3.73.64 \(\int \frac {900 e^{\frac {450}{-1+2 x}}}{1-4 x+4 x^2} \, dx\)

Optimal. Leaf size=15 \[ 2-e^{\frac {450}{-1+2 x}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 27, 2209} \begin {gather*} -e^{-\frac {450}{1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(900*E^(450/(-1 + 2*x)))/(1 - 4*x + 4*x^2),x]

[Out]

-E^(-450/(1 - 2*x))

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2209

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=900 \int \frac {e^{\frac {450}{-1+2 x}}}{1-4 x+4 x^2} \, dx\\ &=900 \int \frac {e^{\frac {450}{-1+2 x}}}{(-1+2 x)^2} \, dx\\ &=-e^{-\frac {450}{1-2 x}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 13, normalized size = 0.87 \begin {gather*} -e^{\frac {450}{-1+2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(900*E^(450/(-1 + 2*x)))/(1 - 4*x + 4*x^2),x]

[Out]

-E^(450/(-1 + 2*x))

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 12, normalized size = 0.80 \begin {gather*} -e^{\left (\frac {450}{2 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*exp(450/(2*x-1))/(4*x^2-4*x+1),x, algorithm="fricas")

[Out]

-e^(450/(2*x - 1))

________________________________________________________________________________________

giac [A]  time = 0.20, size = 12, normalized size = 0.80 \begin {gather*} -e^{\left (\frac {450}{2 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*exp(450/(2*x-1))/(4*x^2-4*x+1),x, algorithm="giac")

[Out]

-e^(450/(2*x - 1))

________________________________________________________________________________________

maple [A]  time = 0.16, size = 13, normalized size = 0.87

method	result	size
gosper	\(-{\mathrm e}^{\frac {450}{2 x -1}}\)	\(13\)
derivativedivides	\(-{\mathrm e}^{\frac {450}{2 x -1}}\)	\(13\)
default	\(-{\mathrm e}^{\frac {450}{2 x -1}}\)	\(13\)
risch	\(-{\mathrm e}^{\frac {450}{2 x -1}}\)	\(13\)
norman	\(\frac {-2 x \,{\mathrm e}^{\frac {450}{2 x -1}}+{\mathrm e}^{\frac {450}{2 x -1}}}{2 x -1}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(900*exp(450/(2*x-1))/(4*x^2-4*x+1),x,method=_RETURNVERBOSE)

[Out]

-exp(450/(2*x-1))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 12, normalized size = 0.80 \begin {gather*} -e^{\left (\frac {450}{2 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*exp(450/(2*x-1))/(4*x^2-4*x+1),x, algorithm="maxima")

[Out]

-e^(450/(2*x - 1))

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 12, normalized size = 0.80 \begin {gather*} -{\mathrm {e}}^{\frac {450}{2\,x-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((900*exp(450/(2*x - 1)))/(4*x^2 - 4*x + 1),x)

[Out]

-exp(450/(2*x - 1))

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 8, normalized size = 0.53 \begin {gather*} - e^{\frac {450}{2 x - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*exp(450/(2*x-1))/(4*x**2-4*x+1),x)

[Out]

-exp(450/(2*x - 1))

________________________________________________________________________________________