3.688 \(\int \frac{e^{2 x}}{\sqrt [4]{1+e^x}} \, dx\)

Optimal. Leaf size=27 \[ \frac{4}{7} \left (e^x+1\right )^{7/4}-\frac{4}{3} \left (e^x+1\right )^{3/4} \]

[Out]

(-4*(1 + E^x)^(3/4))/3 + (4*(1 + E^x)^(7/4))/7

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Rubi [A]  time = 0.0258179, 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} \[ \frac{4}{7} \left (e^x+1\right )^{7/4}-\frac{4}{3} \left (e^x+1\right )^{3/4} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*x)/(1 + E^x)^(1/4),x]

[Out]

(-4*(1 + E^x)^(3/4))/3 + (4*(1 + E^x)^(7/4))/7

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 [4]{1+e^x}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1+x}} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt [4]{1+x}}+(1+x)^{3/4}\right ) \, dx,x,e^x\right )\\ &=-\frac{4}{3} \left (1+e^x\right )^{3/4}+\frac{4}{7} \left (1+e^x\right )^{7/4}\\ \end{align*}

Mathematica [A]  time = 0.0099306, size = 20, normalized size = 0.74 \[ \frac{4}{21} \left (e^x+1\right )^{3/4} \left (3 e^x-4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)/(1 + E^x)^(1/4),x]

[Out]

(4*(1 + E^x)^(3/4)*(-4 + 3*E^x))/21

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Maple [A]  time = 0.021, size = 18, normalized size = 0.7 \begin{align*} -{\frac{4}{3} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{3}{4}}}}+{\frac{4}{7} \left ( 1+{{\rm e}^{x}} \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/(1+exp(x))^(1/4),x)

[Out]

-4/3*(1+exp(x))^(3/4)+4/7*(1+exp(x))^(7/4)

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Maxima [A]  time = 0.959522, size = 23, normalized size = 0.85 \begin{align*} \frac{4}{7} \,{\left (e^{x} + 1\right )}^{\frac{7}{4}} - \frac{4}{3} \,{\left (e^{x} + 1\right )}^{\frac{3}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(1+exp(x))^(1/4),x, algorithm="maxima")

[Out]

4/7*(e^x + 1)^(7/4) - 4/3*(e^x + 1)^(3/4)

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Fricas [A]  time = 0.771953, size = 46, normalized size = 1.7 \begin{align*} \frac{4}{21} \,{\left (3 \, e^{x} - 4\right )}{\left (e^{x} + 1\right )}^{\frac{3}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(1+exp(x))^(1/4),x, algorithm="fricas")

[Out]

4/21*(3*e^x - 4)*(e^x + 1)^(3/4)

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Sympy [A]  time = 2.62772, size = 22, normalized size = 0.81 \begin{align*} \frac{4 \left (e^{x} + 1\right )^{\frac{7}{4}}}{7} - \frac{4 \left (e^{x} + 1\right )^{\frac{3}{4}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(1+exp(x))**(1/4),x)

[Out]

4*(exp(x) + 1)**(7/4)/7 - 4*(exp(x) + 1)**(3/4)/3

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Giac [A]  time = 1.23547, size = 23, normalized size = 0.85 \begin{align*} \frac{4}{7} \,{\left (e^{x} + 1\right )}^{\frac{7}{4}} - \frac{4}{3} \,{\left (e^{x} + 1\right )}^{\frac{3}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(1+exp(x))^(1/4),x, algorithm="giac")

[Out]

4/7*(e^x + 1)^(7/4) - 4/3*(e^x + 1)^(3/4)