\(\int -\frac {84 x}{(20 e^6+21 x^2) \log (20 e^6+21 x^2) \log ^3(\log (20 e^6+21 x^2))} \, dx\) [1726]

Optimal result

Integrand size = 45, antiderivative size = 21 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log ^2\left (\log \left (x^2-20 \left (-e^6-x^2\right )\right )\right )} \]

[Out]

1/ln(ln(20*exp(2)*exp(4)+21*x^2))^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 6818} \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log ^2\left (\log \left (21 x^2+20 e^6\right )\right )} \]

[In]

Int[(-84*x)/((20*E^6 + 21*x^2)*Log[20*E^6 + 21*x^2]*Log[Log[20*E^6 + 21*x^2]]^3),x]

[Out]

Log[Log[20*E^6 + 21*x^2]]^(-2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\left (84 \int \frac {x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx\right ) \\ & = \frac {1}{\log ^2\left (\log \left (20 e^6+21 x^2\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log ^2\left (\log \left (20 e^6+21 x^2\right )\right )} \]

[In]

Integrate[(-84*x)/((20*E^6 + 21*x^2)*Log[20*E^6 + 21*x^2]*Log[Log[20*E^6 + 21*x^2]]^3),x]

[Out]

Log[Log[20*E^6 + 21*x^2]]^(-2)

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {1}{{\ln \left (\ln \left (20 \,{\mathrm e}^{6}+21 x^{2}\right )\right )}^{2}}\)	\(15\)
derivativedivides	\(\frac {1}{{\ln \left (\ln \left (20 \,{\mathrm e}^{2} {\mathrm e}^{4}+21 x^{2}\right )\right )}^{2}}\)	\(17\)
default	\(\frac {1}{{\ln \left (\ln \left (20 \,{\mathrm e}^{2} {\mathrm e}^{4}+21 x^{2}\right )\right )}^{2}}\)	\(17\)
parallelrisch	\(\frac {1}{{\ln \left (\ln \left (20 \,{\mathrm e}^{2} {\mathrm e}^{4}+21 x^{2}\right )\right )}^{2}}\)	\(17\)

[In]

int(-84*x/(20*exp(2)*exp(4)+21*x^2)/ln(20*exp(2)*exp(4)+21*x^2)/ln(ln(20*exp(2)*exp(4)+21*x^2))^3,x,method=_RE
TURNVERBOSE)

[Out]

1/ln(ln(20*exp(6)+21*x^2))^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log \left (\log \left (21 \, x^{2} + 20 \, e^{6}\right )\right )^{2}} \]

[In]

integrate(-84*x/(20*exp(2)*exp(4)+21*x^2)/log(20*exp(2)*exp(4)+21*x^2)/log(log(20*exp(2)*exp(4)+21*x^2))^3,x,
algorithm="fricas")

[Out]

log(log(21*x^2 + 20*e^6))^(-2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log {\left (\log {\left (21 x^{2} + 20 e^{6} \right )} \right )}^{2}} \]

[In]

integrate(-84*x/(20*exp(2)*exp(4)+21*x**2)/ln(20*exp(2)*exp(4)+21*x**2)/ln(ln(20*exp(2)*exp(4)+21*x**2))**3,x)

[Out]

log(log(21*x**2 + 20*exp(6)))**(-2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log \left (\log \left (21 \, x^{2} + 20 \, e^{6}\right )\right )^{2}} \]

[In]

integrate(-84*x/(20*exp(2)*exp(4)+21*x^2)/log(20*exp(2)*exp(4)+21*x^2)/log(log(20*exp(2)*exp(4)+21*x^2))^3,x,
algorithm="maxima")

[Out]

log(log(21*x^2 + 20*e^6))^(-2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{\log \left (\log \left (21 \, x^{2} + 20 \, e^{6}\right )\right )^{2}} \]

[In]

integrate(-84*x/(20*exp(2)*exp(4)+21*x^2)/log(20*exp(2)*exp(4)+21*x^2)/log(log(20*exp(2)*exp(4)+21*x^2))^3,x,
algorithm="giac")

[Out]

log(log(21*x^2 + 20*e^6))^(-2)

Mupad [B] (verification not implemented)

Time = 9.97 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int -\frac {84 x}{\left (20 e^6+21 x^2\right ) \log \left (20 e^6+21 x^2\right ) \log ^3\left (\log \left (20 e^6+21 x^2\right )\right )} \, dx=\frac {1}{{\ln \left (\ln \left (21\,x^2+20\,{\mathrm {e}}^6\right )\right )}^2} \]

[In]

int(-(84*x)/(log(log(20*exp(6) + 21*x^2))^3*log(20*exp(6) + 21*x^2)*(20*exp(6) + 21*x^2)),x)

[Out]

1/log(log(20*exp(6) + 21*x^2))^2