### 3.335 $$\int \frac{1+\cos ^2(x)}{1-\cos ^2(x)} \, dx$$

Optimal. Leaf size=8 $-x-2 \cot (x)$

[Out]

-x - 2*Cot[x]

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Rubi [A]  time = 0.0371258, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.235, Rules used = {3171, 3175, 3767, 8} $-x-2 \cot (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + Cos[x]^2)/(1 - Cos[x]^2),x]

[Out]

-x - 2*Cot[x]

Rule 3171

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(B*x
)/b, x] + Dist[(A*b - a*B)/b, Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.007607, size = 8, normalized size = 1. $-x-2 \cot (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + Cos[x]^2)/(1 - Cos[x]^2),x]

[Out]

-x - 2*Cot[x]

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Maple [A]  time = 0.032, size = 11, normalized size = 1.4 \begin{align*} -2\, \left ( \tan \left ( x \right ) \right ) ^{-1}-x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/tan(x)-x

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Maxima [A]  time = 1.40983, size = 14, normalized size = 1.75 \begin{align*} -x - \frac{2}{\tan \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x)^2)/(-cos(x)^2+1),x, algorithm="maxima")

[Out]

-x - 2/tan(x)

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Fricas [A]  time = 2.15822, size = 42, normalized size = 5.25 \begin{align*} -\frac{x \sin \left (x\right ) + 2 \, \cos \left (x\right )}{\sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x)^2)/(-cos(x)^2+1),x, algorithm="fricas")

[Out]

-(x*sin(x) + 2*cos(x))/sin(x)

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Sympy [A]  time = 1.31058, size = 12, normalized size = 1.5 \begin{align*} - x + \tan{\left (\frac{x}{2} \right )} - \frac{1}{\tan{\left (\frac{x}{2} \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x)**2)/(-cos(x)**2+1),x)

[Out]

-x + tan(x/2) - 1/tan(x/2)

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Giac [A]  time = 1.06638, size = 22, normalized size = 2.75 \begin{align*} -x - \frac{1}{\tan \left (\frac{1}{2} \, x\right )} + \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x)^2)/(-cos(x)^2+1),x, algorithm="giac")

[Out]

-x - 1/tan(1/2*x) + tan(1/2*x)