Integrand size = 17, antiderivative size = 73 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\frac {x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \] Output:
1/2*x/a/(a+b*(c*x^n)^(2/n))+1/2*x*arctan(b^(1/2)*(c*x^n)^(1/n)/a^(1/2))/a^ (3/2)/b^(1/2)/((c*x^n)^(1/n))
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\frac {x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \] Input:
Integrate[(a + b*(c*x^n)^(2/n))^(-2),x]
Output:
x/(2*a*(a + b*(c*x^n)^(2/n))) + (x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a] ])/(2*a^(3/2)*Sqrt[b]*(c*x^n)^n^(-1))
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {786, 215, 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.
\(\displaystyle \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx\) |
\(\Big \downarrow \) 786 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \int \frac {1}{\left (b \left (c x^n\right )^{2/n}+a\right )^2}d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\int \frac {1}{b \left (c x^n\right )^{2/n}+a}d\left (c x^n\right )^{\frac {1}{n}}}{2 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}\right )\) |
Input:
Int[(a + b*(c*x^n)^(2/n))^(-2),x]
Output:
(x*((c*x^n)^n^(-1)/(2*a*(a + b*(c*x^n)^(2/n))) + ArcTan[(Sqrt[b]*(c*x^n)^n ^(-1))/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])))/(c*x^n)^n^(-1)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> Simp[x/(c*x^q )^(1/q) Subst[Int[(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{ a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.18
method | result | size |
risch | \(\frac {x}{2 a \left (b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}+a \right )}+\frac {\arctan \left (\frac {b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x \sqrt {\frac {a b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\right )}{2 a \sqrt {\frac {a b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\) | \(305\) |
Input:
int(1/(a+b*(c*x^n)^(2/n))^2,x,method=_RETURNVERBOSE)
Output:
1/2/a*x/(b*c^(2/n)*(x^n)^(2/n)*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I *c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)+a)+1/2/a/(a*b/x^2*c^(2/n)*(x^n)^(2/n )*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c* x^n))/n))^(1/2)*arctan(b/x*c^(2/n)*(x^n)^(2/n)*exp(I*Pi*csgn(I*c*x^n)*(-cs gn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)/(a*b/x^2*c^(2/n)*(x^ n)^(2/n)*exp(I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-cs gn(I*c*x^n))/n))^(1/2))
Time = 0.11 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.99 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\left [\frac {2 \, a b c^{\frac {2}{n}} x - {\left (b c^{\frac {2}{n}} x^{2} + a\right )} \sqrt {-a b c^{\frac {2}{n}}} \log \left (\frac {b c^{\frac {2}{n}} x^{2} - 2 \, \sqrt {-a b c^{\frac {2}{n}}} x - a}{b c^{\frac {2}{n}} x^{2} + a}\right )}{4 \, {\left (a^{2} b^{2} c^{\frac {4}{n}} x^{2} + a^{3} b c^{\frac {2}{n}}\right )}}, \frac {a b c^{\frac {2}{n}} x + {\left (b c^{\frac {2}{n}} x^{2} + a\right )} \sqrt {a b c^{\frac {2}{n}}} \arctan \left (\frac {\sqrt {a b c^{\frac {2}{n}}} x}{a}\right )}{2 \, {\left (a^{2} b^{2} c^{\frac {4}{n}} x^{2} + a^{3} b c^{\frac {2}{n}}\right )}}\right ] \] Input:
integrate(1/(a+b*(c*x^n)^(2/n))^2,x, algorithm="fricas")
Output:
[1/4*(2*a*b*c^(2/n)*x - (b*c^(2/n)*x^2 + a)*sqrt(-a*b*c^(2/n))*log((b*c^(2 /n)*x^2 - 2*sqrt(-a*b*c^(2/n))*x - a)/(b*c^(2/n)*x^2 + a)))/(a^2*b^2*c^(4/ n)*x^2 + a^3*b*c^(2/n)), 1/2*(a*b*c^(2/n)*x + (b*c^(2/n)*x^2 + a)*sqrt(a*b *c^(2/n))*arctan(sqrt(a*b*c^(2/n))*x/a))/(a^2*b^2*c^(4/n)*x^2 + a^3*b*c^(2 /n))]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {2}{n}}\right )^{2}}\, dx \] Input:
integrate(1/(a+b*(c*x**n)**(2/n))**2,x)
Output:
Integral((a + b*(c*x**n)**(2/n))**(-2), x)
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*(c*x^n)^(2/n))^2,x, algorithm="maxima")
Output:
1/2*x/(a*b*c^(2/n)*(x^n)^(2/n) + a^2) + integrate(1/2/(a*b*c^(2/n)*(x^n)^( 2/n) + a^2), x)
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*(c*x^n)^(2/n))^2,x, algorithm="giac")
Output:
integrate(((c*x^n)^(2/n)*b + a)^(-2), x)
Timed out. \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{2/n}\right )}^2} \,d x \] Input:
int(1/(a + b*(c*x^n)^(2/n))^2,x)
Output:
int(1/(a + b*(c*x^n)^(2/n))^2, x)
Time = 0.15 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\frac {c^{\frac {2}{n}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{n}} b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{n}} b x}{\sqrt {b}\, \sqrt {a}}\right ) a +c^{\frac {1}{n}} a b x}{2 c^{\frac {1}{n}} a^{2} b \left (c^{\frac {2}{n}} b \,x^{2}+a \right )} \] Input:
int(1/(a+b*(c*x^n)^(2/n))^2,x)
Output:
(c**(2/n)*sqrt(b)*sqrt(a)*atan((c**(1/n)*b*x)/(sqrt(b)*sqrt(a)))*b*x**2 + sqrt(b)*sqrt(a)*atan((c**(1/n)*b*x)/(sqrt(b)*sqrt(a)))*a + c**(1/n)*a*b*x) /(2*c**(1/n)*a**2*b*(c**(2/n)*b*x**2 + a))