3.101 \(\int \frac{1}{1+x^2+2 x \cos (\frac{1}{7})} \, dx\)

Optimal. Leaf size=17 \[ \csc \left (\frac{1}{7}\right ) \tan ^{-1}\left (\csc \left (\frac{1}{7}\right ) \left (x+\cos \left (\frac{1}{7}\right )\right )\right ) \]

[Out]

ArcTan[(x + Cos[1/7])*Csc[1/7]]*Csc[1/7]

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Rubi [A]  time = 0.0175999, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {618, 204} \[ \csc \left (\frac{1}{7}\right ) \tan ^{-1}\left (\csc \left (\frac{1}{7}\right ) \left (x+\cos \left (\frac{1}{7}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2 + 2*x*Cos[1/7])^(-1),x]

[Out]

ArcTan[(x + Cos[1/7])*Csc[1/7]]*Csc[1/7]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{1+x^2+2 x \cos \left (\frac{1}{7}\right )} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \sin ^2\left (\frac{1}{7}\right )} \, dx,x,2 x+2 \cos \left (\frac{1}{7}\right )\right )\right )\\ &=\tan ^{-1}\left (\left (x+\cos \left (\frac{1}{7}\right )\right ) \csc \left (\frac{1}{7}\right )\right ) \csc \left (\frac{1}{7}\right )\\ \end{align*}

Mathematica [A]  time = 0.0203081, size = 19, normalized size = 1.12 \[ \csc \left (\frac{1}{7}\right ) \tan ^{-1}\left (\frac{(x-1) \tan \left (\frac{1}{14}\right )}{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2 + 2*x*Cos[1/7])^(-1),x]

[Out]

ArcTan[((-1 + x)*Tan[1/14])/(1 + x)]*Csc[1/7]

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Maple [B]  time = 0.064, size = 33, normalized size = 1.9 \begin{align*}{\frac{1}{\sqrt{1- \left ( \cos \left ({\frac{1}{7}} \right ) \right ) ^{2}}}\arctan \left ({\frac{2\,x+2\,\cos \left ( 1/7 \right ) }{2\,\sqrt{1- \left ( \cos \left ( 1/7 \right ) \right ) ^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x^2+2*x*cos(1/7)),x)

[Out]

1/(1-cos(1/7)^2)^(1/2)*arctan(1/2*(2*x+2*cos(1/7))/(1-cos(1/7)^2)^(1/2))

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Maxima [B]  time = 1.79494, size = 36, normalized size = 2.12 \begin{align*} \frac{\arctan \left (\frac{x + \cos \left (\frac{1}{7}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x^2+2*x*cos(1/7)),x, algorithm="maxima")

[Out]

arctan((x + cos(1/7))/sqrt(-cos(1/7)^2 + 1))/sqrt(-cos(1/7)^2 + 1)

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Fricas [A]  time = 2.25011, size = 57, normalized size = 3.35 \begin{align*} \frac{\arctan \left (\frac{x + \cos \left (\frac{1}{7}\right )}{\sin \left (\frac{1}{7}\right )}\right )}{\sin \left (\frac{1}{7}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x^2+2*x*cos(1/7)),x, algorithm="fricas")

[Out]

arctan((x + cos(1/7))/sin(1/7))/sin(1/7)

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Sympy [C]  time = 0.243859, size = 165, normalized size = 9.71 \begin{align*} - \frac{i \log{\left (x + \cos{\left (\frac{1}{7} \right )} - \frac{i}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} + \frac{i \cos ^{2}{\left (\frac{1}{7} \right )}}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} \right )}}{2 \sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} + \frac{i \log{\left (x + \cos{\left (\frac{1}{7} \right )} - \frac{i \cos ^{2}{\left (\frac{1}{7} \right )}}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} + \frac{i}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} \right )}}{2 \sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x**2+2*x*cos(1/7)),x)

[Out]

-I*log(x + cos(1/7) - I/(sqrt(1 - cos(1/7))*sqrt(cos(1/7) + 1)) + I*cos(1/7)**2/(sqrt(1 - cos(1/7))*sqrt(cos(1
/7) + 1)))/(2*sqrt(1 - cos(1/7))*sqrt(cos(1/7) + 1)) + I*log(x + cos(1/7) - I*cos(1/7)**2/(sqrt(1 - cos(1/7))*
sqrt(cos(1/7) + 1)) + I/(sqrt(1 - cos(1/7))*sqrt(cos(1/7) + 1)))/(2*sqrt(1 - cos(1/7))*sqrt(cos(1/7) + 1))

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Giac [B]  time = 1.40627, size = 36, normalized size = 2.12 \begin{align*} \frac{\arctan \left (\frac{x + \cos \left (\frac{1}{7}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x^2+2*x*cos(1/7)),x, algorithm="giac")

[Out]

arctan((x + cos(1/7))/sqrt(-cos(1/7)^2 + 1))/sqrt(-cos(1/7)^2 + 1)