3.1.6 \(\int \sec (x) \tan (x) \, dx\) [6]

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

[Out]

sec(x)

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Rubi [A]
time = 0.00, 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 = {2686, 8} \begin {gather*} \sec (x) \end {gather*}

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 2686

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 &=\text {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 \begin {gather*} \sec (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[x]

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Maple [A]
time = 0.01, size = 3, normalized size = 1.50

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

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 = 1.78, size = 4, normalized size = 2.00 \begin {gather*} \frac {1}{\cos \left (x\right )} \end {gather*}

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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Fricas [A]
time = 0.54, size = 4, normalized size = 2.00 \begin {gather*} \frac {1}{\cos \left (x\right )} \end {gather*}

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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Sympy [A]
time = 0.01, size = 3, normalized size = 1.50 \begin {gather*} \frac {1}{\cos {\left (x \right )}} \end {gather*}

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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Giac [A]
time = 1.04, size = 4, normalized size = 2.00 \begin {gather*} \frac {1}{\cos \left (x\right )} \end {gather*}

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

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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