3.81.88 \(\int \frac {1-6 x}{x \log (e^{\frac {3-6 e^{15/2} x}{e^{15/2}}} x) \log (\log (e^{\frac {3-6 e^{15/2} x}{e^{15/2}}} x))} \, dx\)

Optimal. Leaf size=20 \[ 5+\log \left (\log \left (\log \left (e^{3 \left (\frac {1}{e^{15/2}}-2 x\right )} x\right )\right )\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.20, number of steps used = 1, number of rules used = 1, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6684} \begin {gather*} \log \left (\log \left (\log \left (e^{\frac {3 \left (1-2 e^{15/2} x\right )}{e^{15/2}}} x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 6*x)/(x*Log[E^((3 - 6*E^(15/2)*x)/E^(15/2))*x]*Log[Log[E^((3 - 6*E^(15/2)*x)/E^(15/2))*x]]),x]

[Out]

Log[Log[Log[E^((3*(1 - 2*E^(15/2)*x))/E^(15/2))*x]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (\log \left (e^{\frac {3 \left (1-2 e^{15/2} x\right )}{e^{15/2}}} x\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 18, normalized size = 0.90 \begin {gather*} \log \left (\log \left (\log \left (e^{\frac {3}{e^{15/2}}-6 x} x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 6*x)/(x*Log[E^((3 - 6*E^(15/2)*x)/E^(15/2))*x]*Log[Log[E^((3 - 6*E^(15/2)*x)/E^(15/2))*x]]),x]

[Out]

Log[Log[Log[E^(3/E^(15/2) - 6*x)*x]]]

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fricas [A]  time = 0.94, size = 17, normalized size = 0.85 \begin {gather*} \log \left (\log \left (\log \left (x e^{\left (-3 \, {\left (2 \, x e^{\frac {15}{2}} - 1\right )} e^{\left (-\frac {15}{2}\right )}\right )}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-6*x)/x/log(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))/log(log(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))),x,
algorithm="fricas")

[Out]

log(log(log(x*e^(-3*(2*x*e^(15/2) - 1)*e^(-15/2)))))

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giac [A]  time = 0.14, size = 12, normalized size = 0.60 \begin {gather*} \log \left (\log \left (-6 \, x + 3 \, e^{\left (-\frac {15}{2}\right )} + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-6*x)/x/log(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))/log(log(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))),x,
algorithm="giac")

[Out]

log(log(-6*x + 3*e^(-15/2) + log(x)))

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maple [C]  time = 0.13, size = 101, normalized size = 5.05

method	result	size
risch	\(\ln \left (\ln \left (\ln \relax (x )+\ln \left ({\mathrm e}^{-3 \left (2 x \,{\mathrm e}^{\frac {15}{2}}-1\right ) {\mathrm e}^{-\frac {15}{2}}}\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-3 \left (2 x \,{\mathrm e}^{\frac {15}{2}}-1\right ) {\mathrm e}^{-\frac {15}{2}}}\right ) \left (-\mathrm {csgn}\left (i x \,{\mathrm e}^{-3 \left (2 x \,{\mathrm e}^{\frac {15}{2}}-1\right ) {\mathrm e}^{-\frac {15}{2}}}\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \,{\mathrm e}^{-3 \left (2 x \,{\mathrm e}^{\frac {15}{2}}-1\right ) {\mathrm e}^{-\frac {15}{2}}}\right )+\mathrm {csgn}\left (i {\mathrm e}^{-3 \left (2 x \,{\mathrm e}^{\frac {15}{2}}-1\right ) {\mathrm e}^{-\frac {15}{2}}}\right )\right )}{2}\right )\right )\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-6*x)/x/ln(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))/ln(ln(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))),x,method=_RE
TURNVERBOSE)

[Out]

ln(ln(ln(x)+ln(exp(-3*(2*x*exp(15/2)-1)*exp(-15/2)))-1/2*I*Pi*csgn(I*x*exp(-3*(2*x*exp(15/2)-1)*exp(-15/2)))*(
-csgn(I*x*exp(-3*(2*x*exp(15/2)-1)*exp(-15/2)))+csgn(I*x))*(-csgn(I*x*exp(-3*(2*x*exp(15/2)-1)*exp(-15/2)))+cs
gn(I*exp(-3*(2*x*exp(15/2)-1)*exp(-15/2))))))

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maxima [A]  time = 0.40, size = 25, normalized size = 1.25 \begin {gather*} e^{\left (-\frac {1}{2}\right )} \log \left (\log \left (-{\left (6 \, x e^{7} - e^{7} \log \relax (x)\right )} e^{\frac {1}{2}} + 3\right ) - \frac {15}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-6*x)/x/log(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))/log(log(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))),x,
algorithm="maxima")

[Out]

e^(-1/2)*log(log(-(6*x*e^7 - e^7*log(x))*e^(1/2) + 3) - 15/2)

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mupad [B]  time = 8.40, size = 12, normalized size = 0.60 \begin {gather*} \ln \left (\ln \left (3\,{\mathrm {e}}^{-\frac {15}{2}}-6\,x+\ln \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x - 1)/(x*log(x*exp(-exp(-15/2)*(6*x*exp(15/2) - 3)))*log(log(x*exp(-exp(-15/2)*(6*x*exp(15/2) - 3))))
),x)

[Out]

log(log(3*exp(-15/2) - 6*x + log(x)))

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sympy [A]  time = 0.28, size = 22, normalized size = 1.10 \begin {gather*} \log {\left (\log {\left (\log {\left (x e^{\frac {- 6 x e^{\frac {15}{2}} + 3}{e^{\frac {15}{2}}}} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-6*x)/x/ln(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))/ln(ln(x*exp((-6*x*exp(15/2)+3)/exp(15/2)))),x)

[Out]

log(log(log(x*exp((-6*x*exp(15/2) + 3)*exp(-15/2)))))

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