\(\int \sec (x) \tan (x) \, dx\) [9]

Optimal result

Integrand size = 5, antiderivative size = 2 \[ \int \sec (x) \tan (x) \, dx=\sec (x) \]

[Out]

sec(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2686, 8} \[ \int \sec (x) \tan (x) \, dx=\sec (x) \]

[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*} \text {integral}& = \text {Subst}(\int 1 \, dx,x,\sec (x)) \\ & = \sec (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sec (x) \tan (x) \, dx=\sec (x) \]

[In]

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

[Out]

Sec[x]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50

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

[In]

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

[Out]

sec(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos \left (x\right )} \]

[In]

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

[Out]

1/cos(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos {\left (x \right )}} \]

[In]

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

[Out]

1/cos(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos \left (x\right )} \]

[In]

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

[Out]

1/cos(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos \left (x\right )} \]

[In]

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

[Out]

1/cos(x)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 6.00 \[ \int \sec (x) \tan (x) \, dx=-\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2-1} \]

[In]

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

[Out]

-2/(tan(x/2)^2 - 1)