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

Optimal. Leaf size=26 \[ 2-e^x+x-(-3+x) x-x^{\frac {4}{\log \left (x^2\right )}} \]

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Rubi [F]  time = 0.53, 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 {x^{\frac {4}{\log \left (x^2\right )}} \left (8 \log (x)-4 \log \left (x^2\right )\right )+\left (4 x-e^x x-2 x^2\right ) \log ^2\left (x^2\right )}{x \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 26, normalized size = 1.00 \begin {gather*} -e^x+4 x-x^2-x^{\frac {4}{\log \left (x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^x + 4*x - x^2 - x^(4/Log[x^2])

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fricas [A]  time = 0.57, size = 13, normalized size = 0.50 \begin {gather*} -x^{2} + 4 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x^2)+8*log(x))*exp(4*log(x)/log(x^2))+(-exp(x)*x-2*x^2+4*x)*log(x^2)^2)/x/log(x^2)^2,x, alg
orithm="fricas")

[Out]

-x^2 + 4*x - e^x

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giac [A]  time = 0.25, size = 13, normalized size = 0.50 \begin {gather*} -x^{2} + 4 \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x^2)+8*log(x))*exp(4*log(x)/log(x^2))+(-exp(x)*x-2*x^2+4*x)*log(x^2)^2)/x/log(x^2)^2,x, alg
orithm="giac")

[Out]

-x^2 + 4*x - e^x

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maple [A]  time = 0.12, size = 36, normalized size = 1.38

method	result	size
default	\(4 x -{\mathrm e}^{2-\frac {2 \left (\ln \left (x^{2}\right )-2 \ln \relax (x )\right )}{\ln \left (x^{2}\right )}}-x^{2}-{\mathrm e}^{x}\)	\(36\)
risch	\(-x^{2}+4 x -{\mathrm e}^{x}-x^{\frac {4}{i \pi \,\mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i x^{2}\right )+2 \ln \relax (x )}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*ln(x^2)+8*ln(x))*exp(4*ln(x)/ln(x^2))+(-exp(x)*x-2*x^2+4*x)*ln(x^2)^2)/x/ln(x^2)^2,x,method=_RETURNVE
RBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 17, normalized size = 0.65 \begin {gather*} -x^{2} + 4 \, x - 2 \, e^{2} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x^2)+8*log(x))*exp(4*log(x)/log(x^2))+(-exp(x)*x-2*x^2+4*x)*log(x^2)^2)/x/log(x^2)^2,x, alg
orithm="maxima")

[Out]

-x^2 + 4*x - 2*e^2 - e^x

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mupad [B]  time = 4.25, size = 25, normalized size = 0.96 \begin {gather*} 4\,x-{\mathrm {e}}^x-x^2-x^{\frac {4}{\ln \left (x^2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - exp(x) - x^2 - x^(4/log(x^2))

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sympy [A]  time = 0.25, size = 8, normalized size = 0.31 \begin {gather*} - x^{2} + 4 x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + 4*x - exp(x)

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