3.31.93 \(\int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} (-6 e^4-x^7)+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+(-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)) \log (3-e^{\frac {e^4}{x^6}}+x+\log (4))} \, dx\)

Optimal. Leaf size=24 \[ 4+\log \left (x-\log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )\right ) \]

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Rubi [A]  time = 0.67, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6, 6741, 6684} \begin {gather*} \log \left (x-\log \left (-e^{\frac {e^4}{x^6}}+x+3+\log (4)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^7 + x^8 + E^(E^4/x^6)*(-6*E^4 - x^7) + x^7*Log[4])/(3*x^8 - E^(E^4/x^6)*x^8 + x^9 + x^8*Log[4] + (-3*
x^7 + E^(E^4/x^6)*x^7 - x^8 - x^7*Log[4])*Log[3 - E^(E^4/x^6) + x + Log[4]]),x]

[Out]

Log[x - Log[3 - E^(E^4/x^6) + x + Log[4]]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 6684

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 (2+\log (4))}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx\\ &=\int \frac {x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 (2+\log (4))}{-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 (3+\log (4))+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx\\ &=\int \frac {-x^8-e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )-x^7 (2+\log (4))}{x^7 \left (e^{\frac {e^4}{x^6}}-x-3 \left (1+\frac {2 \log (2)}{3}\right )\right ) \left (x-\log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )\right )} \, dx\\ &=\log \left (x-\log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 22, normalized size = 0.92 \begin {gather*} \log \left (x-\log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^7 + x^8 + E^(E^4/x^6)*(-6*E^4 - x^7) + x^7*Log[4])/(3*x^8 - E^(E^4/x^6)*x^8 + x^9 + x^8*Log[4]
+ (-3*x^7 + E^(E^4/x^6)*x^7 - x^8 - x^7*Log[4])*Log[3 - E^(E^4/x^6) + x + Log[4]]),x]

[Out]

Log[x - Log[3 - E^(E^4/x^6) + x + Log[4]]]

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fricas [A]  time = 0.82, size = 22, normalized size = 0.92 \begin {gather*} \log \left (-x + \log \left (x - e^{\left (\frac {e^{4}}{x^{6}}\right )} + 2 \, \log \relax (2) + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*exp(exp(4)/x^6)-2*x^7*log(2)-x^8-3*x^
7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)-x^8*exp(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x, algorithm="fricas")

[Out]

log(-x + log(x - e^(e^4/x^6) + 2*log(2) + 3))

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giac [A]  time = 1.21, size = 39, normalized size = 1.62 \begin {gather*} \log \left (x - \log \left ({\left (x e^{4} + 2 \, e^{4} \log \relax (2) + 3 \, e^{4} - e^{\left (\frac {4 \, x^{6} + e^{4}}{x^{6}}\right )}\right )} e^{\left (-4\right )}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*exp(exp(4)/x^6)-2*x^7*log(2)-x^8-3*x^
7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)-x^8*exp(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x, algorithm="giac")

[Out]

log(x - log((x*e^4 + 2*e^4*log(2) + 3*e^4 - e^((4*x^6 + e^4)/x^6))*e^(-4)))

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maple [A]  time = 0.04, size = 23, normalized size = 0.96

method	result	size
risch	\(\ln \left (\ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{6}}}+2 \ln \relax (2)+3+x \right )-x \right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*ln(2)+x^8+2*x^7)/((x^7*exp(exp(4)/x^6)-2*x^7*ln(2)-x^8-3*x^7)*ln(-e
xp(exp(4)/x^6)+2*ln(2)+3+x)-x^8*exp(exp(4)/x^6)+2*x^8*ln(2)+x^9+3*x^8),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-exp(exp(4)/x^6)+2*ln(2)+3+x)-x)

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maxima [A]  time = 0.87, size = 22, normalized size = 0.92 \begin {gather*} \log \left (-x + \log \left (x - e^{\left (\frac {e^{4}}{x^{6}}\right )} + 2 \, \log \relax (2) + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*exp(exp(4)/x^6)-2*x^7*log(2)-x^8-3*x^
7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)-x^8*exp(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x, algorithm="maxima")

[Out]

log(-x + log(x - e^(e^4/x^6) + 2*log(2) + 3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,x^7\,\ln \relax (2)-{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}\,\left (x^7+6\,{\mathrm {e}}^4\right )+2\,x^7+x^8}{2\,x^8\,\ln \relax (2)-x^8\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}-\ln \left (x-{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}+2\,\ln \relax (2)+3\right )\,\left (2\,x^7\,\ln \relax (2)-x^7\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}+3\,x^7+x^8\right )+3\,x^8+x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^7*log(2) - exp(exp(4)/x^6)*(6*exp(4) + x^7) + 2*x^7 + x^8)/(2*x^8*log(2) - x^8*exp(exp(4)/x^6) - log(
x - exp(exp(4)/x^6) + 2*log(2) + 3)*(2*x^7*log(2) - x^7*exp(exp(4)/x^6) + 3*x^7 + x^8) + 3*x^8 + x^9),x)

[Out]

int((2*x^7*log(2) - exp(exp(4)/x^6)*(6*exp(4) + x^7) + 2*x^7 + x^8)/(2*x^8*log(2) - x^8*exp(exp(4)/x^6) - log(
x - exp(exp(4)/x^6) + 2*log(2) + 3)*(2*x^7*log(2) - x^7*exp(exp(4)/x^6) + 3*x^7 + x^8) + 3*x^8 + x^9), x)

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sympy [A]  time = 0.78, size = 20, normalized size = 0.83 \begin {gather*} \log {\left (- x + \log {\left (x - e^{\frac {e^{4}}{x^{6}}} + 2 \log {\relax (2 )} + 3 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(4)-x**7)*exp(exp(4)/x**6)+2*x**7*ln(2)+x**8+2*x**7)/((x**7*exp(exp(4)/x**6)-2*x**7*ln(2)-x*
*8-3*x**7)*ln(-exp(exp(4)/x**6)+2*ln(2)+3+x)-x**8*exp(exp(4)/x**6)+2*x**8*ln(2)+x**9+3*x**8),x)

[Out]

log(-x + log(x - exp(exp(4)/x**6) + 2*log(2) + 3))

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