3.97.18 \(\int \frac {-12-21 x+e^{2 x} (-27 x+6 x^2)-3 x \log (x)}{32 x+2 e^{4 x} x+16 x^2+2 x^3+e^{2 x} (16 x+4 x^2)+(16 x+4 e^{2 x} x+4 x^2) \log (x)+2 x \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ 3 \left (\log (4)+\frac {4-x}{2 \left (4+e^{2 x}+x+\log (x)\right )}\right ) \]

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Rubi [F]  time = 2.30, 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 {-12-21 x+e^{2 x} \left (-27 x+6 x^2\right )-3 x \log (x)}{32 x+2 e^{4 x} x+16 x^2+2 x^3+e^{2 x} \left (16 x+4 x^2\right )+\left (16 x+4 e^{2 x} x+4 x^2\right ) \log (x)+2 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12 - 21*x + E^(2*x)*(-27*x + 6*x^2) - 3*x*Log[x])/(32*x + 2*E^(4*x)*x + 16*x^2 + 2*x^3 + E^(2*x)*(16*x +
4*x^2) + (16*x + 4*E^(2*x)*x + 4*x^2)*Log[x] + 2*x*Log[x]^2),x]

[Out]

(87*Defer[Int][(4 + E^(2*x) + x + Log[x])^(-2), x])/2 - 6*Defer[Int][1/(x*(4 + E^(2*x) + x + Log[x])^2), x] +
(3*Defer[Int][x/(4 + E^(2*x) + x + Log[x])^2, x])/2 - 3*Defer[Int][x^2/(4 + E^(2*x) + x + Log[x])^2, x] + 12*D
efer[Int][Log[x]/(4 + E^(2*x) + x + Log[x])^2, x] - 3*Defer[Int][(x*Log[x])/(4 + E^(2*x) + x + Log[x])^2, x] -
(27*Defer[Int][(4 + E^(2*x) + x + Log[x])^(-1), x])/2 + 3*Defer[Int][x/(4 + E^(2*x) + x + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-4-\left (7+9 e^{2 x}\right ) x+2 e^{2 x} x^2-x \log (x)\right )}{2 x \left (4+e^{2 x}+x+\log (x)\right )^2} \, dx\\ &=\frac {3}{2} \int \frac {-4-\left (7+9 e^{2 x}\right ) x+2 e^{2 x} x^2-x \log (x)}{x \left (4+e^{2 x}+x+\log (x)\right )^2} \, dx\\ &=\frac {3}{2} \int \left (\frac {-9+2 x}{4+e^{2 x}+x+\log (x)}-\frac {(-4+x) \left (-1+7 x+2 x^2+2 x \log (x)\right )}{x \left (4+e^{2 x}+x+\log (x)\right )^2}\right ) \, dx\\ &=\frac {3}{2} \int \frac {-9+2 x}{4+e^{2 x}+x+\log (x)} \, dx-\frac {3}{2} \int \frac {(-4+x) \left (-1+7 x+2 x^2+2 x \log (x)\right )}{x \left (4+e^{2 x}+x+\log (x)\right )^2} \, dx\\ &=\frac {3}{2} \int \left (-\frac {9}{4+e^{2 x}+x+\log (x)}+\frac {2 x}{4+e^{2 x}+x+\log (x)}\right ) \, dx-\frac {3}{2} \int \left (\frac {-1+7 x+2 x^2+2 x \log (x)}{\left (4+e^{2 x}+x+\log (x)\right )^2}-\frac {4 \left (-1+7 x+2 x^2+2 x \log (x)\right )}{x \left (4+e^{2 x}+x+\log (x)\right )^2}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {-1+7 x+2 x^2+2 x \log (x)}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx\right )+3 \int \frac {x}{4+e^{2 x}+x+\log (x)} \, dx+6 \int \frac {-1+7 x+2 x^2+2 x \log (x)}{x \left (4+e^{2 x}+x+\log (x)\right )^2} \, dx-\frac {27}{2} \int \frac {1}{4+e^{2 x}+x+\log (x)} \, dx\\ &=-\left (\frac {3}{2} \int \left (-\frac {1}{\left (4+e^{2 x}+x+\log (x)\right )^2}+\frac {7 x}{\left (4+e^{2 x}+x+\log (x)\right )^2}+\frac {2 x^2}{\left (4+e^{2 x}+x+\log (x)\right )^2}+\frac {2 x \log (x)}{\left (4+e^{2 x}+x+\log (x)\right )^2}\right ) \, dx\right )+3 \int \frac {x}{4+e^{2 x}+x+\log (x)} \, dx+6 \int \left (\frac {7}{\left (4+e^{2 x}+x+\log (x)\right )^2}-\frac {1}{x \left (4+e^{2 x}+x+\log (x)\right )^2}+\frac {2 x}{\left (4+e^{2 x}+x+\log (x)\right )^2}+\frac {2 \log (x)}{\left (4+e^{2 x}+x+\log (x)\right )^2}\right ) \, dx-\frac {27}{2} \int \frac {1}{4+e^{2 x}+x+\log (x)} \, dx\\ &=\frac {3}{2} \int \frac {1}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx-3 \int \frac {x^2}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx-3 \int \frac {x \log (x)}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx+3 \int \frac {x}{4+e^{2 x}+x+\log (x)} \, dx-6 \int \frac {1}{x \left (4+e^{2 x}+x+\log (x)\right )^2} \, dx-\frac {21}{2} \int \frac {x}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx+12 \int \frac {x}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx+12 \int \frac {\log (x)}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx-\frac {27}{2} \int \frac {1}{4+e^{2 x}+x+\log (x)} \, dx+42 \int \frac {1}{\left (4+e^{2 x}+x+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 21, normalized size = 0.81 \begin {gather*} \frac {3 (4-x)}{2 \left (4+e^{2 x}+x+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 - 21*x + E^(2*x)*(-27*x + 6*x^2) - 3*x*Log[x])/(32*x + 2*E^(4*x)*x + 16*x^2 + 2*x^3 + E^(2*x)*(
16*x + 4*x^2) + (16*x + 4*E^(2*x)*x + 4*x^2)*Log[x] + 2*x*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.69, size = 16, normalized size = 0.62 \begin {gather*} -\frac {3 \, {\left (x - 4\right )}}{2 \, {\left (x + e^{\left (2 \, x\right )} + \log \relax (x) + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x)+(6*x^2-27*x)*exp(x)^2-21*x-12)/(2*x*log(x)^2+(4*x*exp(x)^2+4*x^2+16*x)*log(x)+2*x*exp(x
)^4+(4*x^2+16*x)*exp(x)^2+2*x^3+16*x^2+32*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 16, normalized size = 0.62 \begin {gather*} -\frac {3 \, {\left (x - 4\right )}}{2 \, {\left (x + e^{\left (2 \, x\right )} + \log \relax (x) + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x)+(6*x^2-27*x)*exp(x)^2-21*x-12)/(2*x*log(x)^2+(4*x*exp(x)^2+4*x^2+16*x)*log(x)+2*x*exp(x
)^4+(4*x^2+16*x)*exp(x)^2+2*x^3+16*x^2+32*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 17, normalized size = 0.65

method	result	size
risch	\(-\frac {3 \left (x -4\right )}{2 \left ({\mathrm e}^{2 x}+x +\ln \relax (x )+4\right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x*ln(x)+(6*x^2-27*x)*exp(x)^2-21*x-12)/(2*x*ln(x)^2+(4*x*exp(x)^2+4*x^2+16*x)*ln(x)+2*x*exp(x)^4+(4*x^
2+16*x)*exp(x)^2+2*x^3+16*x^2+32*x),x,method=_RETURNVERBOSE)

[Out]

-3/2*(x-4)/(exp(2*x)+x+ln(x)+4)

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maxima [A]  time = 0.41, size = 16, normalized size = 0.62 \begin {gather*} -\frac {3 \, {\left (x - 4\right )}}{2 \, {\left (x + e^{\left (2 \, x\right )} + \log \relax (x) + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x)+(6*x^2-27*x)*exp(x)^2-21*x-12)/(2*x*log(x)^2+(4*x*exp(x)^2+4*x^2+16*x)*log(x)+2*x*exp(x
)^4+(4*x^2+16*x)*exp(x)^2+2*x^3+16*x^2+32*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.34, size = 22, normalized size = 0.85 \begin {gather*} -\frac {3\,\left (x-4\right )}{2\,\left (x+{\mathrm {e}}^{2\,x}+\ln \relax (x)+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(21*x + exp(2*x)*(27*x - 6*x^2) + 3*x*log(x) + 12)/(32*x + exp(2*x)*(16*x + 4*x^2) + 2*x*exp(4*x) + 2*x*l
og(x)^2 + log(x)*(16*x + 4*x*exp(2*x) + 4*x^2) + 16*x^2 + 2*x^3),x)

[Out]

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

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sympy [A]  time = 0.27, size = 20, normalized size = 0.77 \begin {gather*} \frac {12 - 3 x}{2 x + 2 e^{2 x} + 2 \log {\relax (x )} + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*ln(x)+(6*x**2-27*x)*exp(x)**2-21*x-12)/(2*x*ln(x)**2+(4*x*exp(x)**2+4*x**2+16*x)*ln(x)+2*x*exp
(x)**4+(4*x**2+16*x)*exp(x)**2+2*x**3+16*x**2+32*x),x)

[Out]

(12 - 3*x)/(2*x + 2*exp(2*x) + 2*log(x) + 8)

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