3.1.96 \(\int \sec ^2(x) \tan (x) \, dx\) [96]

Optimal. Leaf size=8 \[ \frac {\sec ^2(x)}{2} \]

[Out]

1/2*sec(x)^2

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 30} \begin {gather*} \frac {\sec ^2(x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[x]^2/2

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*} \int \sec ^2(x) \tan (x) \, dx &=\text {Subst}(\int x \, dx,x,\sec (x))\\ &=\frac {\sec ^2(x)}{2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\sec ^2(x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[x]^2/2

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 7, normalized size = 0.88

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2/cot(x),x,method=_RETURNVERBOSE)

[Out]

1/2*sec(x)^2

________________________________________________________________________________________

Maxima [A]
time = 1.23, size = 10, normalized size = 1.25 \begin {gather*} -\frac {1}{2 \, {\left (\sin \left (x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/cot(x),x, algorithm="maxima")

[Out]

-1/2/(sin(x)^2 - 1)

________________________________________________________________________________________

Fricas [A]
time = 0.60, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2 \, \cos \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/cot(x),x, algorithm="fricas")

[Out]

1/2/cos(x)^2

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 7, normalized size = 0.88 \begin {gather*} \frac {1}{2 \cos ^{2}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2/cot(x),x)

[Out]

1/(2*cos(x)**2)

________________________________________________________________________________________

Giac [A]
time = 1.09, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2 \, \cos \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/cot(x),x, algorithm="giac")

[Out]

1/2/cos(x)^2

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 6, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2*cot(x)),x)

[Out]

tan(x)^2/2

________________________________________________________________________________________