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

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 = 0.00902192, 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 \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[-Sec[Pi/4 + 2*x],x]

[Out]

-ArcTanh[Sin[Pi/4 + 2*x]]/2

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Rubi in Sympy [A]  time = 0.703907, size = 12, normalized size = 0.8 \[ - \frac{\operatorname{atanh}{\left (\sin{\left (2 x + \frac{\pi }{4} \right )} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-1/cos(1/4*pi+2*x),x)

[Out]

-atanh(sin(2*x + pi/4))/2

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Mathematica [B]  time = 0.0129273, size = 55, normalized size = 3.67 \[ \frac{1}{2} \log \left (\cos \left (\frac{1}{8} (8 x+\pi )\right )-\sin \left (\frac{1}{8} (8 x+\pi )\right )\right )-\frac{1}{2} \log \left (\sin \left (\frac{1}{8} (8 x+\pi )\right )+\cos \left (\frac{1}{8} (8 x+\pi )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[-Sec[Pi/4 + 2*x],x]

[Out]

Log[Cos[(Pi + 8*x)/8] - Sin[(Pi + 8*x)/8]]/2 - Log[Cos[(Pi + 8*x)/8] + Sin[(Pi +
8*x)/8]]/2

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Maple [A]  time = 0.01, size = 21, normalized size = 1.4 \[ -{\frac{1}{2}\ln \left ( \sec \left ({\frac{\pi }{4}}+2\,x \right ) +\tan \left ({\frac{\pi }{4}}+2\,x \right ) \right ) } \]

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 [A]  time = 1.34341, size = 36, normalized size = 2.4 \[ -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) - 1\right ) \]

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 [A]  time = 0.223654, size = 39, normalized size = 2.6 \[ -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (-\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) \]

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 = 0.251277, size = 22, normalized size = 1.47 \[ \frac{\log{\left (\tan{\left (x + \frac{\pi }{8} \right )} - 1 \right )}}{2} - \frac{\log{\left (\tan{\left (x + \frac{\pi }{8} \right )} + 1 \right )}}{2} \]

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/XCAS [A]  time = 0.203297, size = 39, normalized size = 2.6 \[ -\frac{1}{4} \,{\rm ln}\left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \,{\rm ln}\left (-\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) \]

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