3.17.79 $\int \frac{-1+e^{2}x+\frac{2(-1+e^{2}x{)}^2}{e^4}}{-1+e^{2}x}dx$

Optimal. Leaf size=22 $$-1-e^5+e^{2(-3+\log (-1+e^{2}x))}+x$$

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.77, number of steps used = 2, number of rules used = 1, integrand size = 31, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.032, Rules used = {1586} $$\begin{array}{c}\frac{x^2}{e^2}+(1-\frac{2}{e^4})x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-1 + E^2*x + (2*(-1 + E^2*x)^2)/E^4)/(-1 + E^2*x),x]

[Out]

(1 - 2/E^4)*x + x^2/E^2

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.68 $$\begin{array}{c}x-\frac{2x}{e^4}+\frac{x^2}{e^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^2*x + (2*(-1 + E^2*x)^2)/E^4)/(-1 + E^2*x),x]

[Out]

x - (2*x)/E^4 + x^2/E^2

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fricas [A]  time = 0.82, size = 17, normalized size = 0.77 $$\begin{array}{c}(x^2{e}^2+x{e}^4-2x)e^{(-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*exp(2*log(exp(2)*x-1)-6)+exp(2)*x-1)/(exp(2)*x-1),x, algorithm="fricas")

[Out]

(x^2*e^2 + x*e^4 - 2*x)*e^(-4)

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giac [A]  time = 0.41, size = 17, normalized size = 0.77 $$\begin{array}{c}(x^2{e}^2+x{e}^4-2x)e^{(-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*exp(2*log(exp(2)*x-1)-6)+exp(2)*x-1)/(exp(2)*x-1),x, algorithm="giac")

[Out]

(x^2*e^2 + x*e^4 - 2*x)*e^(-4)

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maple [A]  time = 0.30, size = 14, normalized size = 0.64

method	result	size
risch	$-2x{\mathrm{e}}^{-4}+x^2{\mathrm{e}}^{-2}+x$	$14$
default	$x+{\mathrm{e}}^{2\ln ({\mathrm{e}}^{2}x-1)-6}$	$15$
norman	${\mathrm{e}}^4{\mathrm{e}}^{-6}{x}^2-(2{\mathrm{e}}^2-{\mathrm{e}}^6){\mathrm{e}}^{-6}x$	$30$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(2)*exp(2*ln(exp(2)*x-1)-6)+exp(2)*x-1)/(exp(2)*x-1),x,method=_RETURNVERBOSE)

[Out]

-2*x*exp(-4)+x^2*exp(-2)+x

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maxima [A]  time = 0.39, size = 16, normalized size = 0.73 $$\begin{array}{c}(x^2{e}^2+x(e^4-2))e^{(-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*exp(2*log(exp(2)*x-1)-6)+exp(2)*x-1)/(exp(2)*x-1),x, algorithm="maxima")

[Out]

(x^2*e^2 + x*(e^4 - 2))*e^(-4)

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mupad [B]  time = 0.07, size = 12, normalized size = 0.55 $$\begin{array}{c}x(x{\mathrm{e}}^{-2}-2{\mathrm{e}}^{-4}+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(2) + 2*exp(2*log(x*exp(2) - 1) - 6)*exp(2) - 1)/(x*exp(2) - 1),x)

[Out]

x*(x*exp(-2) - 2*exp(-4) + 1)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.68 $$\begin{array}{c}\frac{x^2}{e^2}+\frac{x(-2+e^4)}{e^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)*exp(2*ln(exp(2)*x-1)-6)+exp(2)*x-1)/(exp(2)*x-1),x)

[Out]

x**2*exp(-2) + x*(-2 + exp(4))*exp(-4)

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