3.6 \(\int \sec (x) \tan (x) \, dx\)

Optimal. Leaf size=2 \[ \sec (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2606, 8} \[ \sec (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]*Tan[x],x]

[Out]

Sec[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {align*} \int \sec (x) \tan (x) \, dx &=\operatorname {Subst}(\int 1 \, dx,x,\sec (x))\\ &=\sec (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \[ \sec (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]*Tan[x],x]

[Out]

Sec[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec (x) \tan (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sec[x]*Tan[x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.15, size = 4, normalized size = 2.00 \[ \frac {1}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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giac [A]  time = 0.93, size = 4, normalized size = 2.00 \[ \frac {1}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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maple [A]  time = 0.07, size = 3, normalized size = 1.50

method	result	size
derivativedivides	\(\sec \relax (x )\)	\(3\)
default	\(\sec \relax (x )\)	\(3\)
risch	\(\frac {2 \,{\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x),x,method=_RETURNVERBOSE)

[Out]

sec(x)

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maxima [A]  time = 0.42, size = 4, normalized size = 2.00 \[ \frac {1}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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mupad [B]  time = 0.25, size = 12, normalized size = 6.00 \[ -\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/cos(x),x)

[Out]

-2/(tan(x/2)^2 - 1)

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sympy [A]  time = 0.07, size = 3, normalized size = 1.50 \[ \frac {1}{\cos {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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