3.2.2 \(\int \frac {1}{1+x^2+2 x \cos (\frac {\pi }{7})} \, dx\) [102]

Optimal. Leaf size=23 \[ \tan ^{-1}\left (\cot \left (\frac {\pi }{7}\right )+x \csc \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right ) \]

[Out]

arctan(cot(1/7*Pi)+x*csc(1/7*Pi))*csc(1/7*Pi)

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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {632, 210} \begin {gather*} \csc \left (\frac {\pi }{7}\right ) \text {ArcTan}\left (\csc \left (\frac {\pi }{7}\right ) \left (x+\cos \left (\frac {\pi }{7}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^2 + 2*x*Cos[Pi/7])^(-1),x]

[Out]

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

Rule 210

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

Rule 632

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(23)=46\).
time = 0.03, size = 56, normalized size = 2.43 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [7]{-1}-(-1)^{6/7}+2 x}{\sqrt {2-(-1)^{2/7}+(-1)^{5/7}}}\right )}{\sqrt {2-(-1)^{2/7}+(-1)^{5/7}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*ArcTan[((-1)^(1/7) - (-1)^(6/7) + 2*x)/Sqrt[2 - (-1)^(2/7) + (-1)^(5/7)]])/Sqrt[2 - (-1)^(2/7) + (-1)^(5/7)
]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(17)=34\).
time = 0.94, size = 39, normalized size = 1.70

method	result	size
default	\(\frac {\arctan \left (\frac {2 x +2 \cos \left (\frac {\pi }{7}\right )}{2 \sqrt {-\left (\cos ^{2}\left (\frac {\pi }{7}\right )\right )+1}}\right )}{\sqrt {-\left (\cos ^{2}\left (\frac {\pi }{7}\right )\right )+1}}\)	\(39\)
norman	\(\left (-\frac {4 \left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{5}}{7}+\frac {\left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{4}}{7}-\frac {5 \left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{3}}{7}+\frac {2 \left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{2}}{7}-\frac {6 \cos \left (\frac {\pi }{7}\right )}{7}-\frac {6 i \sin \left (\frac {\pi }{7}\right )}{7}+\frac {3}{7}\right ) \ln \left (\left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{5}-\left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{4}+\left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{3}-\left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{2}+\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )-x -1\right )+\left (\frac {4 \left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{5}}{7}-\frac {\left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{4}}{7}+\frac {5 \left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{3}}{7}-\frac {2 \left (\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )^{2}}{7}+\frac {6 \cos \left (\frac {\pi }{7}\right )}{7}+\frac {6 i \sin \left (\frac {\pi }{7}\right )}{7}-\frac {3}{7}\right ) \ln \left (x +\cos \left (\frac {\pi }{7}\right )+i \sin \left (\frac {\pi }{7}\right )\right )\)	\(253\)
risch	\(\text {Expression too large to display}\)	\(2335\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x^2+2*x*cos(1/7*Pi)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 33, normalized size = 1.43 \begin {gather*} \frac {\arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}}\right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.87, size = 21, normalized size = 0.91 \begin {gather*} \frac {\arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sin \left (\frac {1}{7} \, \pi \right )}\right )}{\sin \left (\frac {1}{7} \, \pi \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.31, size = 70, normalized size = 3.04 \begin {gather*} - \frac {i \log {\left (x + \cos {\left (\frac {\pi }{7} \right )} - \frac {i \left (2 - 2 \cos ^{2}{\left (\frac {\pi }{7} \right )}\right )}{2 \sin {\left (\frac {\pi }{7} \right )}} \right )}}{2 \sin {\left (\frac {\pi }{7} \right )}} + \frac {i \log {\left (x + \cos {\left (\frac {\pi }{7} \right )} + \frac {i \left (2 - 2 \cos ^{2}{\left (\frac {\pi }{7} \right )}\right )}{2 \sin {\left (\frac {\pi }{7} \right )}} \right )}}{2 \sin {\left (\frac {\pi }{7} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(x + cos(pi/7) - I*(2 - 2*cos(pi/7)**2)/(2*sin(pi/7)))/(2*sin(pi/7)) + I*log(x + cos(pi/7) + I*(2 - 2*co
s(pi/7)**2)/(2*sin(pi/7)))/(2*sin(pi/7))

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Giac [A]
time = 1.60, size = 33, normalized size = 1.43 \begin {gather*} \frac {\arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}}\right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.30, size = 42, normalized size = 1.83 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x+\cos \left (\frac {\Pi }{7}\right )}{\sqrt {\cos \left (\frac {\Pi }{7}\right )-1}\,\sqrt {\cos \left (\frac {\Pi }{7}\right )+1}}\right )}{\sqrt {\cos \left (\frac {\Pi }{7}\right )-1}\,\sqrt {\cos \left (\frac {\Pi }{7}\right )+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 + 2*x*cos(Pi/7) + 1),x)

[Out]

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

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