3.26.34 \(\int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx\) [2534]

3.26.34.1 Optimal result

Integrand size = 22, antiderivative size = 13 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {1}{e^{16}+\frac {x^2}{64}} \]

output

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

3.26.34.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {64}{64 e^{16}+x^2} \]

input

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

output

64/(64*E^16 + x^2)
 

3.26.34.3 Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {27, 1380, 241}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1380

\(\displaystyle -128 \int \frac {x}{\left (x^2+64 e^{16}\right )^2}dx\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {64}{x^2+64 e^{16}}\)

input

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

output

64/(64*E^16 + x^2)
 

3.26.34.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 1380

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

3.26.34.4 Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 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	\(\frac {64}{x^{2}+64 \,{\mathrm e}^{16}}\)	\(13\)
parallelrisch	\(\frac {64}{x^{2}+64 \,{\mathrm e}^{16}}\)	\(13\)

input

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

output

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

3.26.34.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {64}{x^{2} + 64 \, e^{16}} \]

input

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

output

64/(x^2 + 64*e^16)
 

3.26.34.6 Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {128}{2 x^{2} + 128 e^{16}} \]

input

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

output

128/(2*x**2 + 128*exp(16))
 

3.26.34.7 Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {64}{x^{2} + 64 \, e^{16}} \]

input

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

output

64/(x^2 + 64*e^16)
 

3.26.34.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {64}{x^{2} + 64 \, e^{16}} \]

input

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

output

64/(x^2 + 64*e^16)
 

3.26.34.9 Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int -\frac {128 x}{4096 e^{32}+128 e^{16} x^2+x^4} \, dx=\frac {64}{x^2+64\,{\mathrm {e}}^{16}} \]

input

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

output

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