3.714 \(\int e^{\tan (x)} \sec ^2(x) \, dx\)

Optimal. Leaf size=4 \[ e^{\tan (x)} \]

[Out]

exp(tan(x))

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Rubi [A]  time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4342, 2194} \[ e^{\tan (x)} \]

Antiderivative was successfully verified.

[In]

Int[E^Tan[x]*Sec[x]^2,x]

[Out]

E^Tan[x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4342

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/
(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])

Rubi steps

\begin {align*} \int e^{\tan (x)} \sec ^2(x) \, dx &=\operatorname {Subst}\left (\int e^x \, dx,x,\tan (x)\right )\\ &=e^{\tan (x)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 4, normalized size = 1.00 \[ e^{\tan (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^Tan[x]*Sec[x]^2,x]

[Out]

E^Tan[x]

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fricas [B]  time = 0.52, size = 8, normalized size = 2.00 \[ e^{\frac {\sin \relax (x)}{\cos \relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(tan(x))*sec(x)^2,x, algorithm="fricas")

[Out]

e^(sin(x)/cos(x))

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giac [A]  time = 0.13, size = 3, normalized size = 0.75 \[ e^{\tan \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(tan(x))*sec(x)^2,x, algorithm="giac")

[Out]

e^tan(x)

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maple [A]  time = 0.05, size = 4, normalized size = 1.00 \[ {\mathrm e}^{\tan \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(tan(x))*sec(x)^2,x)

[Out]

exp(tan(x))

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maxima [A]  time = 0.32, size = 3, normalized size = 0.75 \[ e^{\tan \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(tan(x))*sec(x)^2,x, algorithm="maxima")

[Out]

e^tan(x)

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mupad [B]  time = 3.10, size = 3, normalized size = 0.75 \[ {\mathrm {e}}^{\mathrm {tan}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(tan(x))/cos(x)^2,x)

[Out]

exp(tan(x))

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sympy [A]  time = 0.95, size = 3, normalized size = 0.75 \[ e^{\tan {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(tan(x))*sec(x)**2,x)

[Out]

exp(tan(x))

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