\(\int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx\) [7960]

Optimal result

Integrand size = 17, antiderivative size = 7 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{e^5+x} \]

[Out]

1/(exp(5)+x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 32} \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{x+e^5} \]

[In]

Int[-(E^10 + 2*E^5*x + x^2)^(-1),x]

[Out]

(E^5 + x)^(-1)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{e^5+x} \]

[In]

Integrate[-(E^10 + 2*E^5*x + x^2)^(-1),x]

[Out]

(E^5 + x)^(-1)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {1}{{\mathrm e}^{5}+x}\)	\(7\)
norman	\(\frac {1}{{\mathrm e}^{5}+x}\)	\(7\)
risch	\(\frac {1}{{\mathrm e}^{5}+x}\)	\(7\)
parallelrisch	\(\frac {1}{{\mathrm e}^{5}+x}\)	\(7\)
meijerg	\(-\frac {{\mathrm e}^{-10} x}{1+x \,{\mathrm e}^{-5}}\)	\(14\)

[In]

int(-1/(exp(5)^2+2*x*exp(5)+x^2),x,method=_RETURNVERBOSE)

[Out]

1/(exp(5)+x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{x + e^{5}} \]

[In]

integrate(-1/(exp(5)^2+2*x*exp(5)+x^2),x, algorithm="fricas")

[Out]

1/(x + e^5)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{x + e^{5}} \]

[In]

integrate(-1/(exp(5)**2+2*x*exp(5)+x**2),x)

[Out]

1/(x + exp(5))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{x + e^{5}} \]

[In]

integrate(-1/(exp(5)^2+2*x*exp(5)+x^2),x, algorithm="maxima")

[Out]

1/(x + e^5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{x + e^{5}} \]

[In]

integrate(-1/(exp(5)^2+2*x*exp(5)+x^2),x, algorithm="giac")

[Out]

1/(x + e^5)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86 \[ \int -\frac {1}{e^{10}+2 e^5 x+x^2} \, dx=\frac {1}{x+{\mathrm {e}}^5} \]

[In]

int(-1/(exp(10) + 2*x*exp(5) + x^2),x)

[Out]

1/(x + exp(5))