3.8.52 \(\int -\frac {1+e^x}{\sqrt [3]{e^x+x}} \, dx\) [752]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = {6818} \begin {gather*} -\frac {3}{2} \left (x+e^x\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-((1 + E^x)/(E^x + x)^(1/3)),x]

[Out]

(-3*(E^x + x)^(2/3))/2

Rule 6818

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*}

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Mathematica [A]
time = 0.02, size = 13, normalized size = 1.00 \begin {gather*} -\frac {3}{2} \left (e^x+x\right )^{2/3} \end {gather*}

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 = 0.01, size = 9, normalized size = 0.69

method	result	size
risch	\(-\frac {3 \left ({\mathrm e}^{x}+x \right )^{\frac {2}{3}}}{2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1-exp(x))/(exp(x)+x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 8, normalized size = 0.62 \begin {gather*} -\frac {3}{2} \, {\left (x + e^{x}\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x + e^x)^(2/3)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [A]
time = 0.07, size = 12, normalized size = 0.92 \begin {gather*} - \frac {3 \left (x + e^{x}\right )^{\frac {2}{3}}}{2} \end {gather*}

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 = 5.40, size = 8, normalized size = 0.62 \begin {gather*} -\frac {3}{2} \, {\left (x + e^{x}\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x + e^x)^(2/3)

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Mupad [B]
time = 3.37, size = 8, normalized size = 0.62 \begin {gather*} -\frac {3\,{\left (x+{\mathrm {e}}^x\right )}^{2/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(x + exp(x))^(2/3))/2

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