3.28.39 \(\int \frac {(-28-8 x+8 x^2) \log (16)+(-7-4 x+6 x^2) \log (16) \log (x^4)}{(49 x^2+28 x^3-24 x^4-8 x^5+4 x^6) \log ^2(x^4)} \, dx\)

Optimal. Leaf size=26 \[ \frac {\log (16)}{x \left (-1+2 \left (4+x-x^2\right )\right ) \log \left (x^4\right )} \]

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Rubi [F]  time = 0.59, 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 {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-28 - 8*x + 8*x^2)*Log[16] + (-7 - 4*x + 6*x^2)*Log[16]*Log[x^4])/((49*x^2 + 28*x^3 - 24*x^4 - 8*x^5 + 4
*x^6)*Log[x^4]^2),x]

[Out]

4*Log[16]*Defer[Int][1/(x^2*(-7 - 2*x + 2*x^2)*Log[x^4]^2), x] + Log[16]*Defer[Int][(-7 - 4*x + 6*x^2)/(x^2*(-
7 - 2*x + 2*x^2)^2*Log[x^4]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (16) \left (-28-8 x+8 x^2+\left (-7-4 x+6 x^2\right ) \log \left (x^4\right )\right )}{x^2 \left (7+2 x-2 x^2\right )^2 \log ^2\left (x^4\right )} \, dx\\ &=\log (16) \int \frac {-28-8 x+8 x^2+\left (-7-4 x+6 x^2\right ) \log \left (x^4\right )}{x^2 \left (7+2 x-2 x^2\right )^2 \log ^2\left (x^4\right )} \, dx\\ &=\log (16) \int \left (\frac {4}{x^2 \left (-7-2 x+2 x^2\right ) \log ^2\left (x^4\right )}+\frac {-7-4 x+6 x^2}{x^2 \left (-7-2 x+2 x^2\right )^2 \log \left (x^4\right )}\right ) \, dx\\ &=\log (16) \int \frac {-7-4 x+6 x^2}{x^2 \left (-7-2 x+2 x^2\right )^2 \log \left (x^4\right )} \, dx+(4 \log (16)) \int \frac {1}{x^2 \left (-7-2 x+2 x^2\right ) \log ^2\left (x^4\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 25, normalized size = 0.96 \begin {gather*} -\frac {\log (16)}{x \left (-7-2 x+2 x^2\right ) \log \left (x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-28 - 8*x + 8*x^2)*Log[16] + (-7 - 4*x + 6*x^2)*Log[16]*Log[x^4])/((49*x^2 + 28*x^3 - 24*x^4 - 8*x
^5 + 4*x^6)*Log[x^4]^2),x]

[Out]

-(Log[16]/(x*(-7 - 2*x + 2*x^2)*Log[x^4]))

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fricas [A]  time = 0.58, size = 26, normalized size = 1.00 \begin {gather*} -\frac {4 \, \log \relax (2)}{{\left (2 \, x^{3} - 2 \, x^{2} - 7 \, x\right )} \log \left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(6*x^2-4*x-7)*log(2)*log(x^4)+4*(8*x^2-8*x-28)*log(2))/(4*x^6-8*x^5-24*x^4+28*x^3+49*x^2)/log(x^4
)^2,x, algorithm="fricas")

[Out]

-4*log(2)/((2*x^3 - 2*x^2 - 7*x)*log(x^4))

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giac [A]  time = 0.32, size = 32, normalized size = 1.23 \begin {gather*} -\frac {4 \, \log \relax (2)}{2 \, x^{3} \log \left (x^{4}\right ) - 2 \, x^{2} \log \left (x^{4}\right ) - 7 \, x \log \left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(6*x^2-4*x-7)*log(2)*log(x^4)+4*(8*x^2-8*x-28)*log(2))/(4*x^6-8*x^5-24*x^4+28*x^3+49*x^2)/log(x^4
)^2,x, algorithm="giac")

[Out]

-4*log(2)/(2*x^3*log(x^4) - 2*x^2*log(x^4) - 7*x*log(x^4))

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maple [A]  time = 0.09, size = 26, normalized size = 1.00

method	result	size
norman	\(-\frac {4 \ln \relax (2)}{x \left (2 x^{2}-2 x -7\right ) \ln \left (x^{4}\right )}\)	\(26\)
risch	\(-\frac {4 \ln \relax (2)}{x \left (2 x^{2}-2 x -7\right ) \ln \left (x^{4}\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(6*x^2-4*x-7)*ln(2)*ln(x^4)+4*(8*x^2-8*x-28)*ln(2))/(4*x^6-8*x^5-24*x^4+28*x^3+49*x^2)/ln(x^4)^2,x,meth
od=_RETURNVERBOSE)

[Out]

-4*ln(2)/x/(2*x^2-2*x-7)/ln(x^4)

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maxima [A]  time = 0.58, size = 24, normalized size = 0.92 \begin {gather*} -\frac {\log \relax (2)}{{\left (2 \, x^{3} - 2 \, x^{2} - 7 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(6*x^2-4*x-7)*log(2)*log(x^4)+4*(8*x^2-8*x-28)*log(2))/(4*x^6-8*x^5-24*x^4+28*x^3+49*x^2)/log(x^4
)^2,x, algorithm="maxima")

[Out]

-log(2)/((2*x^3 - 2*x^2 - 7*x)*log(x))

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mupad [B]  time = 2.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {4\,\ln \relax (2)}{x\,\ln \left (x^4\right )\,\left (-2\,x^2+2\,x+7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(2)*(8*x - 8*x^2 + 28) + 4*log(x^4)*log(2)*(4*x - 6*x^2 + 7))/(log(x^4)^2*(49*x^2 + 28*x^3 - 24*x^4
- 8*x^5 + 4*x^6)),x)

[Out]

(4*log(2))/(x*log(x^4)*(2*x - 2*x^2 + 7))

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sympy [A]  time = 0.13, size = 24, normalized size = 0.92 \begin {gather*} - \frac {4 \log {\relax (2 )}}{\left (2 x^{3} - 2 x^{2} - 7 x\right ) \log {\left (x^{4} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(6*x**2-4*x-7)*ln(2)*ln(x**4)+4*(8*x**2-8*x-28)*ln(2))/(4*x**6-8*x**5-24*x**4+28*x**3+49*x**2)/ln
(x**4)**2,x)

[Out]

-4*log(2)/((2*x**3 - 2*x**2 - 7*x)*log(x**4))

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