3.70 \(\int e^x \sec (e^x) \, dx\)

Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]

[Out]

arctanh(sin(exp(x)))

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Rubi [A]  time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 3770} \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[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 3770

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

Rubi steps

\begin {align*} \int e^x \sec \left (e^x\right ) \, dx &=\operatorname {Subst}\left (\int \sec (x) \, dx,x,e^x\right )\\ &=\tanh ^{-1}\left (\sin \left (e^x\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 5, normalized size = 1.00 \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[E^x]]

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fricas [B]  time = 0.53, size = 19, normalized size = 3.80 \[ \frac {1}{2} \, \log \left (\sin \left (e^{x}\right ) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \left (e^{x}\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(sin(e^x) + 1) - 1/2*log(-sin(e^x) + 1)

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giac [B]  time = 0.12, size = 29, normalized size = 5.80 \[ \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) + 2 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(1/sin(e^x) + sin(e^x) + 2)) - 1/4*log(abs(1/sin(e^x) + sin(e^x) - 2))

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maple [A]  time = 0.00, size = 9, normalized size = 1.80 \[ \ln \left (\sec \left ({\mathrm e}^{x}\right )+\tan \left ({\mathrm e}^{x}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sec(exp(x)),x)

[Out]

ln(sec(exp(x))+tan(exp(x)))

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maxima [A]  time = 0.31, size = 8, normalized size = 1.60 \[ \log \left (\sec \left (e^{x}\right ) + \tan \left (e^{x}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sec(e^x) + tan(e^x))

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mupad [B]  time = 2.79, size = 10, normalized size = 2.00 \[ -\mathrm {atan}\left ({\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\right )\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/cos(exp(x)),x)

[Out]

-atan(exp(exp(x)*1i))*2i

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sympy [A]  time = 1.02, size = 10, normalized size = 2.00 \[ \log {\left (\tan {\left (e^{x} \right )} + \sec {\left (e^{x} \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sec(exp(x)),x)

[Out]

log(tan(exp(x)) + sec(exp(x)))

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