3.80 \(\int \frac{1}{(1+\tan (x))^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \log (\sin (x)+\cos (x))-\frac{1}{2 (\tan (x)+1)} \]

[Out]

Log[Cos[x] + Sin[x]]/2 - 1/(2*(1 + Tan[x]))

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Rubi [A]  time = 0.0256371, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3483, 3530} \[ \frac{1}{2} \log (\sin (x)+\cos (x))-\frac{1}{2 (\tan (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Tan[x])^(-2),x]

[Out]

Log[Cos[x] + Sin[x]]/2 - 1/(2*(1 + Tan[x]))

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0404895, size = 27, normalized size = 1.29 \[ \frac{\tan (x)+\log (\sin (x)+\cos (x))+\tan (x) \log (\sin (x)+\cos (x))}{2 \tan (x)+2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Tan[x])^(-2),x]

[Out]

(Log[Cos[x] + Sin[x]] + Tan[x] + Log[Cos[x] + Sin[x]]*Tan[x])/(2 + 2*Tan[x])

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Maple [A]  time = 0.014, size = 26, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }{4}}-{\frac{1}{2+2\,\tan \left ( x \right ) }}+{\frac{\ln \left ( 1+\tan \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*ln(tan(x)^2+1)-1/2/(1+tan(x))+1/2*ln(1+tan(x))

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Maxima [A]  time = 1.42644, size = 34, normalized size = 1.62 \begin{align*} -\frac{1}{2 \,{\left (\tan \left (x\right ) + 1\right )}} - \frac{1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \, \log \left (\tan \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(tan(x) + 1) - 1/4*log(tan(x)^2 + 1) + 1/2*log(tan(x) + 1)

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Fricas [B]  time = 2.06109, size = 124, normalized size = 5.9 \begin{align*} \frac{{\left (\tan \left (x\right ) + 1\right )} \log \left (\frac{\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) + \tan \left (x\right ) - 1}{4 \,{\left (\tan \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((tan(x) + 1)*log((tan(x)^2 + 2*tan(x) + 1)/(tan(x)^2 + 1)) + tan(x) - 1)/(tan(x) + 1)

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Sympy [B]  time = 0.365338, size = 75, normalized size = 3.57 \begin{align*} \frac{2 \log{\left (\tan{\left (x \right )} + 1 \right )} \tan{\left (x \right )}}{4 \tan{\left (x \right )} + 4} + \frac{2 \log{\left (\tan{\left (x \right )} + 1 \right )}}{4 \tan{\left (x \right )} + 4} - \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan{\left (x \right )}}{4 \tan{\left (x \right )} + 4} - \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{4 \tan{\left (x \right )} + 4} - \frac{2}{4 \tan{\left (x \right )} + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(tan(x) + 1)*tan(x)/(4*tan(x) + 4) + 2*log(tan(x) + 1)/(4*tan(x) + 4) - log(tan(x)**2 + 1)*tan(x)/(4*tan(
x) + 4) - log(tan(x)**2 + 1)/(4*tan(x) + 4) - 2/(4*tan(x) + 4)

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Giac [A]  time = 1.10039, size = 35, normalized size = 1.67 \begin{align*} -\frac{1}{2 \,{\left (\tan \left (x\right ) + 1\right )}} - \frac{1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(tan(x) + 1) - 1/4*log(tan(x)^2 + 1) + 1/2*log(abs(tan(x) + 1))