3.1.82 \(\int \frac {1}{1+\tan ^2(x)} \, dx\) [82]

Optimal. Leaf size=14 \[ \frac {x}{2}+\frac {1}{2} \cos (x) \sin (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3738, 2715, 8} \begin {gather*} \frac {x}{2}+\frac {1}{2} \sin (x) \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Tan[x]^2)^(-1),x]

[Out]

x/2 + (Cos[x]*Sin[x])/2

Rule 8

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3738

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{2}+\frac {1}{4} \sin (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Tan[x]^2)^(-1),x]

[Out]

x/2 + Sin[2*x]/4

________________________________________________________________________________________

Mathics [A]
time = 2.24, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2}+\frac {\text {Sin}\left [2 x\right ]}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(1 + Tan[x]^2),x]')

[Out]

x / 2 + Sin[2 x] / 4

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 19, normalized size = 1.36

method	result	size
risch	\(\frac {x}{2}+\frac {\sin \left (2 x \right )}{4}\)	\(11\)
derivativedivides	\(\frac {\tan \left (x \right )}{2+2 \left (\tan ^{2}\left (x \right )\right )}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}\)	\(19\)
default	\(\frac {\tan \left (x \right )}{2+2 \left (\tan ^{2}\left (x \right )\right )}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}\)	\(19\)
norman	\(\frac {\frac {x}{2}+\frac {x \left (\tan ^{2}\left (x \right )\right )}{2}+\frac {\tan \left (x \right )}{2}}{1+\tan ^{2}\left (x \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+tan(x)^2),x,method=_RETURNVERBOSE)

[Out]

1/2/(1+tan(x)^2)*tan(x)+1/2*arctan(tan(x))

________________________________________________________________________________________

Maxima [A]
time = 0.35, size = 16, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, x + \frac {\tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 1/2*tan(x)/(tan(x)^2 + 1)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 20, normalized size = 1.43 \begin {gather*} \frac {x \tan \left (x\right )^{2} + x + \tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x*tan(x)^2 + x + tan(x))/(tan(x)^2 + 1)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (10) = 20\)
time = 0.19, size = 36, normalized size = 2.57 \begin {gather*} \frac {x \tan ^{2}{\left (x \right )}}{2 \tan ^{2}{\left (x \right )} + 2} + \frac {x}{2 \tan ^{2}{\left (x \right )} + 2} + \frac {\tan {\left (x \right )}}{2 \tan ^{2}{\left (x \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+tan(x)**2),x)

[Out]

x*tan(x)**2/(2*tan(x)**2 + 2) + x/(2*tan(x)**2 + 2) + tan(x)/(2*tan(x)**2 + 2)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 17, normalized size = 1.21 \begin {gather*} \frac {\tan x}{2 \left (\tan ^{2}x+1\right )}+\frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+tan(x)^2),x)

[Out]

1/2*x + 1/2*tan(x)/(tan(x)^2 + 1)

________________________________________________________________________________________

Mupad [B]
time = 0.18, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2}+\frac {\sin \left (2\,x\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(x)^2 + 1),x)

[Out]

x/2 + sin(2*x)/4

________________________________________________________________________________________