\(\int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx\) [5088]

Optimal result

Integrand size = 25, antiderivative size = 15 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log \left (21-10 e^5 (2-x+\log (4))\right ) \]

[Out]

ln(21-10*exp(5)*(2-x+2*ln(2)))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 33, 31} \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log \left (21-10 e^5 (-x+2+\log (4))\right ) \]

[In]

Int[(-10*E^5)/(-21 + E^5*(20 - 10*x) + 10*E^5*Log[4]),x]

[Out]

Log[21 - 10*E^5*(2 - x + Log[4])]

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = -\left (\left (10 e^5\right ) \int \frac {1}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx\right ) \\ & = e^5 \text {Subst}\left (\int \frac {1}{-21+e^5 x+10 e^5 \log (4)} \, dx,x,20-10 x\right ) \\ & = \log \left (21-10 e^5 (2-x+\log (4))\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log \left (21+10 e^5 (-2+x-\log (4))\right ) \]

[In]

Integrate[(-10*E^5)/(-21 + E^5*(20 - 10*x) + 10*E^5*Log[4]),x]

[Out]

Log[21 + 10*E^5*(-2 + x - Log[4])]

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27

method	result	size
default	\(\ln \left (20 \,{\mathrm e}^{5} \ln \left (2\right )-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right )\)	\(19\)
norman	\(\ln \left (20 \,{\mathrm e}^{5} \ln \left (2\right )-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right )\)	\(19\)
risch	\(\ln \left (20 \,{\mathrm e}^{5} \ln \left (2\right )-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right )\)	\(19\)
meijerg	\(\ln \left (1-\frac {10 x \,{\mathrm e}^{5}}{20 \,{\mathrm e}^{5} \ln \left (2\right )+20 \,{\mathrm e}^{5}-21}\right )\)	\(23\)
parallelrisch	\(\ln \left (-\frac {\left (20 \,{\mathrm e}^{5} \ln \left (2\right )-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right ) {\mathrm e}^{-5}}{10}\right )\)	\(25\)

[In]

int(-10*exp(5)/(20*exp(5)*ln(2)+(20-10*x)*exp(5)-21),x,method=_RETURNVERBOSE)

[Out]

ln(20*exp(5)*ln(2)-10*x*exp(5)+20*exp(5)-21)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log \left (10 \, {\left (x - 2\right )} e^{5} - 20 \, e^{5} \log \left (2\right ) + 21\right ) \]

[In]

integrate(-10*exp(5)/(20*exp(5)*log(2)+(20-10*x)*exp(5)-21),x, algorithm="fricas")

[Out]

log(10*(x - 2)*e^5 - 20*e^5*log(2) + 21)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log {\left (10 x e^{5} - 20 e^{5} - 20 e^{5} \log {\left (2 \right )} + 21 \right )} \]

[In]

integrate(-10*exp(5)/(20*exp(5)*ln(2)+(20-10*x)*exp(5)-21),x)

[Out]

log(10*x*exp(5) - 20*exp(5) - 20*exp(5)*log(2) + 21)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log \left (10 \, {\left (x - 2\right )} e^{5} - 20 \, e^{5} \log \left (2\right ) + 21\right ) \]

[In]

integrate(-10*exp(5)/(20*exp(5)*log(2)+(20-10*x)*exp(5)-21),x, algorithm="maxima")

[Out]

log(10*(x - 2)*e^5 - 20*e^5*log(2) + 21)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\log \left ({\left | 10 \, {\left (x - 2\right )} e^{5} - 20 \, e^{5} \log \left (2\right ) + 21 \right |}\right ) \]

[In]

integrate(-10*exp(5)/(20*exp(5)*log(2)+(20-10*x)*exp(5)-21),x, algorithm="giac")

[Out]

log(abs(10*(x - 2)*e^5 - 20*e^5*log(2) + 21))

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx=\ln \left (x+\frac {21\,{\mathrm {e}}^{-5}}{10}-\ln \left (4\right )-2\right ) \]

[In]

int((10*exp(5))/(exp(5)*(10*x - 20) - 20*exp(5)*log(2) + 21),x)

[Out]

log(x + (21*exp(-5))/10 - log(4) - 2)