3.8.3 \(\int \frac {e^{3 x}}{1+e^{2 x}} \, dx\) [703]

Optimal. Leaf size=10 \[ e^x-\tan ^{-1}\left (e^x\right ) \]

[Out]

exp(x)-arctan(exp(x))

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2280, 327, 209} \begin {gather*} e^x-\text {ArcTan}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*x)/(1 + E^(2*x)),x]

[Out]

E^x - ArcTan[E^x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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*} \int \frac {e^{3 x}}{1+e^{2 x}} \, dx &=\text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,e^x\right )\\ &=e^x-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=e^x-\tan ^{-1}\left (e^x\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} e^x-\tan ^{-1}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(3*x)/(1 + E^(2*x)),x]

[Out]

E^x - ArcTan[E^x]

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 9, normalized size = 0.90

method	result	size
default	\({\mathrm e}^{x}-\arctan \left ({\mathrm e}^{x}\right )\)	\(9\)
risch	\({\mathrm e}^{x}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(3*x)/(1+exp(2*x)),x,method=_RETURNVERBOSE)

[Out]

exp(x)-arctan(exp(x))

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 8, normalized size = 0.80 \begin {gather*} -\arctan \left (e^{x}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(e^x) + e^x

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 8, normalized size = 0.80 \begin {gather*} -\arctan \left (e^{x}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(e^x) + e^x

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
time = 0.04, size = 19, normalized size = 1.90 \begin {gather*} e^{x} + \operatorname {RootSum} {\left (4 z^{2} + 1, \left ( i \mapsto i \log {\left (- 2 i + e^{x} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x) + RootSum(4*_z**2 + 1, Lambda(_i, _i*log(-2*_i + exp(x))))

________________________________________________________________________________________

Giac [A]
time = 4.14, size = 8, normalized size = 0.80 \begin {gather*} -\arctan \left (e^{x}\right ) + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(e^x) + e^x

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 8, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^x-\mathrm {atan}\left ({\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(3*x)/(exp(2*x) + 1),x)

[Out]

exp(x) - atan(exp(x))

________________________________________________________________________________________