3.86.90 \(\int \frac {e^{-x} ((-24 x^2+8 x^6) \log ^3(\frac {3+x^4}{x^2})+(9 x^2-3 x^3+3 x^6-x^7) \log ^4(\frac {3+x^4}{x^2}))}{3+x^4} \, dx\)

Optimal. Leaf size=21 \[ e^{-x} x^3 \log ^4\left (\frac {3}{x^2}+x^2\right ) \]

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Rubi [F]  time = 2.86, 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^{-x} \left (\left (-24 x^2+8 x^6\right ) \log ^3\left (\frac {3+x^4}{x^2}\right )+\left (9 x^2-3 x^3+3 x^6-x^7\right ) \log ^4\left (\frac {3+x^4}{x^2}\right )\right )}{3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-24*x^2 + 8*x^6)*Log[(3 + x^4)/x^2]^3 + (9*x^2 - 3*x^3 + 3*x^6 - x^7)*Log[(3 + x^4)/x^2]^4)/(E^x*(3 + x^
4)),x]

[Out]

-4*(-3)^(3/4)*Defer[Int][Log[(3 + x^4)/x^2]^3/(E^x*((-3)^(1/4) - x)), x] + 4*(-1)^(1/4)*3^(3/4)*Defer[Int][Log
[(3 + x^4)/x^2]^3/(E^x*(-((-1)^(3/4)*3^(1/4)) - x)), x] + 8*Defer[Int][(x^2*Log[(3 + x^4)/x^2]^3)/E^x, x] - 4*
(-3)^(3/4)*Defer[Int][Log[(3 + x^4)/x^2]^3/(E^x*((-3)^(1/4) + x)), x] + 4*(-1)^(1/4)*3^(3/4)*Defer[Int][Log[(3
+ x^4)/x^2]^3/(E^x*(-((-1)^(3/4)*3^(1/4)) + x)), x] + 3*Defer[Int][(x^2*Log[(3 + x^4)/x^2]^4)/E^x, x] - Defer
[Int][(x^3*Log[(3 + x^4)/x^2]^4)/E^x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {8 e^{-x} x^2 \left (-3+x^4\right ) \log ^3\left (\frac {3+x^4}{x^2}\right )}{3+x^4}-e^{-x} (-3+x) x^2 \log ^4\left (\frac {3+x^4}{x^2}\right )\right ) \, dx\\ &=8 \int \frac {e^{-x} x^2 \left (-3+x^4\right ) \log ^3\left (\frac {3+x^4}{x^2}\right )}{3+x^4} \, dx-\int e^{-x} (-3+x) x^2 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx\\ &=8 \int \left (e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right )-\frac {6 e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right )}{3+x^4}\right ) \, dx-\int \left (-3 e^{-x} x^2 \log ^4\left (\frac {3+x^4}{x^2}\right )+e^{-x} x^3 \log ^4\left (\frac {3+x^4}{x^2}\right )\right ) \, dx\\ &=3 \int e^{-x} x^2 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx+8 \int e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right ) \, dx-48 \int \frac {e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right )}{3+x^4} \, dx-\int e^{-x} x^3 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx\\ &=3 \int e^{-x} x^2 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx+8 \int e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right ) \, dx-48 \int \left (-\frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{2 \left (i \sqrt {3}-x^2\right )}+\frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{2 \left (i \sqrt {3}+x^2\right )}\right ) \, dx-\int e^{-x} x^3 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx\\ &=3 \int e^{-x} x^2 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx+8 \int e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right ) \, dx+24 \int \frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{i \sqrt {3}-x^2} \, dx-24 \int \frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{i \sqrt {3}+x^2} \, dx-\int e^{-x} x^3 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx\\ &=3 \int e^{-x} x^2 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx+8 \int e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right ) \, dx+24 \int \left (-\frac {(-1)^{3/4} e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{2 \sqrt [4]{3} \left (\sqrt [4]{-3}-x\right )}-\frac {(-1)^{3/4} e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{2 \sqrt [4]{3} \left (\sqrt [4]{-3}+x\right )}\right ) \, dx-24 \int \left (-\frac {\sqrt [4]{-\frac {1}{3}} e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{2 \left (-(-1)^{3/4} \sqrt [4]{3}-x\right )}-\frac {\sqrt [4]{-\frac {1}{3}} e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{2 \left (-(-1)^{3/4} \sqrt [4]{3}+x\right )}\right ) \, dx-\int e^{-x} x^3 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx\\ &=3 \int e^{-x} x^2 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx+8 \int e^{-x} x^2 \log ^3\left (\frac {3+x^4}{x^2}\right ) \, dx-\left (4 (-3)^{3/4}\right ) \int \frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{\sqrt [4]{-3}-x} \, dx-\left (4 (-3)^{3/4}\right ) \int \frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{\sqrt [4]{-3}+x} \, dx+\left (4 \sqrt [4]{-1} 3^{3/4}\right ) \int \frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{-(-1)^{3/4} \sqrt [4]{3}-x} \, dx+\left (4 \sqrt [4]{-1} 3^{3/4}\right ) \int \frac {e^{-x} \log ^3\left (\frac {3+x^4}{x^2}\right )}{-(-1)^{3/4} \sqrt [4]{3}+x} \, dx-\int e^{-x} x^3 \log ^4\left (\frac {3+x^4}{x^2}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-x} \left (\left (-24 x^2+8 x^6\right ) \log ^3\left (\frac {3+x^4}{x^2}\right )+\left (9 x^2-3 x^3+3 x^6-x^7\right ) \log ^4\left (\frac {3+x^4}{x^2}\right )\right )}{3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-24*x^2 + 8*x^6)*Log[(3 + x^4)/x^2]^3 + (9*x^2 - 3*x^3 + 3*x^6 - x^7)*Log[(3 + x^4)/x^2]^4)/(E^x*(
3 + x^4)),x]

[Out]

Integrate[((-24*x^2 + 8*x^6)*Log[(3 + x^4)/x^2]^3 + (9*x^2 - 3*x^3 + 3*x^6 - x^7)*Log[(3 + x^4)/x^2]^4)/(E^x*(
3 + x^4)), x]

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fricas [A]  time = 0.63, size = 20, normalized size = 0.95 \begin {gather*} x^{3} e^{\left (-x\right )} \log \left (\frac {x^{4} + 3}{x^{2}}\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^7+3*x^6-3*x^3+9*x^2)*log((x^4+3)/x^2)^4+(8*x^6-24*x^2)*log((x^4+3)/x^2)^3)/(x^4+3)/exp(x),x, al
gorithm="fricas")

[Out]

x^3*e^(-x)*log((x^4 + 3)/x^2)^4

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giac [A]  time = 4.48, size = 20, normalized size = 0.95 \begin {gather*} x^{3} e^{\left (-x\right )} \log \left (\frac {x^{4} + 3}{x^{2}}\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^7+3*x^6-3*x^3+9*x^2)*log((x^4+3)/x^2)^4+(8*x^6-24*x^2)*log((x^4+3)/x^2)^3)/(x^4+3)/exp(x),x, al
gorithm="giac")

[Out]

x^3*e^(-x)*log((x^4 + 3)/x^2)^4

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maple [C]  time = 1.93, size = 15428, normalized size = 734.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(15428\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^7+3*x^6-3*x^3+9*x^2)*ln((x^4+3)/x^2)^4+(8*x^6-24*x^2)*ln((x^4+3)/x^2)^3)/(x^4+3)/exp(x),x,method=_RET
URNVERBOSE)

[Out]

result too large to display

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maxima [B]  time = 0.51, size = 89, normalized size = 4.24 \begin {gather*} x^{3} e^{\left (-x\right )} \log \left (x^{4} + 3\right )^{4} - 8 \, x^{3} e^{\left (-x\right )} \log \left (x^{4} + 3\right )^{3} \log \relax (x) + 24 \, x^{3} e^{\left (-x\right )} \log \left (x^{4} + 3\right )^{2} \log \relax (x)^{2} - 32 \, x^{3} e^{\left (-x\right )} \log \left (x^{4} + 3\right ) \log \relax (x)^{3} + 16 \, x^{3} e^{\left (-x\right )} \log \relax (x)^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^7+3*x^6-3*x^3+9*x^2)*log((x^4+3)/x^2)^4+(8*x^6-24*x^2)*log((x^4+3)/x^2)^3)/(x^4+3)/exp(x),x, al
gorithm="maxima")

[Out]

x^3*e^(-x)*log(x^4 + 3)^4 - 8*x^3*e^(-x)*log(x^4 + 3)^3*log(x) + 24*x^3*e^(-x)*log(x^4 + 3)^2*log(x)^2 - 32*x^
3*e^(-x)*log(x^4 + 3)*log(x)^3 + 16*x^3*e^(-x)*log(x)^4

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mupad [B]  time = 5.36, size = 20, normalized size = 0.95 \begin {gather*} x^3\,{\mathrm {e}}^{-x}\,{\ln \left (\frac {x^4+3}{x^2}\right )}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(log((x^4 + 3)/x^2)^3*(24*x^2 - 8*x^6) - log((x^4 + 3)/x^2)^4*(9*x^2 - 3*x^3 + 3*x^6 - x^7)))/(x
^4 + 3),x)

[Out]

x^3*exp(-x)*log((x^4 + 3)/x^2)^4

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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(((-x**7+3*x**6-3*x**3+9*x**2)*ln((x**4+3)/x**2)**4+(8*x**6-24*x**2)*ln((x**4+3)/x**2)**3)/(x**4+3)/e
xp(x),x)

[Out]

Timed out

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