Integrand size = 26, antiderivative size = 169 \[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\frac {2 \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} n}-\frac {2 \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} n} \]
2*arctan(x^(1/2*n)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c ^(1/2)/n/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-2*arctan(x^(1/2*n )*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)/n/(-4*a*c+ b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.86 \[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\frac {2 \sqrt {2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} n} \]
(2*Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b - Sqrt[b^2 - 4 *a*c]]]/Sqrt[b - Sqrt[b^2 - 4*a*c]] - ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqr t[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[b^2 - 4*a*c] *n)
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1717, 1406, 218}
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.
| \(\displaystyle \int \frac {x^{\frac {n}{2}-1}}{a+b x^n+c x^{2 n}} \, dx\) |
| \(\Big \downarrow \) 1717 |
| \(\displaystyle \frac {2 \int \frac {1}{b x^n+c x^{2 n}+a}dx^{n/2}}{n}\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \frac {2 \left (\frac {c \int \frac {1}{c x^n+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx^{n/2}}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{c x^n+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx^{n/2}}{\sqrt {b^2-4 a c}}\right )}{n}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{n}\) |
(2*((Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[ c]*ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^ 2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/n
3.6.58.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) Subst[Int[(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[ 2*(n/(m + 1))])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && ! IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{3} c^{2} n^{4}-8 a^{2} b^{2} c \,n^{4}+a \,b^{4} n^{4}\right ) \textit {\_Z}^{4}+\left (-4 a b c \,n^{2}+b^{3} n^{2}\right ) \textit {\_Z}^{2}+c \right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{2}}+\left (4 n^{3} b \,a^{2}-\frac {n^{3} b^{3} a}{c}\right ) \textit {\_R}^{3}+\left (2 a n -\frac {n \,b^{2}}{c}\right ) \textit {\_R} \right )\) | \(114\) |
sum(_R*ln(x^(1/2*n)+(4*n^3*b*a^2-1/c*n^3*b^3*a)*_R^3+(2*a*n-1/c*n*b^2)*_R) ,_R=RootOf((16*a^3*c^2*n^4-8*a^2*b^2*c*n^4+a*b^4*n^4)*_Z^4+(-4*a*b*c*n^2+b ^3*n^2)*_Z^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 801, normalized size of antiderivative = 4.74 \[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) \]
1/2*sqrt(2)*sqrt(-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2))*log((4*c*x*x^(1/2*n - 1) + sqrt(2)*((a*b^3 - 4*a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - (b^2 - 4*a*c)*n)*sqrt (-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2)))/x) - 1/2*sqrt(2)*sqrt(-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2 *b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2))*log((4*c*x*x^(1/2*n - 1) - sqrt(2)*((a*b^3 - 4*a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - (b^2 - 4*a*c)*n)*sqrt(-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)* n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2)))/x) - 1/2*sqrt(2)*sqrt(((a*b^2 - 4*a^2 *c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2))*lo g((4*c*x*x^(1/2*n - 1) + sqrt(2)*((a*b^3 - 4*a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + (b^2 - 4*a*c)*n)*sqrt(((a*b^2 - 4*a^2*c)*n^2*sqrt(1/(( a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2)))/x) + 1/2*sqrt(2)*s qrt(((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2))*log((4*c*x*x^(1/2*n - 1) - sqrt(2)*((a*b^3 - 4*a^2*b*c)*n ^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + (b^2 - 4*a*c)*n)*sqrt(((a*b^2 - 4*a ^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2))) /x)
\[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{2} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \]
\[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{2} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1037 vs. \(2 (129) = 258\).
Time = 0.68 (sec) , antiderivative size = 1037, normalized size of antiderivative = 6.14 \[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\frac {\frac {{\left (\sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} + 16 \, a b^{2} c^{2} + 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 32 \, a^{2} c^{3} - 8 \, a b c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{n}}}{\sqrt {\frac {b + \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{{\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}} + \frac {{\left (\sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} c + 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} - 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a c^{3} + 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{n}}}{\sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{{\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}}}{2 \, n} \]
1/2*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 3*c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqr t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 16*a*b^2*c^2 + 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 - 2*(b^ 2 - 4*a*c)*b*c^2)*arctan(2*sqrt(1/2)*sqrt(x^n)/sqrt((b + sqrt(b^2 - 4*a*c) )/c))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2 *c^2 - 4*a^2*c^3)*abs(c)) + (sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4* a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt (2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 16*a*b^2*c^2 - 2*b^3*c^2 - 4 *sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 + 8*a*b*c^3 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2)* sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c - 2*sqrt(2)*sqr...
Timed out. \[ \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{2}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]