3.54.12 \(\int \frac {x^3+4 \log (3)}{x^3 \log (3)} \, dx\)

Optimal. Leaf size=18 \[ 1-e^{10}-\frac {2}{x^2}+\frac {x}{\log (3)} \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 14} \begin {gather*} \frac {x}{\log (3)}-\frac {\log (81)}{2 x^2 \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3 + 4*Log[3])/(x^3*Log[3]),x]

[Out]

x/Log[3] - Log[81]/(2*x^2*Log[3])

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {x^3+4 \log (3)}{x^3} \, dx}{\log (3)}\\ &=\frac {\int \left (1+\frac {\log (81)}{x^3}\right ) \, dx}{\log (3)}\\ &=\frac {x}{\log (3)}-\frac {\log (81)}{2 x^2 \log (3)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} \frac {x-\frac {\log (81)}{2 x^2}}{\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3 + 4*Log[3])/(x^3*Log[3]),x]

[Out]

(x - Log[81]/(2*x^2))/Log[3]

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fricas [A]  time = 0.83, size = 16, normalized size = 0.89 \begin {gather*} \frac {x^{3} - 2 \, \log \relax (3)}{x^{2} \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(3)+x^3)/x^3/log(3),x, algorithm="fricas")

[Out]

(x^3 - 2*log(3))/(x^2*log(3))

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giac [A]  time = 0.16, size = 14, normalized size = 0.78 \begin {gather*} \frac {x - \frac {2 \, \log \relax (3)}{x^{2}}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(3)+x^3)/x^3/log(3),x, algorithm="giac")

[Out]

(x - 2*log(3)/x^2)/log(3)

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maple [A]  time = 0.05, size = 13, normalized size = 0.72

method	result	size
risch	\(\frac {x}{\ln \relax (3)}-\frac {2}{x^{2}}\)	\(13\)
default	\(\frac {x -\frac {2 \ln \relax (3)}{x^{2}}}{\ln \relax (3)}\)	\(15\)
norman	\(\frac {-2+\frac {x^{3}}{\ln \relax (3)}}{x^{2}}\)	\(15\)
gosper	\(-\frac {-x^{3}+2 \ln \relax (3)}{x^{2} \ln \relax (3)}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(3)+x^3)/x^3/ln(3),x,method=_RETURNVERBOSE)

[Out]

x/ln(3)-2/x^2

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maxima [A]  time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} \frac {x - \frac {2 \, \log \relax (3)}{x^{2}}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(3)+x^3)/x^3/log(3),x, algorithm="maxima")

[Out]

(x - 2*log(3)/x^2)/log(3)

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mupad [B]  time = 0.03, size = 12, normalized size = 0.67 \begin {gather*} \frac {x}{\ln \relax (3)}-\frac {2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(3) + x^3)/(x^3*log(3)),x)

[Out]

x/log(3) - 2/x^2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.67 \begin {gather*} \frac {x - \frac {2 \log {\relax (3 )}}{x^{2}}}{\log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(3)+x**3)/x**3/ln(3),x)

[Out]

(x - 2*log(3)/x**2)/log(3)

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