3.41.5 $\int \frac{4(5+i\pi {)}^2-2e^8{x}^2+4e^{16}{x}^4+(5+i\pi )(-2-8e^8{x}^2)}{(5+i\pi {)}^2-2e^8(5+i\pi )x^2+e^{16}{x}^4}dx$

Optimal. Leaf size=23 $$x(4-\frac{2}{5+i\pi -e^8{x}^2})$$

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 81, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.049, Rules used = {28, 1814, 21, 8} $$\begin{array}{c}4x-\frac{2x}{-e^8{x}^2+i\pi +5}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(4*(5 + I*Pi)^2 - 2*E^8*x^2 + 4*E^16*x^4 + (5 + I*Pi)*(-2 - 8*E^8*x^2))/((5 + I*Pi)^2 - 2*E^8*(5 + I*Pi)*x
^2 + E^16*x^4),x]

[Out]

4*x - (2*x)/(5 + I*Pi - E^8*x^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =e^{16}\int \frac{4(5+i\pi {)}^2-2e^8{x}^2+4e^{16}{x}^4+(5+i\pi )(-2-8e^8{x}^2)}{{(-e^8(5+i\pi )+e^{16}{x}^2)}^2}dx\\ & =-\frac{2x}{5+i\pi -e^8{x}^2}+\frac{e^8\int \frac{8(5i-\pi {)}^2+8e^8(5+i\pi )x^2}{-e^8(5+i\pi )+e^{16}{x}^2}dx}{2(5+i\pi )}\\ & =-\frac{2x}{5+i\pi -e^8{x}^2}+4\int 1dx\\ & =4x-\frac{2x}{5+i\pi -e^8{x}^2}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 $$\begin{array}{c}4x+\frac{2x}{-5-i\pi +e^8{x}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(4*(5 + I*Pi)^2 - 2*E^8*x^2 + 4*E^16*x^4 + (5 + I*Pi)*(-2 - 8*E^8*x^2))/((5 + I*Pi)^2 - 2*E^8*(5 + I
*Pi)*x^2 + E^16*x^4),x]

[Out]

4*x + (2*x)/(-5 - I*Pi + E^8*x^2)

________________________________________________________________________________________

fricas [C]  time = 0.51, size = 31, normalized size = 1.35 $$\begin{array}{c}-\frac{2(2x^3{e}^8-2i\pi x-9x)}{i\pi -x^2{e}^8+5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(-exp(5))^2+(-8*x^2*exp(4)^2-2)*log(-exp(5))+4*x^4*exp(4)^4-2*x^2*exp(4)^2)/(log(-exp(5))^2-2*
x^2*exp(4)^2*log(-exp(5))+x^4*exp(4)^4),x, algorithm="fricas")

[Out]

-2*(2*x^3*e^8 - 2*I*pi*x - 9*x)/(I*pi - x^2*e^8 + 5)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Exception raised: TypeError}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(-exp(5))^2+(-8*x^2*exp(4)^2-2)*log(-exp(5))+4*x^4*exp(4)^4-2*x^2*exp(4)^2)/(log(-exp(5))^2-2*
x^2*exp(4)^2*log(-exp(5))+x^4*exp(4)^4),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument Valueindex.c
c index_m o

________________________________________________________________________________________

maple [C]  time = 0.14, size = 23, normalized size = 1.00

method	result	size
risch	$4x+\frac{2ix}{i{\mathrm{e}}^8{x}^2+\pi -5i}$	$23$
gosper	$\frac{2x(-2x^2{\mathrm{e}}^8+2\ln (-{\mathrm{e}}^5)-1)}{\ln (-{\mathrm{e}}^5)-x^2{\mathrm{e}}^8}$	$39$
default	$4x-\frac{2{\left({\mathrm{e}}^{16}\right)}^2{\mathrm{e}}^{-8}{\mathrm{e}}^{-16}x}{-x^2{\mathrm{e}}^{16}+\ln (-{\mathrm{e}}^5){\mathrm{e}}^8}$	$94$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(-exp(5))^2+(-8*x^2*exp(4)^2-2)*ln(-exp(5))+4*x^4*exp(4)^4-2*x^2*exp(4)^2)/(ln(-exp(5))^2-2*x^2*exp(4
)^2*ln(-exp(5))+x^4*exp(4)^4),x,method=_RETURNVERBOSE)

[Out]

4*x+2*I*x/(I*exp(8)*x^2+Pi-5*I)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 23, normalized size = 1.00 $$\begin{array}{c}4x+\frac{2x}{x^2{e}^8-\log (-e^5)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(-exp(5))^2+(-8*x^2*exp(4)^2-2)*log(-exp(5))+4*x^4*exp(4)^4-2*x^2*exp(4)^2)/(log(-exp(5))^2-2*
x^2*exp(4)^2*log(-exp(5))+x^4*exp(4)^4),x, algorithm="maxima")

[Out]

4*x + 2*x/(x^2*e^8 - log(-e^5))

________________________________________________________________________________________

mupad [B]  time = 0.22, size = 22, normalized size = 0.96 $$\begin{array}{c}4x-\frac{2x}{\ln (-{\mathrm{e}}^5)-x^2{\mathrm{e}}^8}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(-exp(5))*(8*x^2*exp(8) + 2) - 4*log(-exp(5))^2 + 2*x^2*exp(8) - 4*x^4*exp(16))/(log(-exp(5))^2 + x^4
*exp(16) - 2*x^2*log(-exp(5))*exp(8)),x)

[Out]

4*x - (2*x)/(log(-exp(5)) - x^2*exp(8))

________________________________________________________________________________________

sympy [A]  time = 0.40, size = 17, normalized size = 0.74 $$\begin{array}{c}4x+\frac{2x}{x^2{e}^8-5-i\pi }\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(-exp(5))**2+(-8*x**2*exp(4)**2-2)*ln(-exp(5))+4*x**4*exp(4)**4-2*x**2*exp(4)**2)/(ln(-exp(5))*
*2-2*x**2*exp(4)**2*ln(-exp(5))+x**4*exp(4)**4),x)

[Out]

4*x + 2*x/(x**2*exp(8) - 5 - I*pi)

________________________________________________________________________________________