3.693 \(\int \frac{e^{5 x}+e^{7 x}}{e^{-x}+e^x} \, dx\)

Optimal. Leaf size=9 \[ \frac{e^{6 x}}{6} \]

[Out]

E^(6*x)/6

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Rubi [A]  time = 0.0258115, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2282, 30} \[ \frac{e^{6 x}}{6} \]

Antiderivative was successfully verified.

[In]

Int[(E^(5*x) + E^(7*x))/(E^(-x) + E^x),x]

[Out]

E^(6*x)/6

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{5 x}+e^{7 x}}{e^{-x}+e^x} \, dx &=\operatorname{Subst}\left (\int x^5 \, dx,x,e^x\right )\\ &=\frac{e^{6 x}}{6}\\ \end{align*}

Mathematica [A]  time = 0.0010617, size = 9, normalized size = 1. \[ \frac{e^{6 x}}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(5*x) + E^(7*x))/(E^(-x) + E^x),x]

[Out]

E^(6*x)/6

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Maple [A]  time = 0.023, size = 7, normalized size = 0.8 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5*x)+exp(7*x))/(exp(-x)+exp(x)),x)

[Out]

1/6*exp(x)^6

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Maxima [A]  time = 0.989968, size = 8, normalized size = 0.89 \begin{align*} \frac{1}{6} \, e^{\left (6 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5*x)+exp(7*x))/(exp(-x)+exp(x)),x, algorithm="maxima")

[Out]

1/6*e^(6*x)

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Fricas [A]  time = 0.822916, size = 18, normalized size = 2. \begin{align*} \frac{1}{6} \, e^{\left (6 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5*x)+exp(7*x))/(exp(-x)+exp(x)),x, algorithm="fricas")

[Out]

1/6*e^(6*x)

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Sympy [A]  time = 0.116089, size = 5, normalized size = 0.56 \begin{align*} \frac{e^{6 x}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5*x)+exp(7*x))/(exp(-x)+exp(x)),x)

[Out]

exp(6*x)/6

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Giac [A]  time = 1.21292, size = 8, normalized size = 0.89 \begin{align*} \frac{1}{6} \, e^{\left (6 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(5*x)+exp(7*x))/(exp(-x)+exp(x)),x, algorithm="giac")

[Out]

1/6*e^(6*x)