3.60.31 \(\int \frac {e^{9 x} (8-72 x+24 x^2-72 x^3+10 x^4-18 x^5) \log (27)+(16 x^3+8 x^5) \log (27)}{e^{18 x}+2 e^{9 x} x+x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac {2 x \left (2+x^2\right )^2 \log (27)}{e^{9 x}+x} \]

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Rubi [F]  time = 1.24, 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 {e^{9 x} \left (8-72 x+24 x^2-72 x^3+10 x^4-18 x^5\right ) \log (27)+\left (16 x^3+8 x^5\right ) \log (27)}{e^{18 x}+2 e^{9 x} x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(9*x)*(8 - 72*x + 24*x^2 - 72*x^3 + 10*x^4 - 18*x^5)*Log[27] + (16*x^3 + 8*x^5)*Log[27])/(E^(18*x) + 2*
E^(9*x)*x + x^2),x]

[Out]

-8*Log[27]*Defer[Int][x/(E^(9*x) + x)^2, x] + 72*Log[27]*Defer[Int][x^2/(E^(9*x) + x)^2, x] - 8*Log[27]*Defer[
Int][x^3/(E^(9*x) + x)^2, x] + 72*Log[27]*Defer[Int][x^4/(E^(9*x) + x)^2, x] - 2*Log[27]*Defer[Int][x^5/(E^(9*
x) + x)^2, x] + 18*Log[27]*Defer[Int][x^6/(E^(9*x) + x)^2, x] + 8*Log[27]*Defer[Int][(E^(9*x) + x)^(-1), x] -
72*Log[27]*Defer[Int][x/(E^(9*x) + x), x] + 24*Log[27]*Defer[Int][x^2/(E^(9*x) + x), x] - 72*Log[27]*Defer[Int
][x^3/(E^(9*x) + x), x] + 10*Log[27]*Defer[Int][x^4/(E^(9*x) + x), x] - 18*Log[27]*Defer[Int][x^5/(E^(9*x) + x
), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (2+x^2\right ) \left (4 x^3-e^{9 x} \left (-2+18 x-5 x^2+9 x^3\right )\right ) \log (27)}{\left (e^{9 x}+x\right )^2} \, dx\\ &=(2 \log (27)) \int \frac {\left (2+x^2\right ) \left (4 x^3-e^{9 x} \left (-2+18 x-5 x^2+9 x^3\right )\right )}{\left (e^{9 x}+x\right )^2} \, dx\\ &=(2 \log (27)) \int \left (\frac {x (-1+9 x) \left (2+x^2\right )^2}{\left (e^{9 x}+x\right )^2}-\frac {-4+36 x-12 x^2+36 x^3-5 x^4+9 x^5}{e^{9 x}+x}\right ) \, dx\\ &=(2 \log (27)) \int \frac {x (-1+9 x) \left (2+x^2\right )^2}{\left (e^{9 x}+x\right )^2} \, dx-(2 \log (27)) \int \frac {-4+36 x-12 x^2+36 x^3-5 x^4+9 x^5}{e^{9 x}+x} \, dx\\ &=(2 \log (27)) \int \left (-\frac {4 x}{\left (e^{9 x}+x\right )^2}+\frac {36 x^2}{\left (e^{9 x}+x\right )^2}-\frac {4 x^3}{\left (e^{9 x}+x\right )^2}+\frac {36 x^4}{\left (e^{9 x}+x\right )^2}-\frac {x^5}{\left (e^{9 x}+x\right )^2}+\frac {9 x^6}{\left (e^{9 x}+x\right )^2}\right ) \, dx-(2 \log (27)) \int \left (-\frac {4}{e^{9 x}+x}+\frac {36 x}{e^{9 x}+x}-\frac {12 x^2}{e^{9 x}+x}+\frac {36 x^3}{e^{9 x}+x}-\frac {5 x^4}{e^{9 x}+x}+\frac {9 x^5}{e^{9 x}+x}\right ) \, dx\\ &=-\left ((2 \log (27)) \int \frac {x^5}{\left (e^{9 x}+x\right )^2} \, dx\right )-(8 \log (27)) \int \frac {x}{\left (e^{9 x}+x\right )^2} \, dx-(8 \log (27)) \int \frac {x^3}{\left (e^{9 x}+x\right )^2} \, dx+(8 \log (27)) \int \frac {1}{e^{9 x}+x} \, dx+(10 \log (27)) \int \frac {x^4}{e^{9 x}+x} \, dx+(18 \log (27)) \int \frac {x^6}{\left (e^{9 x}+x\right )^2} \, dx-(18 \log (27)) \int \frac {x^5}{e^{9 x}+x} \, dx+(24 \log (27)) \int \frac {x^2}{e^{9 x}+x} \, dx+(72 \log (27)) \int \frac {x^2}{\left (e^{9 x}+x\right )^2} \, dx+(72 \log (27)) \int \frac {x^4}{\left (e^{9 x}+x\right )^2} \, dx-(72 \log (27)) \int \frac {x}{e^{9 x}+x} \, dx-(72 \log (27)) \int \frac {x^3}{e^{9 x}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 x \left (2+x^2\right )^2 \log (27)}{e^{9 x}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(9*x)*(8 - 72*x + 24*x^2 - 72*x^3 + 10*x^4 - 18*x^5)*Log[27] + (16*x^3 + 8*x^5)*Log[27])/(E^(18*x
) + 2*E^(9*x)*x + x^2),x]

[Out]

(2*x*(2 + x^2)^2*Log[27])/(E^(9*x) + x)

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fricas [A]  time = 0.62, size = 24, normalized size = 1.14 \begin {gather*} \frac {6 \, {\left (x^{5} + 4 \, x^{3} + 4 \, x\right )} \log \relax (3)}{x + e^{\left (9 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*(-18*x^5+10*x^4-72*x^3+24*x^2-72*x+8)*log(3)*exp(9*x)+3*(8*x^5+16*x^3)*log(3))/(exp(9*x)^2+2*x*ex
p(9*x)+x^2),x, algorithm="fricas")

[Out]

6*(x^5 + 4*x^3 + 4*x)*log(3)/(x + e^(9*x))

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giac [A]  time = 0.16, size = 29, normalized size = 1.38 \begin {gather*} \frac {6 \, {\left (x^{5} \log \relax (3) + 4 \, x^{3} \log \relax (3) + 4 \, x \log \relax (3)\right )}}{x + e^{\left (9 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*(-18*x^5+10*x^4-72*x^3+24*x^2-72*x+8)*log(3)*exp(9*x)+3*(8*x^5+16*x^3)*log(3))/(exp(9*x)^2+2*x*ex
p(9*x)+x^2),x, algorithm="giac")

[Out]

6*(x^5*log(3) + 4*x^3*log(3) + 4*x*log(3))/(x + e^(9*x))

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maple [A]  time = 0.16, size = 24, normalized size = 1.14

method	result	size
risch	\(\frac {6 \left (x^{4}+4 x^{2}+4\right ) \ln \relax (3) x}{x +{\mathrm e}^{9 x}}\)	\(24\)
norman	\(\frac {-24 \ln \relax (3) {\mathrm e}^{9 x}+24 x^{3} \ln \relax (3)+6 x^{5} \ln \relax (3)}{x +{\mathrm e}^{9 x}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*(-18*x^5+10*x^4-72*x^3+24*x^2-72*x+8)*ln(3)*exp(9*x)+3*(8*x^5+16*x^3)*ln(3))/(exp(9*x)^2+2*x*exp(9*x)+x
^2),x,method=_RETURNVERBOSE)

[Out]

6*(x^4+4*x^2+4)*ln(3)*x/(x+exp(9*x))

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maxima [A]  time = 0.50, size = 29, normalized size = 1.38 \begin {gather*} \frac {6 \, {\left (x^{5} \log \relax (3) + 4 \, x^{3} \log \relax (3) + 4 \, x \log \relax (3)\right )}}{x + e^{\left (9 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*(-18*x^5+10*x^4-72*x^3+24*x^2-72*x+8)*log(3)*exp(9*x)+3*(8*x^5+16*x^3)*log(3))/(exp(9*x)^2+2*x*ex
p(9*x)+x^2),x, algorithm="maxima")

[Out]

6*(x^5*log(3) + 4*x^3*log(3) + 4*x*log(3))/(x + e^(9*x))

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mupad [B]  time = 4.30, size = 20, normalized size = 0.95 \begin {gather*} \frac {6\,x\,\ln \relax (3)\,{\left (x^2+2\right )}^2}{x+{\mathrm {e}}^{9\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*log(3)*(16*x^3 + 8*x^5) - 3*exp(9*x)*log(3)*(72*x - 24*x^2 + 72*x^3 - 10*x^4 + 18*x^5 - 8))/(exp(18*x)
+ 2*x*exp(9*x) + x^2),x)

[Out]

(6*x*log(3)*(x^2 + 2)^2)/(x + exp(9*x))

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sympy [A]  time = 0.12, size = 29, normalized size = 1.38 \begin {gather*} \frac {6 x^{5} \log {\relax (3 )} + 24 x^{3} \log {\relax (3 )} + 24 x \log {\relax (3 )}}{x + e^{9 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*(-18*x**5+10*x**4-72*x**3+24*x**2-72*x+8)*ln(3)*exp(9*x)+3*(8*x**5+16*x**3)*ln(3))/(exp(9*x)**2+2
*x*exp(9*x)+x**2),x)

[Out]

(6*x**5*log(3) + 24*x**3*log(3) + 24*x*log(3))/(x + exp(9*x))

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