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

Optimal. Leaf size=27 \[ \frac{3}{5} \left (e^x+1\right )^{5/3}-\frac{3}{2} \left (e^x+1\right )^{2/3} \]

[Out]

(-3*(1 + E^x)^(2/3))/2 + (3*(1 + E^x)^(5/3))/5

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Rubi [A]  time = 0.0259199, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac{3}{5} \left (e^x+1\right )^{5/3}-\frac{3}{2} \left (e^x+1\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*x)/(1 + E^x)^(1/3),x]

[Out]

(-3*(1 + E^x)^(2/3))/2 + (3*(1 + E^x)^(5/3))/5

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{2 x}}{\sqrt [3]{1+e^x}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\sqrt [3]{1+x}} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt [3]{1+x}}+(1+x)^{2/3}\right ) \, dx,x,e^x\right )\\ &=-\frac{3}{2} \left (1+e^x\right )^{2/3}+\frac{3}{5} \left (1+e^x\right )^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.0101943, size = 20, normalized size = 0.74 \[ \frac{3}{10} \left (e^x+1\right )^{2/3} \left (2 e^x-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)/(1 + E^x)^(1/3),x]

[Out]

(3*(1 + E^x)^(2/3)*(-3 + 2*E^x))/10

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Maple [A]  time = 0.02, size = 18, normalized size = 0.7 \begin{align*} -{\frac{3}{2} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{2}{3}}}}+{\frac{3}{5} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(1+exp(x))^(2/3)+3/5*(1+exp(x))^(5/3)

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Maxima [A]  time = 0.991878, size = 23, normalized size = 0.85 \begin{align*} \frac{3}{5} \,{\left (e^{x} + 1\right )}^{\frac{5}{3}} - \frac{3}{2} \,{\left (e^{x} + 1\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(e^x + 1)^(5/3) - 3/2*(e^x + 1)^(2/3)

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Fricas [A]  time = 0.78088, size = 46, normalized size = 1.7 \begin{align*} \frac{3}{10} \,{\left (2 \, e^{x} - 3\right )}{\left (e^{x} + 1\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(2*e^x - 3)*(e^x + 1)^(2/3)

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Sympy [A]  time = 1.89628, size = 22, normalized size = 0.81 \begin{align*} \frac{3 \left (e^{x} + 1\right )^{\frac{5}{3}}}{5} - \frac{3 \left (e^{x} + 1\right )^{\frac{2}{3}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(exp(x) + 1)**(5/3)/5 - 3*(exp(x) + 1)**(2/3)/2

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Giac [A]  time = 1.19501, size = 23, normalized size = 0.85 \begin{align*} \frac{3}{5} \,{\left (e^{x} + 1\right )}^{\frac{5}{3}} - \frac{3}{2} \,{\left (e^{x} + 1\right )}^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(e^x + 1)^(5/3) - 3/2*(e^x + 1)^(2/3)