3.85.34 $\int \frac{-5-i\pi -\log (4)}{x^2+e^{5+e^5}{x}^2}dx$

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 29, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.103, Rules used = {6, 12, 30} $$\begin{array}{c}\frac{5+i\pi +\log (4)}{(1+e^{5+e^5})x}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

(5 + I*Pi + Log[4])/((1 + E^(5 + E^5))*x)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 24, normalized size = 1.04 $$\begin{array}{c}\frac{5+i\pi +\log (4)}{(1+e^{5+e^5})x}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

(5 + I*Pi + Log[4])/((1 + E^(5 + E^5))*x)

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 21, normalized size = 0.91 $$\begin{array}{c}\frac{i\pi +2\log (2)+5}{x{e}^{(e^5+5)}+x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*pi + 2*log(2) + 5)/(x*e^(e^5 + 5) + x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 22, normalized size = 0.96 $$\begin{array}{c}\frac{i\pi +2\log (2)+5}{x(e^{(e^5+5)}+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*pi + 2*log(2) + 5)/(x*(e^(e^5 + 5) + 1))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 25, normalized size = 1.09

method	result	size
gosper	$\frac{2\ln (2)+i\pi +5}{{\mathrm{e}}^{\ln (x)+{\mathrm{e}}^5}{\mathrm{e}}^5+x}$	$25$
default	$-\frac{-2\ln (2)-i\pi -5}{({\mathrm{e}}^{{\mathrm{e}}^5+5}+1)x}$	$25$
norman	$\frac{2\ln (2)+i\pi +5}{({\mathrm{e}}^5{\mathrm{e}}^{{\mathrm{e}}^5}+1)x}$	$25$
risch	$\frac{i\pi }{({\mathrm{e}}^{{\mathrm{e}}^5+5}+1)x}+\frac{2\ln (2)}{({\mathrm{e}}^{{\mathrm{e}}^5+5}+1)x}+\frac{5}{({\mathrm{e}}^{{\mathrm{e}}^5+5}+1)x}$	$48$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*ln(2)+I*Pi+5)/(exp(ln(x)+exp(5))*exp(5)+x)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 23, normalized size = 1.00 $$\begin{array}{c}-\frac{-i\pi -2\log (2)-5}{x(e^{(e^5+5)}+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-I*pi - 2*log(2) - 5)/(x*(e^(e^5 + 5) + 1))

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 21, normalized size = 0.91 $$\begin{array}{c}\frac{\ln (4)+5+\mathrm{\Pi }1\mathrm{i}}{x({\mathrm{e}}^{{\mathrm{e}}^5+5}+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(Pi*1i + 2*log(2) + 5)/(x^2 + x*exp(exp(5) + log(x))*exp(5)),x)

[Out]

(Pi*1i + log(4) + 5)/(x*(exp(exp(5) + 5) + 1))

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 24, normalized size = 1.04 $$\begin{array}{c}-\frac{2\log (2)+5+i\pi }{x(-e^5{e}^{e^5}-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(2)-I*pi-5)/(x*exp(5)*exp(ln(x)+exp(5))+x**2),x)

[Out]

-(2*log(2) + 5 + I*pi)/(x*(-exp(5)*exp(exp(5)) - 1))

________________________________________________________________________________________