3.69 \(\int e^x \tan (e^x) \, dx\)

Optimal. Leaf size=7 \[ -\log \left (\cos \left (e^x\right )\right ) \]

[Out]

-Log[Cos[E^x]]

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Rubi [A]  time = 0.0101972, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 3475} \[ -\log \left (\cos \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Tan[E^x],x]

[Out]

-Log[Cos[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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int e^x \tan \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \tan (x) \, dx,x,e^x\right )\\ &=-\log \left (\cos \left (e^x\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0081558, size = 7, normalized size = 1. \[ -\log \left (\cos \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Tan[E^x],x]

[Out]

-Log[Cos[E^x]]

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Maple [A]  time = 0.002, size = 7, normalized size = 1. \begin{align*} -\ln \left ( \cos \left ({{\rm e}^{x}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*tan(exp(x)),x)

[Out]

-ln(cos(exp(x)))

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Maxima [A]  time = 0.994762, size = 5, normalized size = 0.71 \begin{align*} \log \left (\sec \left (e^{x}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tan(exp(x)),x, algorithm="maxima")

[Out]

log(sec(e^x))

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Fricas [A]  time = 0.474122, size = 41, normalized size = 5.86 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{\tan \left (e^{x}\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tan(exp(x)),x, algorithm="fricas")

[Out]

-1/2*log(1/(tan(e^x)^2 + 1))

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Sympy [A]  time = 0.643082, size = 10, normalized size = 1.43 \begin{align*} \frac{\log{\left (\tan ^{2}{\left (e^{x} \right )} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tan(exp(x)),x)

[Out]

log(tan(exp(x))**2 + 1)/2

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Giac [A]  time = 1.20248, size = 9, normalized size = 1.29 \begin{align*} -\log \left ({\left | \cos \left (e^{x}\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tan(exp(x)),x, algorithm="giac")

[Out]

-log(abs(cos(e^x)))