\(\int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx\) [6701]

Optimal result

Integrand size = 39, antiderivative size = 22 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{\frac {1}{1+6 x+\frac {1}{3} (1-x-\log (4))}} \]

[Out]

exp(1/(17/3*x+4/3-2/3*ln(2)))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 6820, 2240} \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{\frac {3}{17 x+4-\log (4)}} \]

[In]

Int[-51/(E^(3/(-4 - 17*x + Log[4]))*(16 + 136*x + 289*x^2 + (-8 - 34*x)*Log[4] + Log[4]^2)),x]

[Out]

E^(3/(4 + 17*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 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = -\left (51 \int \frac {e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx\right ) \\ & = -\left (51 \int \frac {e^{\frac {3}{4+17 x-\log (4)}}}{(4+17 x-\log (4))^2} \, dx\right ) \\ & = e^{\frac {3}{4+17 x-\log (4)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{\frac {3}{4+17 x-\log (4)}} \]

[In]

Integrate[-51/(E^(3/(-4 - 17*x + Log[4]))*(16 + 136*x + 289*x^2 + (-8 - 34*x)*Log[4] + Log[4]^2)),x]

[Out]

E^(3/(4 + 17*x - Log[4]))

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68

method	result	size
gosper	\({\mathrm e}^{-\frac {3}{2 \ln \left (2\right )-17 x -4}}\)	\(15\)
derivativedivides	\({\mathrm e}^{\frac {3}{-2 \ln \left (2\right )+17 x +4}}\)	\(15\)
default	\({\mathrm e}^{\frac {3}{-2 \ln \left (2\right )+17 x +4}}\)	\(15\)
risch	\({\mathrm e}^{-\frac {3}{2 \ln \left (2\right )-17 x -4}}\)	\(15\)
parallelrisch	\({\mathrm e}^{-\frac {3}{2 \ln \left (2\right )-17 x -4}}\)	\(15\)
norman	\(\frac {\left (2 \ln \left (2\right )-4\right ) {\mathrm e}^{-\frac {3}{2 \ln \left (2\right )-17 x -4}}-17 x \,{\mathrm e}^{-\frac {3}{2 \ln \left (2\right )-17 x -4}}}{2 \ln \left (2\right )-17 x -4}\)	\(52\)

[In]

int(-51*exp(-3/(2*ln(2)-17*x-4))/(4*ln(2)^2+2*(-34*x-8)*ln(2)+289*x^2+136*x+16),x,method=_RETURNVERBOSE)

[Out]

exp(-3/(2*ln(2)-17*x-4))

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{\left (\frac {3}{17 \, x - 2 \, \log \left (2\right ) + 4}\right )} \]

[In]

integrate(-51*exp(-3/(2*log(2)-17*x-4))/(4*log(2)^2+2*(-34*x-8)*log(2)+289*x^2+136*x+16),x, algorithm="fricas"
)

[Out]

e^(3/(17*x - 2*log(2) + 4))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{- \frac {3}{- 17 x - 4 + 2 \log {\left (2 \right )}}} \]

[In]

integrate(-51*exp(-3/(2*ln(2)-17*x-4))/(4*ln(2)**2+2*(-34*x-8)*ln(2)+289*x**2+136*x+16),x)

[Out]

exp(-3/(-17*x - 4 + 2*log(2)))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{\left (\frac {3}{17 \, x - 2 \, \log \left (2\right ) + 4}\right )} \]

[In]

integrate(-51*exp(-3/(2*log(2)-17*x-4))/(4*log(2)^2+2*(-34*x-8)*log(2)+289*x^2+136*x+16),x, algorithm="maxima"
)

[Out]

e^(3/(17*x - 2*log(2) + 4))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx=e^{\left (\frac {3}{17 \, x - 2 \, \log \left (2\right ) + 4}\right )} \]

[In]

integrate(-51*exp(-3/(2*log(2)-17*x-4))/(4*log(2)^2+2*(-34*x-8)*log(2)+289*x^2+136*x+16),x, algorithm="giac")

[Out]

e^(3/(17*x - 2*log(2) + 4))

Mupad [B] (verification not implemented)

Time = 12.40 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int -\frac {51 e^{-\frac {3}{-4-17 x+\log (4)}}}{16+136 x+289 x^2+(-8-34 x) \log (4)+\log ^2(4)} \, dx={\mathrm {e}}^{\frac {3}{17\,x-\ln \left (4\right )+4}} \]

[In]

int(-(51*exp(3/(17*x - 2*log(2) + 4)))/(136*x - 2*log(2)*(34*x + 8) + 4*log(2)^2 + 289*x^2 + 16),x)

[Out]

exp(3/(17*x - log(4) + 4))