\(\int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx\) [688]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=-\frac {4}{3} \left (1+e^x\right )^{3/4}+\frac {4}{7} \left (1+e^x\right )^{7/4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2280, 45} \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4}{7} \left (e^x+1\right )^{7/4}-\frac {4}{3} \left (e^x+1\right )^{3/4} \]

[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 45

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 2280

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*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt [4]{1+x}} \, dx,x,e^x\right ) \\ & = \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4}{21} \left (1+e^x\right )^{3/4} \left (-7+3 \left (1+e^x\right )\right ) \]

[In]

Integrate[E^(2*x)/(1 + E^x)^(1/4),x]

[Out]

(4*(1 + E^x)^(3/4)*(-7 + 3*(1 + E^x)))/21

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {4 \left (-4+3 \,{\mathrm e}^{x}\right ) \left (1+{\mathrm e}^{x}\right )^{\frac {3}{4}}}{21}\)	\(15\)
default	\(-\frac {4 \left (1+{\mathrm e}^{x}\right )^{\frac {3}{4}}}{3}+\frac {4 \left (1+{\mathrm e}^{x}\right )^{\frac {7}{4}}}{7}\)	\(18\)

[In]

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

[Out]

4/21*(-4+3*exp(x))*(1+exp(x))^(3/4)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4}{21} \, {\left (3 \, e^{x} - 4\right )} {\left (e^{x} + 1\right )}^{\frac {3}{4}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4 \left (e^{x} + 1\right )^{\frac {7}{4}}}{7} - \frac {4 \left (e^{x} + 1\right )^{\frac {3}{4}}}{3} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4}{7} \, {\left (e^{x} + 1\right )}^{\frac {7}{4}} - \frac {4}{3} \, {\left (e^{x} + 1\right )}^{\frac {3}{4}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4}{7} \, {\left (e^{x} + 1\right )}^{\frac {7}{4}} - \frac {4}{3} \, {\left (e^{x} + 1\right )}^{\frac {3}{4}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 x}}{\sqrt [4]{1+e^x}} \, dx=\frac {4\,{\left ({\mathrm {e}}^x+1\right )}^{3/4}\,\left (3\,{\mathrm {e}}^x-4\right )}{21} \]

[In]

int(exp(2*x)/(exp(x) + 1)^(1/4),x)

[Out]

(4*(exp(x) + 1)^(3/4)*(3*exp(x) - 4))/21