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]

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

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.39, size = 14, normalized size = 0.52 \[ \frac {3}{10} \, {\left (2 \, e^{x} - 3\right )} {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]

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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giac [A]  time = 0.21, size = 17, normalized size = 0.63 \[ \frac {3}{5} \, {\left (e^{x} + 1\right )}^{\frac {5}{3}} - \frac {3}{2} \, {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]

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)

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maple [A]  time = 0.03, size = 18, normalized size = 0.67 \[ -\frac {3 \left ({\mathrm e}^{x}+1\right )^{\frac {2}{3}}}{2}+\frac {3 \left ({\mathrm e}^{x}+1\right )^{\frac {5}{3}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.01, size = 17, normalized size = 0.63 \[ \frac {3}{5} \, {\left (e^{x} + 1\right )}^{\frac {5}{3}} - \frac {3}{2} \, {\left (e^{x} + 1\right )}^{\frac {2}{3}} \]

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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mupad [B]  time = 0.06, size = 14, normalized size = 0.52 \[ \frac {3\,{\left ({\mathrm {e}}^x+1\right )}^{2/3}\,\left (2\,{\mathrm {e}}^x-3\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/(exp(x) + 1)^(1/3),x)

[Out]

(3*(exp(x) + 1)^(2/3)*(2*exp(x) - 3))/10

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sympy [A]  time = 1.99, size = 22, normalized size = 0.81 \[ \frac {3 \left (e^{x} + 1\right )^{\frac {5}{3}}}{5} - \frac {3 \left (e^{x} + 1\right )^{\frac {2}{3}}}{2} \]

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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