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

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 = 0.0414829, 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 \[ -\frac{3}{2} \left (x+e^x\right )^{2/3} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(E^x + x)^(2/3))/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1-exp(x))/(exp(x)+x)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.00918479, 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 = 0.015, size = 9, normalized size = 0.7 \[ -{\frac{3}{2} \left ({{\rm e}^{x}}+x \right ) ^{{\frac{2}{3}}}} \]

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 = 0.747994, size = 11, normalized size = 0.85 \[ -\frac{3}{2} \,{\left (x + e^{x}\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e^x + 1)/(x + e^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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e^x + 1)/(x + e^x)^(1/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [A]  time = 0.264623, size = 12, normalized size = 0.92 \[ - \frac{3 \left (x + e^{x}\right )^{\frac{2}{3}}}{2} \]

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/XCAS [A]  time = 0.23143, size = 11, normalized size = 0.85 \[ -\frac{3}{2} \,{\left (x + e^{x}\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e^x + 1)/(x + e^x)^(1/3),x, algorithm="giac")

[Out]

-3/2*(x + e^x)^(2/3)