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

Optimal. Leaf size=15 \[ \frac {6^{\frac {-1+15 x}{x^4}}}{x} \]

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Rubi [B]  time = 0.07, antiderivative size = 39, normalized size of antiderivative = 2.60, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2288} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[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 2288

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {6^{\frac {-1+15 x}{x^4}}}{x} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 1.26, size = 15, normalized size = 1.00 \begin {gather*} \frac {6^{\frac {15 \, x - 1}{x^{4}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{4} + {\left (45 \, x - 4\right )} \log \relax (6)\right )} 6^{\frac {15 \, x - 1}{x^{4}}}}{x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.12, size = 17, normalized size = 1.13

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [B]  time = 0.50, size = 34, normalized size = 2.27 \begin {gather*} \frac {e^{\left (\frac {15 \, \log \relax (3)}{x^{3}} + \frac {15 \, \log \relax (2)}{x^{3}} - \frac {\log \relax (3)}{x^{4}} - \frac {\log \relax (2)}{x^{4}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 6.73, size = 15, normalized size = 1.00 \begin {gather*} \frac {6^{\frac {15\,x-1}{x^4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.20, size = 14, normalized size = 0.93 \begin {gather*} \frac {e^{\frac {\left (15 x - 1\right ) \log {\relax (6 )}}{x^{4}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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