3.5 \(\int -\sec (\frac{\pi }{4}+2 x) \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]

[Out]

-ArcTanh[Sin[Pi/4 + 2*x]]/2

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Rubi [A]  time = 0.0034344, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3770} \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[-Sec[Pi/4 + 2*x],x]

[Out]

-ArcTanh[Sin[Pi/4 + 2*x]]/2

Rule 3770

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

Rubi steps

\begin{align*} \int -\sec \left (\frac{\pi }{4}+2 x\right ) \, dx &=-\frac{1}{2} \tanh ^{-1}\left (\sin \left (\frac{\pi }{4}+2 x\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0058907, size = 15, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[-Sec[Pi/4 + 2*x],x]

[Out]

-ArcTanh[Sin[Pi/4 + 2*x]]/2

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Maple [A]  time = 0.006, size = 21, normalized size = 1.4 \begin{align*} -{\frac{1}{2}\ln \left ( \sec \left ({\frac{\pi }{4}}+2\,x \right ) +\tan \left ({\frac{\pi }{4}}+2\,x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/cos(1/4*Pi+2*x),x)

[Out]

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

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Maxima [B]  time = 0.929331, size = 36, normalized size = 2.4 \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/cos(1/4*pi+2*x),x, algorithm="maxima")

[Out]

-1/4*log(sin(1/4*pi + 2*x) + 1) + 1/4*log(sin(1/4*pi + 2*x) - 1)

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Fricas [B]  time = 2.07409, size = 90, normalized size = 6. \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (-\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/cos(1/4*pi+2*x),x, algorithm="fricas")

[Out]

-1/4*log(sin(1/4*pi + 2*x) + 1) + 1/4*log(-sin(1/4*pi + 2*x) + 1)

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Sympy [A]  time = 0.198141, size = 22, normalized size = 1.47 \begin{align*} \frac{\log{\left (\tan{\left (x + \frac{\pi }{8} \right )} - 1 \right )}}{2} - \frac{\log{\left (\tan{\left (x + \frac{\pi }{8} \right )} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/cos(1/4*pi+2*x),x)

[Out]

log(tan(x + pi/8) - 1)/2 - log(tan(x + pi/8) + 1)/2

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Giac [B]  time = 1.05812, size = 39, normalized size = 2.6 \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (-\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/cos(1/4*pi+2*x),x, algorithm="giac")

[Out]

-1/4*log(sin(1/4*pi + 2*x) + 1) + 1/4*log(-sin(1/4*pi + 2*x) + 1)