3.91.45 \(\int \frac {6 x^2-x^3-48 \log (3)}{x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac {-6+\frac {16 \log (3)}{x^2}-x \log (x)}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {14} \begin {gather*} \frac {16 \log (3)}{x^3}-\frac {6}{x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6*x^2 - x^3 - 48*Log[3])/x^4,x]

[Out]

-6/x + (16*Log[3])/x^3 - Log[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} &=\int \left (\frac {6}{x^2}-\frac {1}{x}-\frac {48 \log (3)}{x^4}\right ) \, dx\\ &=-\frac {6}{x}+\frac {16 \log (3)}{x^3}-\log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} -\frac {6}{x}+\frac {16 \log (3)}{x^3}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^2 - x^3 - 48*Log[3])/x^4,x]

[Out]

-6/x + (16*Log[3])/x^3 - Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 21, normalized size = 1.17 \begin {gather*} -\frac {x^{3} \log \relax (x) + 6 \, x^{2} - 16 \, \log \relax (3)}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*log(3)-x^3+6*x^2)/x^4,x, algorithm="fricas")

[Out]

-(x^3*log(x) + 6*x^2 - 16*log(3))/x^3

________________________________________________________________________________________

giac [A]  time = 0.13, size = 21, normalized size = 1.17 \begin {gather*} -\frac {2 \, {\left (3 \, x^{2} - 8 \, \log \relax (3)\right )}}{x^{3}} - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*log(3)-x^3+6*x^2)/x^4,x, algorithm="giac")

[Out]

-2*(3*x^2 - 8*log(3))/x^3 - log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 18, normalized size = 1.00

method	result	size
default	\(-\ln \relax (x )-\frac {6}{x}+\frac {16 \ln \relax (3)}{x^{3}}\)	\(18\)
norman	\(\frac {-6 x^{2}+16 \ln \relax (3)}{x^{3}}-\ln \relax (x )\)	\(20\)
risch	\(\frac {-6 x^{2}+16 \ln \relax (3)}{x^{3}}-\ln \relax (x )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-48*ln(3)-x^3+6*x^2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-ln(x)-6/x+16*ln(3)/x^3

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 20, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (3 \, x^{2} - 8 \, \log \relax (3)\right )}}{x^{3}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*log(3)-x^3+6*x^2)/x^4,x, algorithm="maxima")

[Out]

-2*(3*x^2 - 8*log(3))/x^3 - log(x)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 19, normalized size = 1.06 \begin {gather*} \frac {16\,\ln \relax (3)-6\,x^2}{x^3}-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*log(3) - 6*x^2 + x^3)/x^4,x)

[Out]

(16*log(3) - 6*x^2)/x^3 - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 17, normalized size = 0.94 \begin {gather*} - \log {\relax (x )} - \frac {6 x^{2} - 16 \log {\relax (3 )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-48*ln(3)-x**3+6*x**2)/x**4,x)

[Out]

-log(x) - (6*x**2 - 16*log(3))/x**3

________________________________________________________________________________________