3.56.34 \(\int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx\)

Optimal. Leaf size=13 \[ \frac {1}{e^{16}+\frac {x^2}{64}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 28, 261} \begin {gather*} \frac {64}{x^2+64 e^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-128*x)/(4096*E^32 + 128*E^16*x^2 + x^4),x]

[Out]

64/(64*E^16 + x^2)

Rule 12

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (128 \int \frac {x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx\right )\\ &=-\left (128 \int \frac {x}{\left (64 e^{16}+x^2\right )^2} \, dx\right )\\ &=\frac {64}{64 e^{16}+x^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {64}{64 e^{16}+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-128*x)/(4096*E^32 + 128*E^16*x^2 + x^4),x]

[Out]

64/(64*E^16 + x^2)

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 12, normalized size = 0.92 \begin {gather*} \frac {64}{x^{2} + 64 \, e^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-128*x/(4096*exp(16)^2+128*x^2*exp(16)+x^4),x, algorithm="fricas")

[Out]

64/(x^2 + 64*e^16)

________________________________________________________________________________________

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(-128*x/(4096*exp(16)^2+128*x^2*exp(16)+x^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -1/2/sqrt(exp(16)^2-exp(32))*ln(sqrt((2*
sageVARx^2+128*exp(16))^2+(-128*sqrt(-exp(16)^2+exp(32)))^2)/sqrt((2*sageVARx^2+128*exp(16))^2+(128*sqrt(-exp(
16)^2+exp(32)))^2))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 11, normalized size = 0.85

method	result	size
risch	\(\frac {1}{{\mathrm e}^{16}+\frac {x^{2}}{64}}\)	\(11\)
gosper	\(\frac {64}{x^{2}+64 \,{\mathrm e}^{16}}\)	\(13\)
norman	\(\frac {64}{x^{2}+64 \,{\mathrm e}^{16}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-128*x/(4096*exp(16)^2+128*x^2*exp(16)+x^4),x,method=_RETURNVERBOSE)

[Out]

1/(exp(16)+1/64*x^2)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 12, normalized size = 0.92 \begin {gather*} \frac {64}{x^{2} + 64 \, e^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-128*x/(4096*exp(16)^2+128*x^2*exp(16)+x^4),x, algorithm="maxima")

[Out]

64/(x^2 + 64*e^16)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 12, normalized size = 0.92 \begin {gather*} \frac {64}{x^2+64\,{\mathrm {e}}^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(128*x)/(4096*exp(32) + 128*x^2*exp(16) + x^4),x)

[Out]

64/(64*exp(16) + x^2)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 10, normalized size = 0.77 \begin {gather*} \frac {128}{2 x^{2} + 128 e^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-128*x/(4096*exp(16)**2+128*x**2*exp(16)+x**4),x)

[Out]

128/(2*x**2 + 128*exp(16))

________________________________________________________________________________________