3.6 \(\int \sec ^2(x) \tan (x) \, dx\)

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

[Out]

1/2*sec(x)^2

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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 = {2606, 30} \[ \frac {\sec ^2(x)}{2} \]

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

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Mathematica [A]  time = 0.01, size = 8, normalized size = 1.00 \[ \frac {\sec ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[x]^2/2

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fricas [A]  time = 0.43, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \cos \relax (x)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x),x, algorithm="fricas")

[Out]

1/2/cos(x)^2

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giac [A]  time = 1.04, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \cos \relax (x)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x),x, algorithm="giac")

[Out]

1/2/cos(x)^2

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maple [A]  time = 0.02, size = 7, normalized size = 0.88 \[ \frac {\left (\sec ^{2}\relax (x )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*tan(x),x)

[Out]

1/2*sec(x)^2

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maxima [A]  time = 0.55, size = 6, normalized size = 0.75 \[ \frac {1}{2} \, \tan \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x),x, algorithm="maxima")

[Out]

1/2*tan(x)^2

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mupad [B]  time = 0.02, size = 6, normalized size = 0.75 \[ \frac {{\mathrm {tan}\relax (x)}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/cos(x)^2,x)

[Out]

tan(x)^2/2

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sympy [A]  time = 0.07, size = 7, normalized size = 0.88 \[ \frac {1}{2 \cos ^{2}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*tan(x),x)

[Out]

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

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