3.113 \(\int \frac{1}{\cos (x)+\sin (x)} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\tanh ^{-1}\left (\frac{\cos (x)-\sin (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTanh[(Cos[x] - Sin[x])/Sqrt[2]]/Sqrt[2])

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Rubi [A]  time = 0.0186006, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{\tanh ^{-1}\left (\frac{\cos (x)-\sin (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(Cos[x] + Sin[x])^(-1),x]

[Out]

-(ArcTanh[(Cos[x] - Sin[x])/Sqrt[2]]/Sqrt[2])

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Rubi in Sympy [A]  time = 0.541286, size = 22, normalized size = 1.05 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (- \sin{\left (x \right )} + \cos{\left (x \right )}\right )}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(cos(x)+sin(x)),x)

[Out]

-sqrt(2)*atanh(sqrt(2)*(-sin(x) + cos(x))/2)/2

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Mathematica [C]  time = 0.0238186, size = 24, normalized size = 1.14 \[ (-1-i) (-1)^{3/4} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )-1}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Cos[x] + Sin[x])^(-1),x]

[Out]

(-1 - I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]]

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Maple [A]  time = 0.033, size = 19, normalized size = 0.9 \[ \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tan \left ( x/2 \right ) -2 \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(cos(x)+sin(x)),x)

[Out]

2^(1/2)*arctanh(1/4*(2*tan(1/2*x)-2)*2^(1/2))

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Maxima [A]  time = 1.61631, size = 54, normalized size = 2.57 \[ -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}}{2 \, \sqrt{2} + \frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(cos(x) + sin(x)),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*log(-2*(sqrt(2) - sin(x)/(cos(x) + 1) + 1)/((2*sqrt(2)) + 2*sin(x)/
(cos(x) + 1) - 2))

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Fricas [A]  time = 0.215725, size = 51, normalized size = 2.43 \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2} - \cos \left (x\right )\right )} \sin \left (x\right ) - 2 \, \sqrt{2} \cos \left (x\right ) + 3}{2 \, \cos \left (x\right ) \sin \left (x\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(cos(x) + sin(x)),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log((2*(sqrt(2) - cos(x))*sin(x) - 2*sqrt(2)*cos(x) + 3)/(2*cos(x)*s
in(x) + 1))

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Sympy [A]  time = 14.5012, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(cos(x)+sin(x)),x)

[Out]

nan

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GIAC/XCAS [A]  time = 0.237818, size = 50, normalized size = 2.38 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \tan \left (\frac{1}{2} \, x\right ) - 2 \right |}}{{\left | 2 \, \sqrt{2} + 2 \, \tan \left (\frac{1}{2} \, x\right ) - 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(cos(x) + sin(x)),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*ln(abs(-2*sqrt(2) + 2*tan(1/2*x) - 2)/abs(2*sqrt(2) + 2*tan(1/2*x)
- 2))