\(\int \frac {9-27 \log (x)}{40 x^4} \, dx\) [9614]

Optimal result

Integrand size = 13, antiderivative size = 9 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9 \log (x)}{40 x^3} \]

[Out]

9/40*ln(x)/x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 2340} \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9 \log (x)}{40 x^3} \]

[In]

Int[(9 - 27*Log[x])/(40*x^4),x]

[Out]

(9*Log[x])/(40*x^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 2340

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[b*(d*x)^(m + 1)*(Log[c*x^n]/(d
*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{40} \int \frac {9-27 \log (x)}{x^4} \, dx \\ & = \frac {9 \log (x)}{40 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9 \log (x)}{40 x^3} \]

[In]

Integrate[(9 - 27*Log[x])/(40*x^4),x]

[Out]

(9*Log[x])/(40*x^3)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {9 \ln \left (x \right )}{40 x^{3}}\)	\(8\)
norman	\(\frac {9 \ln \left (x \right )}{40 x^{3}}\)	\(8\)
risch	\(\frac {9 \ln \left (x \right )}{40 x^{3}}\)	\(8\)
parallelrisch	\(\frac {9 \ln \left (x \right )}{40 x^{3}}\)	\(8\)
parts	\(\frac {9 \ln \left (x \right )}{40 x^{3}}\)	\(8\)

[In]

int(1/40*(-27*ln(x)+9)/x^4,x,method=_RETURNVERBOSE)

[Out]

9/40*ln(x)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9 \, \log \left (x\right )}{40 \, x^{3}} \]

[In]

integrate(1/40*(-27*log(x)+9)/x^4,x, algorithm="fricas")

[Out]

9/40*log(x)/x^3

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9 \log {\left (x \right )}}{40 x^{3}} \]

[In]

integrate(1/40*(-27*ln(x)+9)/x**4,x)

[Out]

9*log(x)/(40*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (7) = 14\).

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.89 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {3 \, {\left (3 \, \log \left (x\right ) + 1\right )}}{40 \, x^{3}} - \frac {3}{40 \, x^{3}} \]

[In]

integrate(1/40*(-27*log(x)+9)/x^4,x, algorithm="maxima")

[Out]

3/40*(3*log(x) + 1)/x^3 - 3/40/x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9 \, \log \left (x\right )}{40 \, x^{3}} \]

[In]

integrate(1/40*(-27*log(x)+9)/x^4,x, algorithm="giac")

[Out]

9/40*log(x)/x^3

Mupad [B] (verification not implemented)

Time = 15.65 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {9-27 \log (x)}{40 x^4} \, dx=\frac {9\,\ln \left (x\right )}{40\,x^3} \]

[In]

int(-((27*log(x))/40 - 9/40)/x^4,x)

[Out]

(9*log(x))/(40*x^3)