3.74.81 \(\int \frac {3 x \log ^5(x)+e^{\frac {2 (81-81 x^2)}{\log ^4(x)}} (-1944+1944 x^2-972 x^2 \log (x))}{x \log ^5(x)} \, dx\)

Optimal. Leaf size=20 \[ 3 \left (-1+e^{\frac {162 \left (1-x^2\right )}{\log ^4(x)}}+x\right ) \]

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Rubi [F]  time = 1.59, 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 {3 x \log ^5(x)+e^{\frac {2 \left (81-81 x^2\right )}{\log ^4(x)}} \left (-1944+1944 x^2-972 x^2 \log (x)\right )}{x \log ^5(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x*Log[x]^5 + E^((2*(81 - 81*x^2))/Log[x]^4)*(-1944 + 1944*x^2 - 972*x^2*Log[x]))/(x*Log[x]^5),x]

[Out]

3*x - 1944*Defer[Int][1/(E^((162*(-1 + x^2))/Log[x]^4)*x*Log[x]^5), x] + 1944*Defer[Int][x/(E^((162*(-1 + x^2)
)/Log[x]^4)*Log[x]^5), x] - 972*Defer[Int][x/(E^((162*(-1 + x^2))/Log[x]^4)*Log[x]^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3+\frac {972 e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} \left (-2+2 x^2-x^2 \log (x)\right )}{x \log ^5(x)}\right ) \, dx\\ &=3 x+972 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} \left (-2+2 x^2-x^2 \log (x)\right )}{x \log ^5(x)} \, dx\\ &=3 x+972 \int \left (\frac {2 e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} \left (-1+x^2\right )}{x \log ^5(x)}-\frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} x}{\log ^4(x)}\right ) \, dx\\ &=3 x-972 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} x}{\log ^4(x)} \, dx+1944 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} \left (-1+x^2\right )}{x \log ^5(x)} \, dx\\ &=3 x-972 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} x}{\log ^4(x)} \, dx+1944 \int \left (-\frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}}}{x \log ^5(x)}+\frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} x}{\log ^5(x)}\right ) \, dx\\ &=3 x-972 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} x}{\log ^4(x)} \, dx-1944 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}}}{x \log ^5(x)} \, dx+1944 \int \frac {e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}} x}{\log ^5(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 19, normalized size = 0.95 \begin {gather*} 3 e^{-\frac {162 \left (-1+x^2\right )}{\log ^4(x)}}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x*Log[x]^5 + E^((2*(81 - 81*x^2))/Log[x]^4)*(-1944 + 1944*x^2 - 972*x^2*Log[x]))/(x*Log[x]^5),x]

[Out]

3/E^((162*(-1 + x^2))/Log[x]^4) + 3*x

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fricas [A]  time = 0.83, size = 18, normalized size = 0.90 \begin {gather*} 3 \, x + 3 \, e^{\left (-\frac {162 \, {\left (x^{2} - 1\right )}}{\log \relax (x)^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-972*x^2*log(x)+1944*x^2-1944)*exp((-81*x^2+81)/log(x)^4)^2+3*x*log(x)^5)/x/log(x)^5,x, algorithm=
"fricas")

[Out]

3*x + 3*e^(-162*(x^2 - 1)/log(x)^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-972*x^2*log(x)+1944*x^2-1944)*exp((-81*x^2+81)/log(x)^4)^2+3*x*log(x)^5)/x/log(x)^5,x, algorithm=
"giac")

[Out]

undef

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maple [A]  time = 0.05, size = 20, normalized size = 1.00

method	result	size
risch	\(3 \,{\mathrm e}^{-\frac {162 \left (x -1\right ) \left (x +1\right )}{\ln \relax (x )^{4}}}+3 x\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-972*x^2*ln(x)+1944*x^2-1944)*exp((-81*x^2+81)/ln(x)^4)^2+3*x*ln(x)^5)/x/ln(x)^5,x,method=_RETURNVERBOSE
)

[Out]

3*exp(-162*(x-1)*(x+1)/ln(x)^4)+3*x

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maxima [A]  time = 31.35, size = 23, normalized size = 1.15 \begin {gather*} 3 \, x + 3 \, e^{\left (-\frac {162 \, x^{2}}{\log \relax (x)^{4}} + \frac {162}{\log \relax (x)^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-972*x^2*log(x)+1944*x^2-1944)*exp((-81*x^2+81)/log(x)^4)^2+3*x*log(x)^5)/x/log(x)^5,x, algorithm=
"maxima")

[Out]

3*x + 3*e^(-162*x^2/log(x)^4 + 162/log(x)^4)

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mupad [B]  time = 5.41, size = 23, normalized size = 1.15 \begin {gather*} 3\,x+3\,{\mathrm {e}}^{\frac {162}{{\ln \relax (x)}^4}}\,{\mathrm {e}}^{-\frac {162\,x^2}{{\ln \relax (x)}^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x*log(x)^5 - exp(-(2*(81*x^2 - 81))/log(x)^4)*(972*x^2*log(x) - 1944*x^2 + 1944))/(x*log(x)^5),x)

[Out]

3*x + 3*exp(162/log(x)^4)*exp(-(162*x^2)/log(x)^4)

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sympy [A]  time = 0.35, size = 17, normalized size = 0.85 \begin {gather*} 3 x + 3 e^{\frac {2 \left (81 - 81 x^{2}\right )}{\log {\relax (x )}^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-972*x**2*ln(x)+1944*x**2-1944)*exp((-81*x**2+81)/ln(x)**4)**2+3*x*ln(x)**5)/x/ln(x)**5,x)

[Out]

3*x + 3*exp(2*(81 - 81*x**2)/log(x)**4)

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