3.101.22 \(\int \frac {-16 x^4+8 x^4 \log (x)+(-8+2 x^4+(8-2 x^4) \log (x)) \log (-4+x^4)}{e (-4+x^4) \log ^2(-4+x^4)} \, dx\)

Optimal. Leaf size=18 \[ -\frac {2 x (-2+\log (x))}{e \log \left (-4+x^4\right )} \]

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Rubi [F]  time = 1.39, 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 {-16 x^4+8 x^4 \log (x)+\left (-8+2 x^4+\left (8-2 x^4\right ) \log (x)\right ) \log \left (-4+x^4\right )}{e \left (-4+x^4\right ) \log ^2\left (-4+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*x^4 + 8*x^4*Log[x] + (-8 + 2*x^4 + (8 - 2*x^4)*Log[x])*Log[-4 + x^4])/(E*(-4 + x^4)*Log[-4 + x^4]^2),
x]

[Out]

(-16*Defer[Int][Log[-4 + x^4]^(-2), x])/E - (16*Defer[Int][1/((-2 + x^2)*Log[-4 + x^4]^2), x])/E + (16*Defer[I
nt][1/((2 + x^2)*Log[-4 + x^4]^2), x])/E + (8*Defer[Int][Log[x]/Log[-4 + x^4]^2, x])/E - ((2*I)*Sqrt[2]*Defer[
Int][Log[x]/((I*Sqrt[2] - x)*Log[-4 + x^4]^2), x])/E - (2*Sqrt[2]*Defer[Int][Log[x]/((Sqrt[2] - x)*Log[-4 + x^
4]^2), x])/E - ((2*I)*Sqrt[2]*Defer[Int][Log[x]/((I*Sqrt[2] + x)*Log[-4 + x^4]^2), x])/E - (2*Sqrt[2]*Defer[In
t][Log[x]/((Sqrt[2] + x)*Log[-4 + x^4]^2), x])/E + (2*Defer[Int][Log[-4 + x^4]^(-1), x])/E - (2*Defer[Int][Log
[x]/Log[-4 + x^4], x])/E

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 x (-2+\log (x))}{e \log \left (-4+x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x^4 + 8*x^4*Log[x] + (-8 + 2*x^4 + (8 - 2*x^4)*Log[x])*Log[-4 + x^4])/(E*(-4 + x^4)*Log[-4 + x^
4]^2),x]

[Out]

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

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fricas [A]  time = 2.48, size = 20, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (x \log \relax (x) - 2 \, x\right )} e^{\left (-1\right )}}{\log \left (x^{4} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+8)*log(x)+2*x^4-8)*log(x^4-4)+8*x^4*log(x)-16*x^4)/(x^4-4)/exp(1)/log(x^4-4)^2,x, algorith
m="fricas")

[Out]

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

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giac [A]  time = 0.23, size = 20, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (x \log \relax (x) - 2 \, x\right )} e^{\left (-1\right )}}{\log \left (x^{4} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+8)*log(x)+2*x^4-8)*log(x^4-4)+8*x^4*log(x)-16*x^4)/(x^4-4)/exp(1)/log(x^4-4)^2,x, algorith
m="giac")

[Out]

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

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maple [A]  time = 0.12, size = 18, normalized size = 1.00

method	result	size
risch	\(-\frac {2 x \,{\mathrm e}^{-1} \left (\ln \relax (x )-2\right )}{\ln \left (x^{4}-4\right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^4+8)*ln(x)+2*x^4-8)*ln(x^4-4)+8*x^4*ln(x)-16*x^4)/(x^4-4)/exp(1)/ln(x^4-4)^2,x,method=_RETURNVERBO
SE)

[Out]

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

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maxima [A]  time = 0.49, size = 27, normalized size = 1.50 \begin {gather*} -\frac {2 \, {\left (x \log \relax (x) - 2 \, x\right )} e^{\left (-1\right )}}{\log \left (x^{2} + 2\right ) + \log \left (x^{2} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+8)*log(x)+2*x^4-8)*log(x^4-4)+8*x^4*log(x)-16*x^4)/(x^4-4)/exp(1)/log(x^4-4)^2,x, algorith
m="maxima")

[Out]

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

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mupad [B]  time = 6.74, size = 17, normalized size = 0.94 \begin {gather*} -\frac {2\,x\,{\mathrm {e}}^{-1}\,\left (\ln \relax (x)-2\right )}{\ln \left (x^4-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-1)*(log(x^4 - 4)*(log(x)*(2*x^4 - 8) - 2*x^4 + 8) - 8*x^4*log(x) + 16*x^4))/(log(x^4 - 4)^2*(x^4 -
4)),x)

[Out]

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

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sympy [A]  time = 0.27, size = 19, normalized size = 1.06 \begin {gather*} \frac {- 2 x \log {\relax (x )} + 4 x}{e \log {\left (x^{4} - 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**4+8)*ln(x)+2*x**4-8)*ln(x**4-4)+8*x**4*ln(x)-16*x**4)/(x**4-4)/exp(1)/ln(x**4-4)**2,x)

[Out]

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

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