3.8.53 \(\int (-\frac {1}{\sqrt [3]{e^x+x}}+\frac {x}{\sqrt [3]{e^x+x}}-(e^x+x)^{2/3}) \, dx\) [753]

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 = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = {2305} \begin {gather*} -\frac {3}{2} \left (x+e^x\right )^{2/3} \end {gather*}

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 2305

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*}

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Mathematica [A]
time = 0.00, 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[-(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 = 0.02, 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)+x)^(1/3)+x/(exp(x)+x)^(1/3)-(exp(x)+x)^(2/3),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)+x)^(1/3)+x/(exp(x)+x)^(1/3)-(exp(x)+x)^(2/3),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)+x)^(1/3)+x/(exp(x)+x)^(1/3)-(exp(x)+x)^(2/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {e^{x}}{\sqrt [3]{x + e^{x}}}\, dx - \int \frac {1}{\sqrt [3]{x + e^{x}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.38, 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(x/(x + exp(x))^(1/3) - (x + exp(x))^(2/3) - 1/(x + exp(x))^(1/3),x)

[Out]

-(3*(x + exp(x))^(2/3))/2

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