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

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 = 0.0096234, 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, Rules used = {3074, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\cos (x)-\sin (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x] + Sin[x])^(-1),x]

[Out]

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

Rule 3074

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Dist[d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
0]

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

Mathematica [C]  time = 0.0202798, 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 = 0.019, size = 19, normalized size = 0.9 \begin{align*} \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tan \left ( x/2 \right ) -2 \right ) } \right ) \end{align*}

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 [B]  time = 1.4082, size = 53, normalized size = 2.52 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1}{\sqrt{2} + \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1}\right ) \end{align*}

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

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Fricas [B]  time = 2.22559, size = 126, normalized size = 6. \begin{align*} \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 ) \end{align*}

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)*sin(x) + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)+sin(x)),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.12603, size = 50, normalized size = 2.38 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \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 ) \end{align*}

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