3.45.90 \(\int \frac {-2+3 e^{4/5}}{3+3 e^{4/5}} \, dx\)

Optimal. Leaf size=22 \[ -\frac {5 (4-x)}{3 \left (-1-e^{4/5}\right )}+x \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {8} \begin {gather*} -\frac {\left (2-3 e^{4/5}\right ) x}{3 \left (1+e^{4/5}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + 3*E^(4/5))/(3 + 3*E^(4/5)),x]

[Out]

-1/3*((2 - 3*E^(4/5))*x)/(1 + E^(4/5))

Rule 8

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\left (2-3 e^{4/5}\right ) x}{3 \left (1+e^{4/5}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 34, normalized size = 1.55 \begin {gather*} -\frac {2 x}{3+3 e^{4/5}}+\frac {3 e^{4/5} x}{3+3 e^{4/5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 3*E^(4/5))/(3 + 3*E^(4/5)),x]

[Out]

(-2*x)/(3 + 3*E^(4/5)) + (3*E^(4/5)*x)/(3 + 3*E^(4/5))

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fricas [A]  time = 0.46, size = 17, normalized size = 0.77 \begin {gather*} \frac {3 \, x e^{\frac {4}{5}} - 2 \, x}{3 \, {\left (e^{\frac {4}{5}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4/5)-2)/(3*exp(4/5)+3),x, algorithm="fricas")

[Out]

1/3*(3*x*e^(4/5) - 2*x)/(e^(4/5) + 1)

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giac [A]  time = 0.21, size = 15, normalized size = 0.68 \begin {gather*} \frac {x {\left (3 \, e^{\frac {4}{5}} - 2\right )}}{3 \, {\left (e^{\frac {4}{5}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4/5)-2)/(3*exp(4/5)+3),x, algorithm="giac")

[Out]

1/3*x*(3*e^(4/5) - 2)/(e^(4/5) + 1)

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maple [A]  time = 0.02, size = 16, normalized size = 0.73

method	result	size
norman	\(\frac {\left (3 \,{\mathrm e}^{\frac {4}{5}}-2\right ) x}{3 \,{\mathrm e}^{\frac {4}{5}}+3}\)	\(16\)
default	\(\frac {\left (3 \,{\mathrm e}^{\frac {4}{5}}-2\right ) x}{3 \,{\mathrm e}^{\frac {4}{5}}+3}\)	\(17\)
risch	\(\frac {3 x \,{\mathrm e}^{\frac {4}{5}}}{3 \,{\mathrm e}^{\frac {4}{5}}+3}-\frac {2 x}{3 \,{\mathrm e}^{\frac {4}{5}}+3}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(4/5)-2)/(3*exp(4/5)+3),x,method=_RETURNVERBOSE)

[Out]

1/3*(3*exp(4/5)-2)/(exp(4/5)+1)*x

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maxima [A]  time = 0.38, size = 15, normalized size = 0.68 \begin {gather*} \frac {x {\left (3 \, e^{\frac {4}{5}} - 2\right )}}{3 \, {\left (e^{\frac {4}{5}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4/5)-2)/(3*exp(4/5)+3),x, algorithm="maxima")

[Out]

1/3*x*(3*e^(4/5) - 2)/(e^(4/5) + 1)

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mupad [B]  time = 0.00, size = 16, normalized size = 0.73 \begin {gather*} \frac {x\,\left (3\,{\mathrm {e}}^{4/5}-2\right )}{3\,{\mathrm {e}}^{4/5}+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(4/5) - 2)/(3*exp(4/5) + 3),x)

[Out]

(x*(3*exp(4/5) - 2))/(3*exp(4/5) + 3)

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sympy [A]  time = 0.05, size = 17, normalized size = 0.77 \begin {gather*} \frac {x \left (-2 + 3 e^{\frac {4}{5}}\right )}{3 + 3 e^{\frac {4}{5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4/5)-2)/(3*exp(4/5)+3),x)

[Out]

x*(-2 + 3*exp(4/5))/(3 + 3*exp(4/5))

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