3.19.77 \(\int \frac {e^{\frac {2 x+12 x^3-3 x^4+(-3-12 x^2+3 x^3) \log (\frac {x}{\log (3)})}{-3-12 x^2+3 x^3}} (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7)}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{-x+\frac {x}{3+3 (4-x) x^2}} x}{\log (3)} \]

________________________________________________________________________________________

Rubi [F]  time = 6.19, 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 {\exp \left (\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}\right ) \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x + 12*x^3 - 3*x^4 + (-3 - 12*x^2 + 3*x^3)*Log[x/Log[3]])/(-3 - 12*x^2 + 3*x^3))*(3 - 2*x + 24*x^2
- 34*x^3 + 56*x^4 - 72*x^5 + 27*x^6 - 3*x^7))/(3*x + 24*x^3 - 6*x^4 + 48*x^5 - 24*x^6 + 3*x^7),x]

[Out]

Defer[Int][E^((x*(2 + 12*x^2 - 3*x^3))/(3*(-1 - 4*x^2 + x^3))), x]/Log[3] - Defer[Int][E^((x*(2 + 12*x^2 - 3*x
^3))/(3*(-1 - 4*x^2 + x^3)))*x, x]/Log[3] + (4*Defer[Int][E^((x*(2 + 12*x^2 - 3*x^3))/(3*(-1 - 4*x^2 + x^3)))/
(-1 - 4*x^2 + x^3)^2, x])/(3*Log[3]) + Defer[Int][(E^((x*(2 + 12*x^2 - 3*x^3))/(3*(-1 - 4*x^2 + x^3)))*x)/(-1
- 4*x^2 + x^3)^2, x]/Log[3] + (16*Defer[Int][(E^((x*(2 + 12*x^2 - 3*x^3))/(3*(-1 - 4*x^2 + x^3)))*x^2)/(-1 - 4
*x^2 + x^3)^2, x])/(3*Log[3]) + (4*Defer[Int][E^((x*(2 + 12*x^2 - 3*x^3))/(3*(-1 - 4*x^2 + x^3)))/(-1 - 4*x^2
+ x^3), x])/(3*Log[3]) + (2*Defer[Int][(E^((x*(2 + 12*x^2 - 3*x^3))/(3*(-1 - 4*x^2 + x^3)))*x)/(-1 - 4*x^2 + x
^3), x])/(3*Log[3])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 \left (1+4 x^2-x^3\right )^2 \log (3)} \, dx\\ &=\frac {\int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{\left (1+4 x^2-x^3\right )^2} \, dx}{3 \log (3)}\\ &=\frac {\int \left (3 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right )-3 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x+\frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \left (4+3 x+16 x^2\right )}{\left (-1-4 x^2+x^3\right )^2}+\frac {2 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) (2+x)}{-1-4 x^2+x^3}\right ) \, dx}{3 \log (3)}\\ &=\frac {\int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \left (4+3 x+16 x^2\right )}{\left (-1-4 x^2+x^3\right )^2} \, dx}{3 \log (3)}+\frac {2 \int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) (2+x)}{-1-4 x^2+x^3} \, dx}{3 \log (3)}+\frac {\int \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \, dx}{\log (3)}-\frac {\int \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x \, dx}{\log (3)}\\ &=\frac {\int \left (\frac {4 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right )}{\left (-1-4 x^2+x^3\right )^2}+\frac {3 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x}{\left (-1-4 x^2+x^3\right )^2}+\frac {16 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x^2}{\left (-1-4 x^2+x^3\right )^2}\right ) \, dx}{3 \log (3)}+\frac {2 \int \left (\frac {2 \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right )}{-1-4 x^2+x^3}+\frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x}{-1-4 x^2+x^3}\right ) \, dx}{3 \log (3)}+\frac {\int \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \, dx}{\log (3)}-\frac {\int \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x \, dx}{\log (3)}\\ &=\frac {2 \int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x}{-1-4 x^2+x^3} \, dx}{3 \log (3)}+\frac {\int \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) \, dx}{\log (3)}-\frac {\int \exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x \, dx}{\log (3)}+\frac {\int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x}{\left (-1-4 x^2+x^3\right )^2} \, dx}{\log (3)}+\frac {4 \int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right )}{\left (-1-4 x^2+x^3\right )^2} \, dx}{3 \log (3)}+\frac {4 \int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right )}{-1-4 x^2+x^3} \, dx}{3 \log (3)}+\frac {16 \int \frac {\exp \left (\frac {x \left (2+12 x^2-3 x^3\right )}{3 \left (-1-4 x^2+x^3\right )}\right ) x^2}{\left (-1-4 x^2+x^3\right )^2} \, dx}{3 \log (3)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 26, normalized size = 0.93 \begin {gather*} \frac {e^{x \left (-1+\frac {1}{3+12 x^2-3 x^3}\right )} x}{\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*x + 12*x^3 - 3*x^4 + (-3 - 12*x^2 + 3*x^3)*Log[x/Log[3]])/(-3 - 12*x^2 + 3*x^3))*(3 - 2*x + 2
4*x^2 - 34*x^3 + 56*x^4 - 72*x^5 + 27*x^6 - 3*x^7))/(3*x + 24*x^3 - 6*x^4 + 48*x^5 - 24*x^6 + 3*x^7),x]

[Out]

(E^(x*(-1 + (3 + 12*x^2 - 3*x^3)^(-1)))*x)/Log[3]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 48, normalized size = 1.71 \begin {gather*} e^{\left (-\frac {3 \, x^{4} - 12 \, x^{3} - 3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )} \log \left (\frac {x}{\log \relax (3)}\right ) - 2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12*x^2-3)*log(x/log(3))-3*x^4+12*x^3+2
*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48*x^5-6*x^4+24*x^3+3*x),x, algorithm="fricas")

[Out]

e^(-1/3*(3*x^4 - 12*x^3 - 3*(x^3 - 4*x^2 - 1)*log(x/log(3)) - 2*x)/(x^3 - 4*x^2 - 1))

________________________________________________________________________________________

giac [B]  time = 0.39, size = 119, normalized size = 4.25 \begin {gather*} e^{\left (-\frac {x^{4}}{x^{3} - 4 \, x^{2} - 1} + \frac {x^{3} \log \left (\frac {x}{\log \relax (3)}\right )}{x^{3} - 4 \, x^{2} - 1} + \frac {4 \, x^{3}}{x^{3} - 4 \, x^{2} - 1} - \frac {4 \, x^{2} \log \left (\frac {x}{\log \relax (3)}\right )}{x^{3} - 4 \, x^{2} - 1} + \frac {2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )}} - \frac {\log \left (\frac {x}{\log \relax (3)}\right )}{x^{3} - 4 \, x^{2} - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12*x^2-3)*log(x/log(3))-3*x^4+12*x^3+2
*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48*x^5-6*x^4+24*x^3+3*x),x, algorithm="giac")

[Out]

e^(-x^4/(x^3 - 4*x^2 - 1) + x^3*log(x/log(3))/(x^3 - 4*x^2 - 1) + 4*x^3/(x^3 - 4*x^2 - 1) - 4*x^2*log(x/log(3)
)/(x^3 - 4*x^2 - 1) + 2/3*x/(x^3 - 4*x^2 - 1) - log(x/log(3))/(x^3 - 4*x^2 - 1))

________________________________________________________________________________________

maple [B]  time = 0.08, size = 63, normalized size = 2.25

method	result	size
gosper	\({\mathrm e}^{\frac {3 \ln \left (\frac {x}{\ln \relax (3)}\right ) x^{3}-3 x^{4}-12 \ln \left (\frac {x}{\ln \relax (3)}\right ) x^{2}+12 x^{3}-3 \ln \left (\frac {x}{\ln \relax (3)}\right )+2 x}{3 x^{3}-12 x^{2}-3}}\)	\(63\)
risch	\({\mathrm e}^{-\frac {-3 \ln \left (\frac {x}{\ln \relax (3)}\right ) x^{3}+3 x^{4}+12 \ln \left (\frac {x}{\ln \relax (3)}\right ) x^{2}-12 x^{3}+3 \ln \left (\frac {x}{\ln \relax (3)}\right )-2 x}{3 \left (x^{3}-4 x^{2}-1\right )}}\)	\(63\)
norman	\(\frac {x^{3} {\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \relax (3)}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}-4 x^{2} {\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \relax (3)}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}-{\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \relax (3)}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}}{x^{3}-4 x^{2}-1}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12*x^2-3)*ln(x/ln(3))-3*x^4+12*x^3+2*x)/(3*x
^3-12*x^2-3))/(3*x^7-24*x^6+48*x^5-6*x^4+24*x^3+3*x),x,method=_RETURNVERBOSE)

[Out]

exp(1/3*(3*ln(x/ln(3))*x^3-3*x^4-12*ln(x/ln(3))*x^2+12*x^3-3*ln(x/ln(3))+2*x)/(x^3-4*x^2-1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{3} \, \int \frac {{\left (3 \, x^{7} - 27 \, x^{6} + 72 \, x^{5} - 56 \, x^{4} + 34 \, x^{3} - 24 \, x^{2} + 2 \, x - 3\right )} e^{\left (-\frac {3 \, x^{4} - 12 \, x^{3} - 3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )} \log \left (\frac {x}{\log \relax (3)}\right ) - 2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )}}\right )}}{x^{7} - 8 \, x^{6} + 16 \, x^{5} - 2 \, x^{4} + 8 \, x^{3} + x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12*x^2-3)*log(x/log(3))-3*x^4+12*x^3+2
*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48*x^5-6*x^4+24*x^3+3*x),x, algorithm="maxima")

[Out]

-1/3*integrate((3*x^7 - 27*x^6 + 72*x^5 - 56*x^4 + 34*x^3 - 24*x^2 + 2*x - 3)*e^(-1/3*(3*x^4 - 12*x^3 - 3*(x^3
- 4*x^2 - 1)*log(x/log(3)) - 2*x)/(x^3 - 4*x^2 - 1))/(x^7 - 8*x^6 + 16*x^5 - 2*x^4 + 8*x^3 + x), x)

________________________________________________________________________________________

mupad [B]  time = 1.49, size = 145, normalized size = 5.18 \begin {gather*} x^{\frac {12\,x^2}{-3\,x^3+12\,x^2+3}-\frac {x^3-1}{-x^3+4\,x^2+1}}\,{\mathrm {e}}^{-\frac {2\,x}{-3\,x^3+12\,x^2+3}}\,{\mathrm {e}}^{\frac {3\,x^4}{-3\,x^3+12\,x^2+3}}\,{\mathrm {e}}^{-\frac {12\,x^3}{-3\,x^3+12\,x^2+3}}\,{\ln \relax (3)}^{\frac {x^3-1}{-x^3+4\,x^2+1}-\frac {12\,x^2}{-3\,x^3+12\,x^2+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*x - log(x/log(3))*(12*x^2 - 3*x^3 + 3) + 12*x^3 - 3*x^4)/(12*x^2 - 3*x^3 + 3))*(2*x - 24*x^2 + 3
4*x^3 - 56*x^4 + 72*x^5 - 27*x^6 + 3*x^7 - 3))/(3*x + 24*x^3 - 6*x^4 + 48*x^5 - 24*x^6 + 3*x^7),x)

[Out]

x^((12*x^2)/(12*x^2 - 3*x^3 + 3) - (x^3 - 1)/(4*x^2 - x^3 + 1))*exp(-(2*x)/(12*x^2 - 3*x^3 + 3))*exp((3*x^4)/(
12*x^2 - 3*x^3 + 3))*exp(-(12*x^3)/(12*x^2 - 3*x^3 + 3))*log(3)^((x^3 - 1)/(4*x^2 - x^3 + 1) - (12*x^2)/(12*x^
2 - 3*x^3 + 3))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**7+27*x**6-72*x**5+56*x**4-34*x**3+24*x**2-2*x+3)*exp(((3*x**3-12*x**2-3)*ln(x/ln(3))-3*x**4+1
2*x**3+2*x)/(3*x**3-12*x**2-3))/(3*x**7-24*x**6+48*x**5-6*x**4+24*x**3+3*x),x)

[Out]

Timed out

________________________________________________________________________________________