )^2\right )\right ) \log \left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx-3 \int \frac {(3+x) \log \left (3 \left (4+e^x+x\right )^2\right )}{x \left (4+e^x+x\right )} \, dx\\ &=\frac {3}{4} \int \left (\frac {4 \log \left (3 \left (4+e^x+x\right )^2\right )}{x}-\frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2}\right ) \, dx+3 \int \frac {2 \left (1+e^x\right ) \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x} \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \frac {\left (1+e^x\right ) \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x} \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx-\frac {(3+x) \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x}\right ) \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \left (\int \frac {1}{4+e^x+x} \, dx\right ) \, dx-6 \int \frac {(3+x) \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x} \, dx+18 \int \left (\int \frac {1}{x \left (4+e^x+x\right )} \, dx\right ) \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \left (\int \frac {1}{4+e^x+x} \, dx\right ) \, dx-6 \int \left (\frac {3 \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x}+\frac {x \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x}\right ) \, dx+18 \int \left (\int \frac {1}{x \left (4+e^x+x\right )} \, dx\right ) \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \left (\int \frac {1}{4+e^x+x} \, dx\right ) \, dx-6 \int \frac {x \left (\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx\right )}{4+e^x+x} \, dx+18 \int \left (\int \frac {1}{x \left (4+e^x+x\right )} \, dx\right ) \, dx-18 \int \frac {\int \frac {1}{4+e^x+x} \, dx+3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx}{4+e^x+x} \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \left (\int \frac {1}{4+e^x+x} \, dx\right ) \, dx-6 \int \left (\frac {x \int \frac {1}{4+e^x+x} \, dx}{4+e^x+x}+\frac {3 x \int \frac {1}{x \left (4+e^x+x\right )} \, dx}{4+e^x+x}\right ) \, dx+18 \int \left (\int \frac {1}{x \left (4+e^x+x\right )} \, dx\right ) \, dx-18 \int \left (\frac {\int \frac {1}{4+e^x+x} \, dx}{4+e^x+x}+\frac {3 \int \frac {1}{x \left (4+e^x+x\right )} \, dx}{4+e^x+x}\right ) \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {\log ^2\left (3 \left (4+e^x+x\right )^2\right )}{x^2} \, dx\right )+3 \int \frac {\log \left (3 \left (4+e^x+x\right )^2\right )}{x} \, dx+6 \int \left (\int \frac {1}{4+e^x+x} \, dx\right ) \, dx-6 \int \frac {x \int \frac {1}{4+e^x+x} \, dx}{4+e^x+x} \, dx-18 \int \frac {\int \frac {1}{4+e^x+x} \, dx}{4+e^x+x} \, dx+18 \int \left (\int \frac {1}{x \left (4+e^x+x\right )} \, dx\right ) \, dx-18 \int \frac {x \int \frac {1}{x \left (4+e^x+x\right )} \, dx}{4+e^x+x} \, dx-54 \int \frac {\int \frac {1}{x \left (4+e^x+x\right )} \, dx}{4+e^x+x} \, dx-\left (3 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{4+e^x+x} \, dx-\left (9 \log \left (3 \left (4+e^x+x\right )^2\right )\right ) \int \frac {1}{x \left (4+e^x+x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \log ^2\left (3 \left (4+e^x+x\right )^2\right )}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((12*x + 12*E^x*x)*Log[48 + 3*E^(2*x) + 24*x + 3*x^2 + E^x*(24 + 6*x)] + (-12 - 3*E^x - 3*x)*Log[48
+ 3*E^(2*x) + 24*x + 3*x^2 + E^x*(24 + 6*x)]^2)/(16*x^2 + 4*E^x*x^2 + 4*x^3),x]

[Out]

(3*Log[3*(4 + E^x + x)^2]^2)/(4*x)

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fricas [A]  time = 0.92, size = 31, normalized size = 1.55 \begin {gather*} \frac {3 \, \log \left (3 \, x^{2} + 6 \, {\left (x + 4\right )} e^{x} + 24 \, x + 3 \, e^{\left (2 \, x\right )} + 48\right )^{2}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)-3*x-12)*log(3*exp(x)^2+(24+6*x)*exp(x)+3*x^2+24*x+48)^2+(12*exp(x)*x+12*x)*log(3*exp(x)^
2+(24+6*x)*exp(x)+3*x^2+24*x+48))/(4*exp(x)*x^2+4*x^3+16*x^2),x, algorithm="fricas")

[Out]

3/4*log(3*x^2 + 6*(x + 4)*e^x + 24*x + 3*e^(2*x) + 48)^2/x

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giac [A]  time = 1.46, size = 33, normalized size = 1.65 \begin {gather*} \frac {3 \, \log \left (3 \, x^{2} + 6 \, x e^{x} + 24 \, x + 3 \, e^{\left (2 \, x\right )} + 24 \, e^{x} + 48\right )^{2}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)-3*x-12)*log(3*exp(x)^2+(24+6*x)*exp(x)+3*x^2+24*x+48)^2+(12*exp(x)*x+12*x)*log(3*exp(x)^
2+(24+6*x)*exp(x)+3*x^2+24*x+48))/(4*exp(x)*x^2+4*x^3+16*x^2),x, algorithm="giac")

[Out]

3/4*log(3*x^2 + 6*x*e^x + 24*x + 3*e^(2*x) + 24*e^x + 48)^2/x

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maple [C]  time = 0.18, size = 319, normalized size = 15.95

method	result	size
risch	\(\frac {3 \ln \left (4+x +{\mathrm e}^{x}\right )^{2}}{x}+\frac {3 \left (-i \pi \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right )^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{3}+2 \ln \relax (3)\right ) \ln \left (4+x +{\mathrm e}^{x}\right )}{2 x}+\frac {\frac {3 \ln \relax (3)^{2}}{4}-\frac {9 \pi ^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right )^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{4}}{8}+\frac {3 \pi ^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{5}}{4}-\frac {3 i \ln \relax (3) \pi \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{3}}{4}-\frac {3 i \ln \relax (3) \pi \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right )^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )}{4}+\frac {3 i \ln \relax (3) \pi \,\mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{2}}{2}-\frac {3 \pi ^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{6}}{16}-\frac {3 \pi ^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right )^{4} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{2}}{16}+\frac {3 \pi ^{2} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )\right )^{3} \mathrm {csgn}\left (i \left (4+x +{\mathrm e}^{x}\right )^{2}\right )^{3}}{4}}{x}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*exp(x)-3*x-12)*ln(3*exp(x)^2+(24+6*x)*exp(x)+3*x^2+24*x+48)^2+(12*exp(x)*x+12*x)*ln(3*exp(x)^2+(24+6*
x)*exp(x)+3*x^2+24*x+48))/(4*exp(x)*x^2+4*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

3/x*ln(4+x+exp(x))^2+3/2*(-I*Pi*csgn(I*(4+x+exp(x)))^2*csgn(I*(4+x+exp(x))^2)+2*I*Pi*csgn(I*(4+x+exp(x)))*csgn
(I*(4+x+exp(x))^2)^2-I*Pi*csgn(I*(4+x+exp(x))^2)^3+2*ln(3))/x*ln(4+x+exp(x))+3/16*(4*ln(3)^2-6*Pi^2*csgn(I*(4+
x+exp(x)))^2*csgn(I*(4+x+exp(x))^2)^4+4*Pi^2*csgn(I*(4+x+exp(x)))*csgn(I*(4+x+exp(x))^2)^5-4*I*ln(3)*Pi*csgn(I
*(4+x+exp(x))^2)^3-4*I*ln(3)*Pi*csgn(I*(4+x+exp(x)))^2*csgn(I*(4+x+exp(x))^2)+8*I*ln(3)*Pi*csgn(I*(4+x+exp(x))
)*csgn(I*(4+x+exp(x))^2)^2-Pi^2*csgn(I*(4+x+exp(x))^2)^6-Pi^2*csgn(I*(4+x+exp(x)))^4*csgn(I*(4+x+exp(x))^2)^2+
4*Pi^2*csgn(I*(4+x+exp(x)))^3*csgn(I*(4+x+exp(x))^2)^3)/x

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maxima [A]  time = 0.49, size = 30, normalized size = 1.50 \begin {gather*} \frac {3 \, {\left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \left (x + e^{x} + 4\right ) + 4 \, \log \left (x + e^{x} + 4\right )^{2}\right )}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)-3*x-12)*log(3*exp(x)^2+(24+6*x)*exp(x)+3*x^2+24*x+48)^2+(12*exp(x)*x+12*x)*log(3*exp(x)^
2+(24+6*x)*exp(x)+3*x^2+24*x+48))/(4*exp(x)*x^2+4*x^3+16*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.44, size = 32, normalized size = 1.60 \begin {gather*} \frac {3\,{\ln \left (24\,x+3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (6\,x+24\right )+3\,x^2+48\right )}^2}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(24*x + 3*exp(2*x) + exp(x)*(6*x + 24) + 3*x^2 + 48)^2*(3*x + 3*exp(x) + 12) - log(24*x + 3*exp(2*x)
+ exp(x)*(6*x + 24) + 3*x^2 + 48)*(12*x + 12*x*exp(x)))/(4*x^2*exp(x) + 16*x^2 + 4*x^3),x)

[Out]

(3*log(24*x + 3*exp(2*x) + exp(x)*(6*x + 24) + 3*x^2 + 48)^2)/(4*x)

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sympy [A]  time = 0.37, size = 32, normalized size = 1.60 \begin {gather*} \frac {3 \log {\left (3 x^{2} + 24 x + \left (6 x + 24\right ) e^{x} + 3 e^{2 x} + 48 \right )}^{2}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)-3*x-12)*ln(3*exp(x)**2+(24+6*x)*exp(x)+3*x**2+24*x+48)**2+(12*exp(x)*x+12*x)*ln(3*exp(x)
**2+(24+6*x)*exp(x)+3*x**2+24*x+48))/(4*exp(x)*x**2+4*x**3+16*x**2),x)

[Out]

3*log(3*x**2 + 24*x + (6*x + 24)*exp(x) + 3*exp(2*x) + 48)**2/(4*x)

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