3.98 \(\int \frac{1}{(\frac{a}{b})^{2/n}+x^2-2 (\frac{a}{b})^{\frac{1}{n}} x \cos (\frac{\pi -2 k \pi }{n})} \, dx\)

Optimal. Leaf size=62 \[ -\left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi -2 \pi k}{n}\right )-x \left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right )\right ) \]

[Out]

-((ArcTan[Cot[(Pi - 2*k*Pi)/n] - (x*Csc[(Pi - 2*k*Pi)/n])/(a/b)^n^(-1)]*Csc[(Pi - 2*k*Pi)/n])/(a/b)^n^(-1))

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Rubi [A]  time = 0.157198, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {618, 204} \[ -\left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi -2 \pi k}{n}\right )-x \left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[((a/b)^(2/n) + x^2 - 2*(a/b)^n^(-1)*x*Cos[(Pi - 2*k*Pi)/n])^(-1),x]

[Out]

-((ArcTan[Cot[(Pi - 2*k*Pi)/n] - (x*Csc[(Pi - 2*k*Pi)/n])/(a/b)^n^(-1)]*Csc[(Pi - 2*k*Pi)/n])/(a/b)^n^(-1))

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}{\left (\frac{a}{b}\right )^{2/n}+x^2-2 \left (\frac{a}{b}\right )^{\frac{1}{n}} x \cos \left (\frac{\pi -2 k \pi }{n}\right )} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \left (\frac{a}{b}\right )^{2/n} \left (1-\cos ^2\left (\frac{\pi -2 k \pi }{n}\right )\right )} \, dx,x,2 x-2 \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos \left (\frac{\pi -2 k \pi }{n}\right )\right )\right )\\ &=-\left (\frac{a}{b}\right )^{-1/n} \tan ^{-1}\left (\cot \left (\frac{\pi -2 k \pi }{n}\right )-\left (\frac{a}{b}\right )^{-1/n} x \csc \left (\frac{\pi -2 k \pi }{n}\right )\right ) \csc \left (\frac{\pi -2 k \pi }{n}\right )\\ \end{align*}

Mathematica [A]  time = 0.0948207, size = 65, normalized size = 1.05 \[ \left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{\pi -2 \pi k}{2 n}\right ) \left (\left (\frac{a}{b}\right )^{\frac{1}{n}}+x\right )}{\left (\frac{a}{b}\right )^{\frac{1}{n}}-x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a/b)^(2/n) + x^2 - 2*(a/b)^n^(-1)*x*Cos[(Pi - 2*k*Pi)/n])^(-1),x]

[Out]

(ArcTan[(((a/b)^n^(-1) + x)*Tan[(Pi - 2*k*Pi)/(2*n)])/((a/b)^n^(-1) - x)]*Csc[(Pi - 2*k*Pi)/n])/(a/b)^n^(-1)

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Maple [A]  time = 0.423, size = 111, normalized size = 1.8 \begin{align*}{\arctan \left ({\frac{1}{2} \left ( 2\,x-2\,\sqrt [n]{{\frac{a}{b}}}\cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ){\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \right ){\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1/b*a)^(2/n)+x^2-2*(1/b*a)^(1/n)*x*cos((-2*Pi*k+Pi)/n)),x)

[Out]

1/(-((1/b*a)^(1/n))^2*cos(Pi*(2*k-1)/n)^2+(1/b*a)^(2/n))^(1/2)*arctan(1/2*(2*x-2*(1/b*a)^(1/n)*cos(Pi*(2*k-1)/
n))/(-((1/b*a)^(1/n))^2*cos(Pi*(2*k-1)/n)^2+(1/b*a)^(2/n))^(1/2))

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Maxima [B]  time = 1.7059, size = 215, normalized size = 3.47 \begin{align*} \frac{\log \left (\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) + \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} - x}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} - x}\right )}{2 \, \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x*cos((-2*pi*k+pi)/n)),x, algorithm="maxima")

[Out]

1/2*log(((a/b)^(1/n)*cos(2*pi*k/n - pi/n) + sqrt(cos(2*pi*k/n - pi/n)^2 - 1)*(a/b)^(1/n) - x)/((a/b)^(1/n)*cos
(2*pi*k/n - pi/n) - sqrt(cos(2*pi*k/n - pi/n)^2 - 1)*(a/b)^(1/n) - x))/(sqrt(cos(2*pi*k/n - pi/n)^2 - 1)*(a/b)
^(1/n))

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Fricas [A]  time = 2.67609, size = 161, normalized size = 2.6 \begin{align*} -\frac{\arctan \left (\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - x}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \sin \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )}\right )}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \sin \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x*cos((-2*pi*k+pi)/n)),x, algorithm="fricas")

[Out]

-arctan(((a/b)^(1/n)*cos(2*pi*k/n - pi/n) - x)/((a/b)^(1/n)*sin(2*pi*k/n - pi/n)))/((a/b)^(1/n)*sin(2*pi*k/n -
pi/n))

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Sympy [B]  time = 1.40133, size = 212, normalized size = 3.42 \begin{align*} - \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \log{\left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} - \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \left (- 2 \left (\frac{a}{b}\right )^{\frac{2}{n}} \cos ^{2}{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} + 2 \left (\frac{a}{b}\right )^{\frac{2}{n}}\right )}{2} \right )}}{2} + \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \log{\left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} + \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \left (- 2 \left (\frac{a}{b}\right )^{\frac{2}{n}} \cos ^{2}{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} + 2 \left (\frac{a}{b}\right )^{\frac{2}{n}}\right )}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a/b)**(2/n)+x**2-2*(a/b)**(1/n)*x*cos((-2*pi*k+pi)/n)),x)

[Out]

-sqrt((a/b)**(-2/n)/(cos(pi*(2*k - 1)/n)**2 - 1))*log(x - (a/b)**(1/n)*cos(2*pi*k/n - pi/n) - sqrt((a/b)**(-2/
n)/(cos(pi*(2*k - 1)/n)**2 - 1))*(-2*(a/b)**(2/n)*cos(2*pi*k/n - pi/n)**2 + 2*(a/b)**(2/n))/2)/2 + sqrt((a/b)*
*(-2/n)/(cos(pi*(2*k - 1)/n)**2 - 1))*log(x - (a/b)**(1/n)*cos(2*pi*k/n - pi/n) + sqrt((a/b)**(-2/n)/(cos(pi*(
2*k - 1)/n)**2 - 1))*(-2*(a/b)**(2/n)*cos(2*pi*k/n - pi/n)**2 + 2*(a/b)**(2/n))/2)/2

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Giac [A]  time = 1.3673, size = 135, normalized size = 2.18 \begin{align*} \frac{\arctan \left (-\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (-\frac{2 \, \pi k}{n} + \frac{\pi }{n}\right ) - x}{\sqrt{-\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} + 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}}\right )}{\sqrt{-\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} + 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x*cos((-2*pi*k+pi)/n)),x, algorithm="giac")

[Out]

arctan(-((a/b)^(1/n)*cos(-2*pi*k/n + pi/n) - x)/(sqrt(-cos(2*pi*k/n - pi/n)^2 + 1)*(a/b)^(1/n)))/(sqrt(-cos(2*
pi*k/n - pi/n)^2 + 1)*(a/b)^(1/n))