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

Optimal. Leaf size=7 \[ \frac {1}{e^5+x} \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 32} \begin {gather*} \frac {1}{x+e^5} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]  time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} \frac {1}{e^5+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^5 + x)^(-1)

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fricas [A]  time = 0.68, size = 6, normalized size = 0.86 \begin {gather*} \frac {1}{x + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + e^5)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -1/2/sqrt(-exp(10)+exp(5)^2)*ln(abs(2*sa
geVARx+2*exp(5)-2*sqrt(-exp(10)+exp(5)^2))/abs(2*sageVARx+2*exp(5)+2*sqrt(-exp(10)+exp(5)^2)))

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maple [A]  time = 0.32, size = 7, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(exp(5)+x)

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maxima [A]  time = 0.39, size = 6, normalized size = 0.86 \begin {gather*} \frac {1}{x + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + e^5)

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mupad [B]  time = 5.53, size = 6, normalized size = 0.86 \begin {gather*} \frac {1}{x+{\mathrm {e}}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + exp(5))

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sympy [A]  time = 0.09, size = 5, normalized size = 0.71 \begin {gather*} \frac {1}{x + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + exp(5))

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