### 3.112 $$\int \cos ^2(x) (1-\tan ^2(x)) \, dx$$

Optimal. Leaf size=5 $\sin (x) \cos (x)$

[Out]

Cos[x]*Sin[x]

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Rubi [A]  time = 0.0220327, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {3675, 383} $\sin (x) \cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Cos[x]^2*(1 - Tan[x]^2),x]

[Out]

Cos[x]*Sin[x]

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

\begin{align*} \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=\cos (x) \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0016832, size = 8, normalized size = 1.6 $\frac{1}{2} \sin (2 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cos[x]^2*(1 - Tan[x]^2),x]

[Out]

Sin[2*x]/2

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Maple [A]  time = 0.016, size = 6, normalized size = 1.2 \begin{align*} \cos \left ( x \right ) \sin \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1-tan(x)^2)/sec(x)^2,x)

[Out]

cos(x)*sin(x)

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Maxima [B]  time = 0.930044, size = 15, normalized size = 3. \begin{align*} \frac{\tan \left (x\right )}{\tan \left (x\right )^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tan(x)^2)/sec(x)^2,x, algorithm="maxima")

[Out]

tan(x)/(tan(x)^2 + 1)

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Fricas [A]  time = 2.12409, size = 20, normalized size = 4. \begin{align*} \cos \left (x\right ) \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tan(x)^2)/sec(x)^2,x, algorithm="fricas")

[Out]

cos(x)*sin(x)

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Sympy [A]  time = 0.414073, size = 7, normalized size = 1.4 \begin{align*} \frac{\tan{\left (x \right )}}{\sec ^{2}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tan(x)**2)/sec(x)**2,x)

[Out]

tan(x)/sec(x)**2

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Giac [A]  time = 1.06252, size = 12, normalized size = 2.4 \begin{align*} \frac{1}{\frac{1}{\tan \left (x\right )} + \tan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tan(x)^2)/sec(x)^2,x, algorithm="giac")

[Out]

1/(1/tan(x) + tan(x))