Integrand size = 66, antiderivative size = 104 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-\cos \left (\frac {\pi -2 k \pi }{n}\right ) \log \left (\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )-2 \arctan \left (\cot \left (\frac {\pi -2 k \pi }{n}\right )-\left (\frac {a}{b}\right )^{-1/n} x \csc \left (\frac {\pi -2 k \pi }{n}\right )\right ) \sin \left (\frac {\pi -2 k \pi }{n}\right ) \] Output:
-cos((-2*Pi*k+Pi)/n)*ln((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x*cos((-2*Pi*k+Pi)/n ))+2*arctan(-cot((-2*Pi*k+Pi)/n)+x*csc((-2*Pi*k+Pi)/n)/((a/b)^(1/n)))*sin( (-2*Pi*k+Pi)/n)
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.04 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=2 \left (-\frac {1}{2} \cos \left (\frac {(-1+2 k) \pi }{n}\right ) \log \left (\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )-\arctan \left (\cot \left (\frac {(-1+2 k) \pi }{n}\right )-\left (\frac {a}{b}\right )^{-1/n} x \csc \left (\frac {(-1+2 k) \pi }{n}\right )\right ) \sin \left (\frac {(-1+2 k) \pi }{n}\right )\right ) \] Input:
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]
Output:
2*(-1/2*(Cos[((-1 + 2*k)*Pi)/n]*Log[(a/b)^(2/n) + x^2 - 2*(a/b)^n^(-1)*x*C os[((-1 + 2*k)*Pi)/n]]) - ArcTan[Cot[((-1 + 2*k)*Pi)/n] - (x*Csc[((-1 + 2* k)*Pi)/n])/(a/b)^n^(-1)]*Sin[((-1 + 2*k)*Pi)/n])
Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {27, 1142, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {\pi (2 k-1)}{n}\right )\right )}{-2 x \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi (2 k-1)}{n}\right )+\left (\frac {a}{b}\right )^{2/n}+x^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(1-2 k) \pi }{n}\right )}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {(1-2 k) \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}} \sin ^2\left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {1}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {\pi -2 k \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx-\frac {1}{2} \cos \left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {2 \left (x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {\pi -2 k \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}} \sin ^2\left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {1}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {\pi -2 k \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx-\cos \left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {\pi -2 k \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 \left (-\cos \left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {\pi -2 k \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \sin ^2\left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {1}{-4 \left (1-\cos ^2\left (\frac {\pi -2 k \pi }{n}\right )\right ) \left (\frac {a}{b}\right )^{2/n}-\left (2 x-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )^2}d\left (2 x-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\sin \left (\frac {\pi -2 \pi k}{n}\right ) \arctan \left (\frac {1}{2} \left (\frac {a}{b}\right )^{-1/n} \csc \left (\frac {\pi -2 \pi k}{n}\right ) \left (2 x-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 \pi k}{n}\right )\right )\right )-\cos \left (\frac {\pi -2 \pi k}{n}\right ) \int \frac {x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )}{\left (\frac {a}{b}\right )^{2/n}-2 x \cos \left (\frac {\pi -2 k \pi }{n}\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}+x^2}dx\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\sin \left (\frac {\pi -2 \pi k}{n}\right ) \arctan \left (\frac {1}{2} \left (\frac {a}{b}\right )^{-1/n} \csc \left (\frac {\pi -2 \pi k}{n}\right ) \left (2 x-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 \pi k}{n}\right )\right )\right )-\frac {1}{2} \cos \left (\frac {\pi -2 \pi k}{n}\right ) \log \left (-2 x \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 \pi k}{n}\right )+\left (\frac {a}{b}\right )^{2/n}+x^2\right )\right )\) |
Input:
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]
Output:
2*(-1/2*(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]]) + ArcTan[((2*x - 2*(a/b)^n^(-1)*Cos[(Pi - 2*k*Pi)/n])* Csc[(Pi - 2*k*Pi)/n])/(2*(a/b)^n^(-1))]*Sin[(Pi - 2*k*Pi)/n])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 2.35 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.96
| method | result | size |
| default | \(-\cos \left (\frac {\pi \left (2 k -1\right )}{n}\right ) \ln \left (2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )-x^{2}-\left (\frac {a}{b}\right )^{\frac {2}{n}}\right )+\frac {2 \left (\cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{n}}-\left (\frac {a}{b}\right )^{\frac {1}{n}}\right ) \arctan \left (\frac {2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )-2 x}{2 \sqrt {-\left (\frac {a}{b}\right )^{\frac {2}{n}} \cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )^{2}+\left (\frac {a}{b}\right )^{\frac {2}{n}}}}\right )}{\sqrt {-\left (\frac {a}{b}\right )^{\frac {2}{n}} \cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )^{2}+\left (\frac {a}{b}\right )^{\frac {2}{n}}}}\) | \(204\) |
| risch | \(\text {Expression too large to display}\) | \(860\) |
Input:
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*c os(Pi*(2*k-1)/n)),x,method=_RETURNVERBOSE)
Output:
-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 *(cos(Pi*(2*k-1)/n)^2*(a/b)^(1/n)-(a/b)^(1/n))/(-((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))
Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.56 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) \log \left (-\frac {2 \, {\left (2 \, x \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) - x^{2} - \left (\frac {a}{b}\right )^{\frac {2}{n}}\right )}}{\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) + 1}\right ) - 2 \, \arctan \left (\frac {\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) - x}{\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \sin \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )}\right ) \sin \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) \] Input:
integrate(2*((a/b)^(1/n)-x*cos((-1+2*k)*pi/n))/((a/b)^(2/n)+x^2-2*(a/b)^(1 /n)*x*cos((-1+2*k)*pi/n)),x, algorithm="fricas")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (83) = 166\).
Time = 0.47 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.70 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=- \left (- \sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \log {\left (x - \left (\frac {a}{b}\right )^{\frac {1}{n}} \left (- \sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \right )} - \left (\sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \log {\left (x - \left (\frac {a}{b}\right )^{\frac {1}{n}} \left (\sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \right )} \] Input:
integrate(2*((a/b)**(1/n)-x*cos((-1+2*k)*pi/n))/((a/b)**(2/n)+x**2-2*(a/b) **(1/n)*x*cos((-1+2*k)*pi/n)),x)
Output:
-(-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)))
Exception generated. \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(2*((a/b)^(1/n)-x*cos((-1+2*k)*pi/n))/((a/b)^(2/n)+x^2-2*(a/b)^(1 /n)*x*cos((-1+2*k)*pi/n)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(1>0)', see `assume?` for more de tails)Is 1
Time = 0.42 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.94 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) \log \left (-2 \, x \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) + x^{2} + \left (\frac {a}{b}\right )^{\frac {2}{n}}\right ) - \frac {2 \, {\left (\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )^{2} - \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )}\right )} \arctan \left (-\frac {\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) - x}{\sqrt {-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )^{2} + 1} \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )}}\right )}{\sqrt {-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )^{2} + 1} \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )}} \] Input:
integrate(2*((a/b)^(1/n)-x*cos((-1+2*k)*pi/n))/((a/b)^(2/n)+x^2-2*(a/b)^(1 /n)*x*cos((-1+2*k)*pi/n)),x, algorithm="giac")
Output:
-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)) - 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))
Time = 5.50 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.71 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-2\,\mathrm {atan}\left (\frac {2\,x\,\sqrt {1-{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2}-2\,\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )\,{\left (\frac {a}{b}\right )}^{1/n}\,\sqrt {1-{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2}}{2\,{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2\,{\left (\frac {a}{b}\right )}^{1/n}-2\,{\left (\frac {a}{b}\right )}^{1/n}}\right )\,\sqrt {1-{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2}-\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )\,\ln \left ({\left (\frac {a}{b}\right )}^{2/n}+x^2-2\,x\,\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )\,{\left (\frac {a}{b}\right )}^{1/n}\right ) \] Input:
int((2*(a/b)^(1/n) - 2*x*cos((Pi*(2*k - 1))/n))/((a/b)^(2/n) + x^2 - 2*x*c os((Pi*(2*k - 1))/n)*(a/b)^(1/n)),x)
Output:
- 2*atan((2*x*(1 - cos((Pi*(2*k - 1))/n)^2)^(1/2) - 2*cos((Pi*(2*k - 1))/n )*(a/b)^(1/n)*(1 - cos((Pi*(2*k - 1))/n)^2)^(1/2))/(2*cos((Pi*(2*k - 1))/n )^2*(a/b)^(1/n) - 2*(a/b)^(1/n)))*(1 - cos((Pi*(2*k - 1))/n)^2)^(1/2) - co s((Pi*(2*k - 1))/n)*log((a/b)^(2/n) + x^2 - 2*x*cos((Pi*(2*k - 1))/n)*(a/b )^(1/n))
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.40 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-2 \sqrt {-\cos \left (\frac {2 k \pi -\pi }{n}\right )^{2}+1}\, \mathit {atan} \left (\frac {a^{\frac {1}{n}} \cos \left (\frac {2 k \pi -\pi }{n}\right )-b^{\frac {1}{n}} x}{a^{\frac {1}{n}} \sqrt {-\cos \left (\frac {2 k \pi -\pi }{n}\right )^{2}+1}}\right )-\cos \left (\frac {2 k \pi -\pi }{n}\right ) \mathrm {log}\left (2 b^{\frac {1}{n}} a^{\frac {1}{n}} \cos \left (\frac {2 k \pi -\pi }{n}\right ) x -a^{\frac {2}{n}}-b^{\frac {2}{n}} x^{2}\right ) \] Input:
int(2*((a/b)^(1/n)-x*cos((-1+2*k)*Pi/n))/((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x* cos((-1+2*k)*Pi/n)),x)
Output:
- 2*sqrt( - cos((2*k*pi - pi)/n)**2 + 1)*atan((a**(1/n)*cos((2*k*pi - pi) /n) - b**(1/n)*x)/(a**(1/n)*sqrt( - cos((2*k*pi - pi)/n)**2 + 1))) - cos(( 2*k*pi - pi)/n)*log(2*b**(1/n)*a**(1/n)*cos((2*k*pi - pi)/n)*x - a**(2/n) - b**(2/n)*x**2)