3.62.95 \(\int \frac {3 x^2+24 x^4+e^2 (2-x^2-8 x^4)+(6 x^2-2 e^2 x^2) \log (x)}{2 e^2 x} \, dx\)

Optimal. Leaf size=29 \[ \frac {3 x \left (x-\frac {e^2 x}{3}\right ) \left (x^2+\frac {\log (x)}{2}\right )}{e^2}+\log (x) \]

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 14, 2304} \begin {gather*} \frac {\left (3-e^2\right ) x^4}{e^2}+\frac {\left (3-e^2\right ) x^2 \log (x)}{2 e^2}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*x^2 + 24*x^4 + E^2*(2 - x^2 - 8*x^4) + (6*x^2 - 2*E^2*x^2)*Log[x])/(2*E^2*x),x]

[Out]

((3 - E^2)*x^4)/E^2 + Log[x] + ((3 - E^2)*x^2*Log[x])/(2*E^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {3 x^2+24 x^4+e^2 \left (2-x^2-8 x^4\right )+\left (6 x^2-2 e^2 x^2\right ) \log (x)}{x} \, dx}{2 e^2}\\ &=\frac {\int \left (\frac {2 e^2+\left (3-e^2\right ) x^2+8 \left (3-e^2\right ) x^4}{x}-2 \left (-3+e^2\right ) x \log (x)\right ) \, dx}{2 e^2}\\ &=\frac {\int \frac {2 e^2+\left (3-e^2\right ) x^2+8 \left (3-e^2\right ) x^4}{x} \, dx}{2 e^2}+\frac {\left (3-e^2\right ) \int x \log (x) \, dx}{e^2}\\ &=-\frac {\left (3-e^2\right ) x^2}{4 e^2}+\frac {\left (3-e^2\right ) x^2 \log (x)}{2 e^2}+\frac {\int \left (\frac {2 e^2}{x}+\left (3-e^2\right ) x+8 \left (3-e^2\right ) x^3\right ) \, dx}{2 e^2}\\ &=\frac {\left (3-e^2\right ) x^4}{e^2}+\log (x)+\frac {\left (3-e^2\right ) x^2 \log (x)}{2 e^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 35, normalized size = 1.21 \begin {gather*} -\frac {\left (-3+e^2\right ) x^4}{e^2}+\log (x)+\frac {\left (3-e^2\right ) x^2 \log (x)}{2 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + 24*x^4 + E^2*(2 - x^2 - 8*x^4) + (6*x^2 - 2*E^2*x^2)*Log[x])/(2*E^2*x),x]

[Out]

-(((-3 + E^2)*x^4)/E^2) + Log[x] + ((3 - E^2)*x^2*Log[x])/(2*E^2)

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fricas [A]  time = 0.89, size = 36, normalized size = 1.24 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x^{4} e^{2} - 6 \, x^{4} - {\left (3 \, x^{2} - {\left (x^{2} - 2\right )} e^{2}\right )} \log \relax (x)\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2*exp(2)+6*x^2)*log(x)+(-8*x^4-x^2+2)*exp(2)+24*x^4+3*x^2)/exp(2)/x,x, algorithm="fricas"
)

[Out]

-1/2*(2*x^4*e^2 - 6*x^4 - (3*x^2 - (x^2 - 2)*e^2)*log(x))*e^(-2)

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giac [A]  time = 0.14, size = 38, normalized size = 1.31 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x^{4} e^{2} - 6 \, x^{4} + x^{2} e^{2} \log \relax (x) - 3 \, x^{2} \log \relax (x) - 2 \, e^{2} \log \relax (x)\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2*exp(2)+6*x^2)*log(x)+(-8*x^4-x^2+2)*exp(2)+24*x^4+3*x^2)/exp(2)/x,x, algorithm="giac")

[Out]

-1/2*(2*x^4*e^2 - 6*x^4 + x^2*e^2*log(x) - 3*x^2*log(x) - 2*e^2*log(x))*e^(-2)

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maple [A]  time = 0.08, size = 29, normalized size = 1.00

method	result	size
risch	\(-\frac {\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-2} x^{2} \ln \relax (x )}{2}-x^{4}+3 \,{\mathrm e}^{-2} x^{4}+\ln \relax (x )\)	\(29\)
norman	\(\ln \relax (x )-\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-2} x^{4}-\frac {\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-2} x^{2} \ln \relax (x )}{2}\)	\(32\)
default	\(\frac {{\mathrm e}^{-2} \left (-2 x^{4} {\mathrm e}^{2}-2 \,{\mathrm e}^{2} \left (\frac {x^{2} \ln \relax (x )}{2}-\frac {x^{2}}{4}\right )+6 x^{4}-\frac {x^{2} {\mathrm e}^{2}}{2}+3 x^{2} \ln \relax (x )+2 \,{\mathrm e}^{2} \ln \relax (x )\right )}{2}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-2*x^2*exp(2)+6*x^2)*ln(x)+(-8*x^4-x^2+2)*exp(2)+24*x^4+3*x^2)/exp(2)/x,x,method=_RETURNVERBOSE)

[Out]

-1/2*(exp(2)-3)*exp(-2)*x^2*ln(x)-x^4+3*exp(-2)*x^4+ln(x)

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maxima [B]  time = 0.40, size = 52, normalized size = 1.79 \begin {gather*} -\frac {1}{4} \, {\left (4 \, x^{4} e^{2} - 12 \, x^{4} + x^{2} e^{2} - 6 \, x^{2} \log \relax (x) + {\left (2 \, x^{2} \log \relax (x) - x^{2}\right )} e^{2} - 4 \, e^{2} \log \relax (x)\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2*exp(2)+6*x^2)*log(x)+(-8*x^4-x^2+2)*exp(2)+24*x^4+3*x^2)/exp(2)/x,x, algorithm="maxima"
)

[Out]

-1/4*(4*x^4*e^2 - 12*x^4 + x^2*e^2 - 6*x^2*log(x) + (2*x^2*log(x) - x^2)*e^2 - 4*e^2*log(x))*e^(-2)

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mupad [B]  time = 4.15, size = 26, normalized size = 0.90 \begin {gather*} \ln \relax (x)+x^4\,\left (3\,{\mathrm {e}}^{-2}-1\right )+\frac {x^2\,\ln \relax (x)\,\left (3\,{\mathrm {e}}^{-2}-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2)*((exp(2)*(x^2 + 8*x^4 - 2))/2 + (log(x)*(2*x^2*exp(2) - 6*x^2))/2 - (3*x^2)/2 - 12*x^4))/x,x)

[Out]

log(x) + x^4*(3*exp(-2) - 1) + (x^2*log(x)*(3*exp(-2) - 1))/2

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sympy [A]  time = 0.18, size = 37, normalized size = 1.28 \begin {gather*} \frac {\left (- x^{2} e^{2} + 3 x^{2}\right ) \log {\relax (x )}}{2 e^{2}} + \frac {x^{4} \left (3 - e^{2}\right ) + e^{2} \log {\relax (x )}}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x**2*exp(2)+6*x**2)*ln(x)+(-8*x**4-x**2+2)*exp(2)+24*x**4+3*x**2)/exp(2)/x,x)

[Out]

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

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