3.2224 \(\int \frac{2 \left (\left (\frac{a}{b}\right )^{\frac{1}{n}}-x \cos \left (\frac{(-1+2 k) \pi }{n}\right )\right )}{\left (\frac{a}{b}\right )^{2/n}+x^2-2 \left (\frac{a}{b}\right )^{\frac{1}{n}} x \cos \left (\frac{(-1+2 k) \pi }{n}\right )} \, dx\)

Optimal. Leaf size=114 \[ 2 \sin \left (\frac{\pi -2 \pi k}{n}\right ) \tan ^{-1}\left (\left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right ) \left (x-\left (\frac{a}{b}\right )^{\frac{1}{n}} \cos \left (\frac{\pi -2 \pi k}{n}\right )\right )\right )-\cos \left (\frac{\pi -2 \pi k}{n}\right ) \log \left (-2 x \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos \left (\frac{\pi -2 \pi k}{n}\right )+\left (\frac{a}{b}\right )^{2/n}+x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.471328, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 66, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.076 \[ 2 \sin \left (\frac{\pi -2 \pi k}{n}\right ) \tan ^{-1}\left (\left (\frac{a}{b}\right )^{-1/n} \csc \left (\frac{\pi -2 \pi k}{n}\right ) \left (x-\left (\frac{a}{b}\right )^{\frac{1}{n}} \cos \left (\frac{\pi -2 \pi k}{n}\right )\right )\right )-\cos \left (\frac{\pi -2 \pi k}{n}\right ) \log \left (-2 x \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos \left (\frac{\pi -2 \pi k}{n}\right )+\left (\frac{a}{b}\right )^{2/n}+x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.121256, size = 111, normalized size = 0.97 \[ 2 \left (\sin \left (\frac{\pi (2 k-1)}{n}\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{\pi (2 k-1)}{2 n}\right ) \left (\left (\frac{a}{b}\right )^{\frac{1}{n}}+x\right )}{\left (\frac{a}{b}\right )^{\frac{1}{n}}-x}\right )-\frac{1}{2} \cos \left (\frac{\pi (2 k-1)}{n}\right ) \log \left (-2 x \left (\frac{a}{b}\right )^{\frac{1}{n}} \cos \left (\frac{\pi (2 k-1)}{n}\right )+\left (\frac{a}{b}\right )^{2/n}+x^2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.044, size = 311, normalized size = 2.7 \[ -\cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \ln \left ( 2\,\sqrt [n]{{\frac{a}{b}}}x\cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) -{x}^{2}- \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}} \right ) +2\,{1\arctan \left ( 1/2\,{1 \left ( 2\,\sqrt [n]{{\frac{a}{b}}}\cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) -2\,x \right ){\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \right ) \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}\sqrt [n]{{\frac{a}{b}}}{\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}}-2\,{1\arctan \left ( 1/2\,{1 \left ( 2\,\sqrt [n]{{\frac{a}{b}}}\cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) -2\,x \right ){\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \right ) \sqrt [n]{{\frac{a}{b}}}{\frac{1}{\sqrt{- \left ( \sqrt [n]{{\frac{a}{b}}} \right ) ^{2} \left ( \cos \left ({\frac{ \left ( 2\,k-1 \right ) \pi }{n}} \right ) \right ) ^{2}+ \left ({\frac{a}{b}} \right ) ^{2\,{n}^{-1}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-cos(Pi*(2*k-1)/n)*ln(2*(a/b)^(1/n)*x*cos(Pi*(2*k-1)/n)-x^2-(a/b)^(2/n))+2/(-((a
/b)^(1/n))^2*cos(Pi*(2*k-1)/n)^2+(a/b)^(2/n))^(1/2)*arctan(1/2*(2*(a/b)^(1/n)*co
s(Pi*(2*k-1)/n)-2*x)/(-((a/b)^(1/n))^2*cos(Pi*(2*k-1)/n)^2+(a/b)^(2/n))^(1/2))*c
os(Pi*(2*k-1)/n)^2*(a/b)^(1/n)-2/(-((a/b)^(1/n))^2*cos(Pi*(2*k-1)/n)^2+(a/b)^(2/
n))^(1/2)*arctan(1/2*(2*(a/b)^(1/n)*cos(Pi*(2*k-1)/n)-2*x)/(-((a/b)^(1/n))^2*cos
(Pi*(2*k-1)/n)^2+(a/b)^(2/n))^(1/2))*(a/b)^(1/n)

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Maxima [A]  time = 0.905746, size = 282, normalized size = 2.47 \[ -\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) \log \left (-2 \, x \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) + x^{2} + \left (\frac{a}{b}\right )^{\frac{2}{n}}\right ) - \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \log \left (\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) + \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} - x}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - \sqrt{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} - x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24208, size = 219, normalized size = 1.92 \[ -\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) \log \left (-\frac{2 \,{\left (2 \, x \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - x^{2} - \left (\frac{a}{b}\right )^{\frac{2}{n}}\right )}}{\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) + 1}\right ) - 2 \, \arctan \left (\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - x}{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \sin \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )}\right ) \sin \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.57287, size = 177, normalized size = 1.55 \[ - \left (- \sqrt{\left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )}\right ) \log{\left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \left (- \sqrt{\left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )}\right ) \right )} - \left (\sqrt{\left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )}\right ) \log{\left (x - \left (\frac{a}{b}\right )^{\frac{1}{n}} \left (\sqrt{\left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos{\left (\frac{\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos{\left (\frac{2 \pi k}{n} - \frac{\pi }{n} \right )}\right ) \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233119, size = 273, normalized size = 2.39 \[ -\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ){\rm ln}\left (-2 \, x \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) + x^{2} + \left (\frac{a}{b}\right )^{\frac{2}{n}}\right ) - \frac{2 \,{\left (\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} - \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}\right )} \arctan \left (-\frac{\left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )} \cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right ) - x}{\sqrt{-\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} + 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}}\right )}{\sqrt{-\cos \left (\frac{2 \, \pi k}{n} - \frac{\pi }{n}\right )^{2} + 1} \left (\frac{a}{b}\right )^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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