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

Optimal. Leaf size=11 \[ \frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x)) \]

[Out]

ArcTanh[2*Cos[x]*Sin[x]]/2

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Rubi [A]  time = 0.0305843, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {206} \[ \frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Tan[x]^2)/(1 - Tan[x]^2),x]

[Out]

ArcTanh[2*Cos[x]*Sin[x]]/2

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [B]  time = 0.0066211, size = 23, normalized size = 2.09 \[ \frac{1}{2} \log (\sin (x)+\cos (x))-\frac{1}{2} \log (\cos (x)-\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[Cos[x] - Sin[x]]/2 + Log[Cos[x] + Sin[x]]/2

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Maple [A]  time = 0.007, size = 4, normalized size = 0.4 \begin{align*}{\it Artanh} \left ( \tan \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(tan(x))

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Maxima [A]  time = 0.943211, size = 20, normalized size = 1.82 \begin{align*} \frac{1}{2} \, \log \left (\tan \left (x\right ) + 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+tan(x)^2)/(1-tan(x)^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.92308, size = 139, normalized size = 12.64 \begin{align*} \frac{1}{4} \, \log \left (\frac{\tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) - \frac{1}{4} \, \log \left (\frac{\tan \left (x\right )^{2} - 2 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.148052, size = 15, normalized size = 1.36 \begin{align*} - \frac{\log{\left (\tan{\left (x \right )} - 1 \right )}}{2} + \frac{\log{\left (\tan{\left (x \right )} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08463, size = 23, normalized size = 2.09 \begin{align*} \frac{1}{2} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\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+tan(x)^2)/(1-tan(x)^2),x, algorithm="giac")

[Out]

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