3.38 \(\int (1+\tan (2 x))^2 \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2} \tan (2 x)-\log (\cos (2 x)) \]

[Out]

-Log[Cos[2*x]] + Tan[2*x]/2

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Rubi [A]  time = 0.0109919, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3477, 3475} \[ \frac{1}{2} \tan (2 x)-\log (\cos (2 x)) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Tan[2*x])^2,x]

[Out]

-Log[Cos[2*x]] + Tan[2*x]/2

Rule 3477

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0107909, size = 26, normalized size = 1.62 \[ x-\frac{1}{2} \tan ^{-1}(\tan (2 x))+\frac{1}{2} \tan (2 x)-\log (\cos (2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Tan[2*x])^2,x]

[Out]

x - ArcTan[Tan[2*x]]/2 - Log[Cos[2*x]] + Tan[2*x]/2

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Maple [A]  time = 0.001, size = 19, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( 2\,x \right ) }{2}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( 2\,x \right ) \right ) ^{2} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41653, size = 16, normalized size = 1. \begin{align*} \log \left (\sec \left (2 \, x\right )\right ) + \frac{1}{2} \, \tan \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sec(2*x)) + 1/2*tan(2*x)

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Fricas [A]  time = 2.00127, size = 61, normalized size = 3.81 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac{1}{2} \, \tan \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.141312, size = 17, normalized size = 1.06 \begin{align*} \frac{\log{\left (\tan ^{2}{\left (2 x \right )} + 1 \right )}}{2} + \frac{\tan{\left (2 x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(2*x)**2 + 1)/2 + tan(2*x)/2

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Giac [A]  time = 1.09449, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{2} \, \log \left (\frac{4}{\tan \left (2 \, x\right )^{2} + 1}\right ) + \frac{1}{2} \, \tan \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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