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

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 = 0.267981, 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 \[ -\left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi -2 \pi k}{n}\right )-x \left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right )\right ) \]

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))

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Rubi in Sympy [A]  time = 80.5349, size = 94, normalized size = 1.52 \[ \frac{\left (\frac{a}{b}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\left (\frac{a}{b}\right )^{- \frac{1}{n}} \left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )}\right )}{\sqrt{- \cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1} \sqrt{\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1}} \right )}}{\sqrt{- \cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1} \sqrt{\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((a/b)**(2/n)+x**2-2*(a/b)**(1/n)*x*cos((-2*pi*k+pi)/n)),x)

[Out]

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

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Mathematica [A]  time = 0.154134, 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[(P
i - 2*k*Pi)/n])/(a/b)^n^(-1)

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Maple [A]  time = 0.044, size = 111, normalized size = 1.8 \[{1\arctan \left ({\frac{1}{2} \left ( 2\,x-2\,\sqrt [n]{{\frac{a}{b}}}\cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ){\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \right ){\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.828108, size = 215, normalized size = 3.47 \[ \frac{\left (\frac{a}{b}\right )^{-\frac{1}{n}} \log \left (\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) + \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} - x}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} - x}\right )}{2 \, \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a/b)^(-1/n)*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)

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Fricas [A]  time = 0.243957, size = 120, normalized size = 1.94 \[ -\frac{\arctan \left (\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - x}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \sin \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )}\right )}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \sin \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(2*x*(a/b)^(1/n)*cos(-2*pi*k/n + pi/n) - x^2 - (a/b)^(2/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 [A]  time = 2.99014, size = 212, normalized size = 3.42 \[ - \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \log{\left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} - \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \left (- 2 \left (\frac{a}{b}\right )^{\frac{2}{n}} \cos ^{2}{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} + 2 \left (\frac{a}{b}\right )^{\frac{2}{n}}\right )}{2} \right )}}{2} + \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \log{\left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} + \frac{\sqrt{\frac{\left (\frac{a}{b}\right )^{- \frac{2}{n}}}{\cos ^{2}{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1}} \left (- 2 \left (\frac{a}{b}\right )^{\frac{2}{n}} \cos ^{2}{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )} + 2 \left (\frac{a}{b}\right )^{\frac{2}{n}}\right )}{2} \right )}}{2} \]

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/XCAS [A]  time = 0.222173, size = 135, normalized size = 2.18 \[ \frac{\arctan \left (-\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (-\frac{2 \, \pi k}{n} + \frac{\pi }{n}\right ) - x}{\sqrt{-\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} + 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}}\right )}{\sqrt{-\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} + 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(2*x*(a/b)^(1/n)*cos(-2*pi*k/n + pi/n) - x^2 - (a/b)^(2/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))