3.80.6 \(\int \frac {-12 x^3-3 x^7+12 x^3 \log (x^2)}{(256+128 x^4+16 x^8) \log ^3(x^2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {3 x^2}{64 \left (\frac {4}{x^2}+x^2\right ) \log ^2\left (x^2\right )} \]

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Rubi [F]  time = 0.56, 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 x^3-3 x^7+12 x^3 \log \left (x^2\right )}{\left (256+128 x^4+16 x^8\right ) \log ^3\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12*x^3 - 3*x^7 + 12*x^3*Log[x^2])/((256 + 128*x^4 + 16*x^8)*Log[x^2]^3),x]

[Out]

(-3*Defer[Subst][Defer[Int][x/((4 + x^2)*Log[x]^3), x], x, x^2])/32 + (3*Defer[Subst][Defer[Int][x/((4 + x^2)^
2*Log[x]^2), x], x, x^2])/8

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=16 \int \frac {-12 x^3-3 x^7+12 x^3 \log \left (x^2\right )}{\left (64+16 x^4\right )^2 \log ^3\left (x^2\right )} \, dx\\ &=16 \int \frac {3 x^3 \left (-4-x^4+4 \log \left (x^2\right )\right )}{\left (64+16 x^4\right )^2 \log ^3\left (x^2\right )} \, dx\\ &=48 \int \frac {x^3 \left (-4-x^4+4 \log \left (x^2\right )\right )}{\left (64+16 x^4\right )^2 \log ^3\left (x^2\right )} \, dx\\ &=24 \operatorname {Subst}\left (\int -\frac {x \left (4+x^2-4 \log (x)\right )}{256 \left (4+x^2\right )^2 \log ^3(x)} \, dx,x,x^2\right )\\ &=-\left (\frac {3}{32} \operatorname {Subst}\left (\int \frac {x \left (4+x^2-4 \log (x)\right )}{\left (4+x^2\right )^2 \log ^3(x)} \, dx,x,x^2\right )\right )\\ &=-\left (\frac {3}{32} \operatorname {Subst}\left (\int \left (\frac {x}{\left (4+x^2\right ) \log ^3(x)}-\frac {4 x}{\left (4+x^2\right )^2 \log ^2(x)}\right ) \, dx,x,x^2\right )\right )\\ &=-\left (\frac {3}{32} \operatorname {Subst}\left (\int \frac {x}{\left (4+x^2\right ) \log ^3(x)} \, dx,x,x^2\right )\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {x}{\left (4+x^2\right )^2 \log ^2(x)} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} \frac {3 x^4}{64 \left (4+x^4\right ) \log ^2\left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x^3 - 3*x^7 + 12*x^3*Log[x^2])/((256 + 128*x^4 + 16*x^8)*Log[x^2]^3),x]

[Out]

(3*x^4)/(64*(4 + x^4)*Log[x^2]^2)

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fricas [A]  time = 2.06, size = 18, normalized size = 0.75 \begin {gather*} \frac {3 \, x^{4}}{64 \, {\left (x^{4} + 4\right )} \log \left (x^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^3*log(x^2)-3*x^7-12*x^3)/(16*x^8+128*x^4+256)/log(x^2)^3,x, algorithm="fricas")

[Out]

3/64*x^4/((x^4 + 4)*log(x^2)^2)

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giac [A]  time = 0.30, size = 26, normalized size = 1.08 \begin {gather*} \frac {3 \, x^{4}}{64 \, {\left (x^{4} \log \left (x^{2}\right )^{2} + 4 \, \log \left (x^{2}\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^3*log(x^2)-3*x^7-12*x^3)/(16*x^8+128*x^4+256)/log(x^2)^3,x, algorithm="giac")

[Out]

3/64*x^4/(x^4*log(x^2)^2 + 4*log(x^2)^2)

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maple [A]  time = 0.04, size = 19, normalized size = 0.79

method	result	size
risch	\(\frac {3 x^{4}}{64 \left (x^{4}+4\right ) \ln \left (x^{2}\right )^{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^3*ln(x^2)-3*x^7-12*x^3)/(16*x^8+128*x^4+256)/ln(x^2)^3,x,method=_RETURNVERBOSE)

[Out]

3/64*x^4/(x^4+4)/ln(x^2)^2

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maxima [A]  time = 0.49, size = 16, normalized size = 0.67 \begin {gather*} \frac {3 \, x^{4}}{256 \, {\left (x^{4} + 4\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^3*log(x^2)-3*x^7-12*x^3)/(16*x^8+128*x^4+256)/log(x^2)^3,x, algorithm="maxima")

[Out]

3/256*x^4/((x^4 + 4)*log(x)^2)

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mupad [B]  time = 5.19, size = 18, normalized size = 0.75 \begin {gather*} \frac {3\,x^4}{64\,{\ln \left (x^2\right )}^2\,\left (x^4+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x^3 - 12*x^3*log(x^2) + 3*x^7)/(log(x^2)^3*(128*x^4 + 16*x^8 + 256)),x)

[Out]

(3*x^4)/(64*log(x^2)^2*(x^4 + 4))

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sympy [A]  time = 0.13, size = 17, normalized size = 0.71 \begin {gather*} \frac {3 x^{4}}{\left (64 x^{4} + 256\right ) \log {\left (x^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**3*ln(x**2)-3*x**7-12*x**3)/(16*x**8+128*x**4+256)/ln(x**2)**3,x)

[Out]

3*x**4/((64*x**4 + 256)*log(x**2)**2)

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