\(\int \frac {6^{\frac {-1+15 x}{x^4}} (-x^4+(4-45 x) \log (6))}{x^6} \, dx\) [9133]

Optimal result

Integrand size = 29, antiderivative size = 15 \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {6^{\frac {-1+15 x}{x^4}}}{x} \]

[Out]

exp((15*x-1)*ln(6)/x^4)/x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(15)=30\).

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2326} \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {6^{-\frac {1-15 x}{x^4}} (4-45 x)}{\left (\frac {4 (1-15 x)}{x^5}+\frac {15}{x^4}\right ) x^6} \]

[In]

Int[(6^((-1 + 15*x)/x^4)*(-x^4 + (4 - 45*x)*Log[6]))/x^6,x]

[Out]

(4 - 45*x)/(6^((1 - 15*x)/x^4)*((4*(1 - 15*x))/x^5 + 15/x^4)*x^6)

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {6^{-\frac {1-15 x}{x^4}} (4-45 x)}{\left (\frac {4 (1-15 x)}{x^5}+\frac {15}{x^4}\right ) x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {6^{\frac {-1+15 x}{x^4}}}{x} \]

[In]

Integrate[(6^((-1 + 15*x)/x^4)*(-x^4 + (4 - 45*x)*Log[6]))/x^6,x]

[Out]

6^((-1 + 15*x)/x^4)/x

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {\left (15 x -1\right ) \ln \left (6\right )}{x^{4}}}}{x}\)	\(17\)
norman	\(\frac {{\mathrm e}^{\frac {\left (15 x -1\right ) \ln \left (6\right )}{x^{4}}}}{x}\)	\(17\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {\left (15 x -1\right ) \ln \left (6\right )}{x^{4}}}}{x}\)	\(17\)
risch	\(\frac {{\mathrm e}^{\frac {\left (15 x -1\right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )}{x^{4}}}}{x}\)	\(20\)

[In]

int(((-45*x+4)*ln(6)-x^4)*exp((15*x-1)*ln(6)/x^4)/x^6,x,method=_RETURNVERBOSE)

[Out]

exp((15*x-1)*ln(6)/x^4)/x

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {6^{\frac {15 \, x - 1}{x^{4}}}}{x} \]

[In]

integrate(((-45*x+4)*log(6)-x^4)*exp((15*x-1)*log(6)/x^4)/x^6,x, algorithm="fricas")

[Out]

6^((15*x - 1)/x^4)/x

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {e^{\frac {\left (15 x - 1\right ) \log {\left (6 \right )}}{x^{4}}}}{x} \]

[In]

integrate(((-45*x+4)*ln(6)-x**4)*exp((15*x-1)*ln(6)/x**4)/x**6,x)

[Out]

exp((15*x - 1)*log(6)/x**4)/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).

Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {e^{\left (\frac {15 \, \log \left (3\right )}{x^{3}} + \frac {15 \, \log \left (2\right )}{x^{3}} - \frac {\log \left (3\right )}{x^{4}} - \frac {\log \left (2\right )}{x^{4}}\right )}}{x} \]

[In]

integrate(((-45*x+4)*log(6)-x^4)*exp((15*x-1)*log(6)/x^4)/x^6,x, algorithm="maxima")

[Out]

e^(15*log(3)/x^3 + 15*log(2)/x^3 - log(3)/x^4 - log(2)/x^4)/x

Giac [F]

\[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\int { -\frac {{\left (x^{4} + {\left (45 \, x - 4\right )} \log \left (6\right )\right )} 6^{\frac {15 \, x - 1}{x^{4}}}}{x^{6}} \,d x } \]

[In]

integrate(((-45*x+4)*log(6)-x^4)*exp((15*x-1)*log(6)/x^4)/x^6,x, algorithm="giac")

[Out]

integrate(-(x^4 + (45*x - 4)*log(6))*6^((15*x - 1)/x^4)/x^6, x)

Mupad [B] (verification not implemented)

Time = 14.51 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {6^{\frac {-1+15 x}{x^4}} \left (-x^4+(4-45 x) \log (6)\right )}{x^6} \, dx=\frac {6^{\frac {15\,x-1}{x^4}}}{x} \]

[In]

int(-(exp((log(6)*(15*x - 1))/x^4)*(log(6)*(45*x - 4) + x^4))/x^6,x)

[Out]

6^((15*x - 1)/x^4)/x