3.753 \(\int (-\frac{1}{\sqrt [3]{e^x+x}}+\frac{x}{\sqrt [3]{e^x+x}}-(e^x+x)^{2/3}) \, dx\)

Optimal. Leaf size=13 \[ -\frac{3}{2} \left (x+e^x\right )^{2/3} \]

[Out]

(-3*(E^x + x)^(2/3))/2

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Rubi [A]  time = 0.0657407, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {2273} \[ -\frac{3}{2} \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[-(E^x + x)^(-1/3) + x/(E^x + x)^(1/3) - (E^x + x)^(2/3),x]

[Out]

(-3*(E^x + x)^(2/3))/2

Rule 2273

Int[(x_)^(m_.)*(E^(x_) + (x_)^(m_.))^(n_), x_Symbol] :> -Simp[(E^x + x^m)^(n + 1)/(n + 1), x] + (Dist[m, Int[x
^(m - 1)*(E^x + x^m)^n, x], x] + Int[(E^x + x^m)^(n + 1), x]) /; RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] &&
NeQ[n, -1]

Rubi steps

\begin{align*} \int \left (-\frac{1}{\sqrt [3]{e^x+x}}+\frac{x}{\sqrt [3]{e^x+x}}-\left (e^x+x\right )^{2/3}\right ) \, dx &=-\int \frac{1}{\sqrt [3]{e^x+x}} \, dx+\int \frac{x}{\sqrt [3]{e^x+x}} \, dx-\int \left (e^x+x\right )^{2/3} \, dx\\ &=-\frac{3}{2} \left (e^x+x\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.0048584, size = 13, normalized size = 1. \[ -\frac{3}{2} \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Integrate[-(E^x + x)^(-1/3) + x/(E^x + x)^(1/3) - (E^x + x)^(2/3),x]

[Out]

(-3*(E^x + x)^(2/3))/2

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Maple [A]  time = 0.042, size = 9, normalized size = 0.7 \begin{align*} -{\frac{3}{2} \left ({{\rm e}^{x}}+x \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(exp(x)+x)^(1/3)+x/(exp(x)+x)^(1/3)-(exp(x)+x)^(2/3),x)

[Out]

-3/2*(exp(x)+x)^(2/3)

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Maxima [A]  time = 1.10009, size = 11, normalized size = 0.85 \begin{align*} -\frac{3}{2} \,{\left (x + e^{x}\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x+exp(x))^(1/3)+x/(x+exp(x))^(1/3)-(x+exp(x))^(2/3),x, algorithm="maxima")

[Out]

-3/2*(x + e^x)^(2/3)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x+exp(x))^(1/3)+x/(x+exp(x))^(1/3)-(x+exp(x))^(2/3),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{e^{x}}{\sqrt [3]{x + e^{x}}}\, dx - \int \frac{1}{\sqrt [3]{x + e^{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(exp(x)+x)**(1/3)+x/(exp(x)+x)**(1/3)-(exp(x)+x)**(2/3),x)

[Out]

-Integral(exp(x)/(x + exp(x))**(1/3), x) - Integral((x + exp(x))**(-1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (x + e^{x}\right )}^{\frac{2}{3}} + \frac{x}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} - \frac{1}{{\left (x + e^{x}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(x+exp(x))^(1/3)+x/(x+exp(x))^(1/3)-(x+exp(x))^(2/3),x, algorithm="giac")

[Out]

integrate(-(x + e^x)^(2/3) + x/(x + e^x)^(1/3) - 1/(x + e^x)^(1/3), x)