3.92.15 \(\int \frac {8-4 x+3 x^3+x^3 \log (3 x^3)}{x^3 \log (4)} \, dx\)

Optimal. Leaf size=22 \[ \frac {x \left (\frac {-4+4 x}{x^3}+\log \left (3 x^3\right )\right )}{\log (4)} \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 14, 2295} \begin {gather*} \frac {x \log \left (3 x^3\right )}{\log (4)}-\frac {4}{x^2 \log (4)}+\frac {4}{x \log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.09 \begin {gather*} \frac {-\frac {4}{x^2}+\frac {4}{x}+x \log \left (3 x^3\right )}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4/x^2 + 4/x + x*Log[3*x^3])/Log[4]

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fricas [A]  time = 0.50, size = 24, normalized size = 1.09 \begin {gather*} \frac {x^{3} \log \left (3 \, x^{3}\right ) + 4 \, x - 4}{2 \, x^{2} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 23, normalized size = 1.05 \begin {gather*} \frac {x \log \left (3 \, x^{3}\right ) + \frac {4 \, {\left (x - 1\right )}}{x^{2}}}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 27, normalized size = 1.23

method	result	size
risch	\(\frac {\ln \left (3 x^{3}\right ) x}{2 \ln \relax (2)}+\frac {2 x -2}{x^{2} \ln \relax (2)}\)	\(27\)
default	\(\frac {\frac {4}{x}-\frac {4}{x^{2}}+x \ln \relax (3)+x \ln \left (x^{3}\right )}{2 \ln \relax (2)}\)	\(28\)
norman	\(\frac {-\frac {2}{\ln \relax (2)}+\frac {2 x}{\ln \relax (2)}+\frac {x^{3} \ln \left (3 x^{3}\right )}{2 \ln \relax (2)}}{x^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.37, size = 25, normalized size = 1.14 \begin {gather*} \frac {x \log \left (3 \, x^{3}\right ) + \frac {4}{x} - \frac {4}{x^{2}}}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.94, size = 28, normalized size = 1.27 \begin {gather*} \frac {4\,x^2-4\,x+x^4\,\ln \left (3\,x^3\right )}{2\,x^3\,\ln \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3)/2 - 2*x + (x^3*log(3*x^3))/2 + 4)/(x^3*log(2)),x)

[Out]

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

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sympy [A]  time = 0.13, size = 24, normalized size = 1.09 \begin {gather*} \frac {x \log {\left (3 x^{3} \right )}}{2 \log {\relax (2 )}} - \frac {2 - 2 x}{x^{2} \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(3*x**3)/(2*log(2)) - (2 - 2*x)/(x**2*log(2))

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