3.606 \(\int \frac{e^x (1+\sinh (x))}{1+\cosh (x)} \, dx\)

Optimal. Leaf size=13 \[ e^x+\frac{2}{e^x+1} \]

[Out]

E^x + 2/(1 + E^x)

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Rubi [A]  time = 0.030643, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2282, 683} \[ e^x+\frac{2}{e^x+1} \]

Antiderivative was successfully verified.

[In]

Int[(E^x*(1 + Sinh[x]))/(1 + Cosh[x]),x]

[Out]

E^x + 2/(1 + E^x)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int \frac{e^x (1+\sinh (x))}{1+\cosh (x)} \, dx &=\operatorname{Subst}\left (\int \frac{-1+2 x+x^2}{(1+x)^2} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (1-\frac{2}{(1+x)^2}\right ) \, dx,x,e^x\right )\\ &=e^x+\frac{2}{1+e^x}\\ \end{align*}

Mathematica [A]  time = 0.0258615, size = 18, normalized size = 1.38 \[ \frac{e^x+e^{2 x}+2}{e^x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(1 + Sinh[x]))/(1 + Cosh[x]),x]

[Out]

(2 + E^x + E^(2*x))/(1 + E^x)

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Maple [A]  time = 0.012, size = 18, normalized size = 1.4 \begin{align*} -\tanh \left ({\frac{x}{2}} \right ) -2\, \left ( -1+\tanh \left ( x/2 \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(1+sinh(x))/(1+cosh(x)),x)

[Out]

-tanh(1/2*x)-2/(-1+tanh(1/2*x))

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Maxima [A]  time = 0.980585, size = 15, normalized size = 1.15 \begin{align*} \frac{2}{e^{x} + 1} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1+sinh(x))/(1+cosh(x)),x, algorithm="maxima")

[Out]

2/(e^x + 1) + e^x

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Fricas [A]  time = 2.03628, size = 69, normalized size = 5.31 \begin{align*} \frac{3 \, \cosh \left (x\right ) - \sinh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1+sinh(x))/(1+cosh(x)),x, algorithm="fricas")

[Out]

(3*cosh(x) - sinh(x) + 1)/(cosh(x) - sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\sinh{\left (x \right )} + 1\right ) e^{x}}{\cosh{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1+sinh(x))/(1+cosh(x)),x)

[Out]

Integral((sinh(x) + 1)*exp(x)/(cosh(x) + 1), x)

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Giac [A]  time = 1.12289, size = 15, normalized size = 1.15 \begin{align*} \frac{2}{e^{x} + 1} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1+sinh(x))/(1+cosh(x)),x, algorithm="giac")

[Out]

2/(e^x + 1) + e^x