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

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 = 0.024423, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {6686} \[ -\frac{3}{2} \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[-((1 + E^x)/(E^x + x)^(1/3)),x]

[Out]

(-3*(E^x + x)^(2/3))/2

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0086212, size = 13, normalized size = 1. \[ -\frac{3}{2} \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(E^x + x)^(2/3))/2

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Maple [A]  time = 0.028, 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))/(exp(x)+x)^(1/3),x)

[Out]

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

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Maxima [A]  time = 0.947088, 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-exp(x))/(x+exp(x))^(1/3),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-exp(x))/(x+exp(x))^(1/3),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.220586, size = 12, normalized size = 0.92 \begin{align*} - \frac{3 \left (x + e^{x}\right )^{\frac{2}{3}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(x + exp(x))**(2/3)/2

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Giac [A]  time = 1.23833, 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-exp(x))/(x+exp(x))^(1/3),x, algorithm="giac")

[Out]

-3/2*(x + e^x)^(2/3)