3.7 \(\int \sec ^2(x) \, dx\)

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

[Out]

tan(x)

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Rubi [A]  time = 0.01, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3767, 8} \[ \tan (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2,x]

[Out]

Tan[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \sec ^2(x) \, dx &=-\operatorname {Subst}(\int 1 \, dx,x,-\tan (x))\\ &=\tan (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \[ \tan (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2,x]

[Out]

Tan[x]

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fricas [B]  time = 0.43, size = 7, normalized size = 3.50 \[ \frac {\sin \relax (x)}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/cos(x)

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giac [A]  time = 1.11, size = 2, normalized size = 1.00 \[ \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)

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maple [A]  time = 0.01, size = 3, normalized size = 1.50 \[ \tan \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2,x)

[Out]

tan(x)

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maxima [A]  time = 0.42, size = 2, normalized size = 1.00 \[ \tan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)

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mupad [B]  time = 0.02, size = 2, normalized size = 1.00 \[ \mathrm {tan}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)

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sympy [B]  time = 0.06, size = 5, normalized size = 2.50 \[ \frac {\sin {\relax (x )}}{\cos {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/cos(x)

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