3.7.93 \(\int \frac {e^{5 x}+e^{7 x}}{e^{-x}+e^x} \, dx\) [693]

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

[Out]

1/6*exp(6*x)

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Rubi [A]
time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 = {2320, 30} \begin {gather*} \frac {e^{6 x}}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(6*x)/6

Rule 30

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

Rule 2320

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]]]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \frac {e^{6 x}}{6} \end {gather*}

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 = 0.02, size = 7, normalized size = 0.78

method	result	size
default	\(\frac {{\mathrm e}^{6 x}}{6}\)	\(7\)
norman	\(\frac {{\mathrm e}^{6 x}}{6}\)	\(7\)
risch	\(\frac {{\mathrm e}^{6 x}}{6}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*exp(x)^6

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Maxima [A]
time = 0.28, size = 6, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, e^{\left (6 \, x\right )} \end {gather*}

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 = 0.37, size = 6, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, e^{\left (6 \, x\right )} \end {gather*}

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 = 0.03, size = 5, normalized size = 0.56 \begin {gather*} \frac {e^{6 x}}{6} \end {gather*}

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 = 2.19, size = 6, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, e^{\left (6 \, x\right )} \end {gather*}

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)

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Mupad [B]
time = 0.04, size = 6, normalized size = 0.67 \begin {gather*} \frac {{\mathrm {e}}^{6\,x}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(6*x)/6

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