\(\int \sec ^3(x) \tan (x) \, dx\) [356]

Optimal result

Integrand size = 7, antiderivative size = 8 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {\sec ^3(x)}{3} \]

[Out]

1/3*sec(x)^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 30} \[ \int \sec ^3(x) \tan (x) \, dx=\frac {\sec ^3(x)}{3} \]

[In]

Int[Sec[x]^3*Tan[x],x]

[Out]

Sec[x]^3/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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}\left (\int x^2 \, dx,x,\sec (x)\right ) \\ & = \frac {\sec ^3(x)}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {\sec ^3(x)}{3} \]

[In]

Integrate[Sec[x]^3*Tan[x],x]

[Out]

Sec[x]^3/3

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\left (\sec ^{3}\left (x \right )\right )}{3}\)	\(7\)
default	\(\frac {\left (\sec ^{3}\left (x \right )\right )}{3}\)	\(7\)
risch	\(\frac {8 \,{\mathrm e}^{3 i x}}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}\)	\(17\)

[In]

int(sec(x)^3*tan(x),x,method=_RETURNVERBOSE)

[Out]

1/3*sec(x)^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {1}{3 \, \cos \left (x\right )^{3}} \]

[In]

integrate(sec(x)^3*tan(x),x, algorithm="fricas")

[Out]

1/3/cos(x)^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {1}{3 \cos ^{3}{\left (x \right )}} \]

[In]

integrate(sec(x)**3*tan(x),x)

[Out]

1/(3*cos(x)**3)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {1}{3 \, \cos \left (x\right )^{3}} \]

[In]

integrate(sec(x)^3*tan(x),x, algorithm="maxima")

[Out]

1/3/cos(x)^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {1}{3 \, \cos \left (x\right )^{3}} \]

[In]

integrate(sec(x)^3*tan(x),x, algorithm="giac")

[Out]

1/3/cos(x)^3

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^3(x) \tan (x) \, dx=\frac {1}{3\,{\cos \left (x\right )}^3} \]

[In]

int(tan(x)/cos(x)^3,x)

[Out]

1/(3*cos(x)^3)