\(\int \frac {1}{a+b (c x^n)^{2/n}} \, dx\) [3031]

Optimal result

Integrand size = 17, antiderivative size = 44 \[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

[Out]

x*arctan((c*x^n)^(1/n)*b^(1/2)/a^(1/2))/((c*x^n)^(1/n))/a^(1/2)/b^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {260, 211} \[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

[In]

Int[(a + b*(c*x^n)^(2/n))^(-1),x]

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[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)]

Rubi steps \begin{align*} \text {integral}& = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

[In]

Integrate[(a + b*(c*x^n)^(2/n))^(-1),x]

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.47 (sec) , antiderivative size = 222, normalized size of antiderivative = 5.05

method	result	size
risch	\(\frac {\arctan \left (\frac {b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\right )}{\sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\)	\(222\)

[In]

int(1/(a+b*(c*x^n)^(2/n)),x,method=_RETURNVERBOSE)

[Out]

1/(a*b/x^2*(x^n)^(2/n)*c^(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*(x^n)^(2/n)*c^(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*b/x^2*(x^n)^(2/n)*c^(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))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.89 \[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\left [-\frac {\sqrt {-a b c^{\frac {2}{n}}} \log \left (\frac {b c^{\frac {2}{n}} x^{2} - 2 \, \sqrt {-a b c^{\frac {2}{n}}} x - a}{b c^{\frac {2}{n}} x^{2} + a}\right )}{2 \, a b c^{\frac {2}{n}}}, \frac {\sqrt {a b c^{\frac {2}{n}}} \arctan \left (\frac {\sqrt {a b c^{\frac {2}{n}}} x}{a}\right )}{a b c^{\frac {2}{n}}}\right ] \]

[In]

integrate(1/(a+b*(c*x^n)^(2/n)),x, algorithm="fricas")

[Out]

[-1/2*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*b*c^(2/n)),
sqrt(a*b*c^(2/n))*arctan(sqrt(a*b*c^(2/n))*x/a)/(a*b*c^(2/n))]

Sympy [F]

\[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\int \frac {1}{a + b \left (c x^{n}\right )^{\frac {2}{n}}}\, dx \]

[In]

integrate(1/(a+b*(c*x**n)**(2/n)),x)

[Out]

Integral(1/(a + b*(c*x**n)**(2/n)), x)

Maxima [F]

\[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\int { \frac {1}{\left (c x^{n}\right )^{\frac {2}{n}} b + a} \,d x } \]

[In]

integrate(1/(a+b*(c*x^n)^(2/n)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^n)^(2/n)*b + a), x)

Giac [F]

\[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\int { \frac {1}{\left (c x^{n}\right )^{\frac {2}{n}} b + a} \,d x } \]

[In]

integrate(1/(a+b*(c*x^n)^(2/n)),x, algorithm="giac")

[Out]

integrate(1/((c*x^n)^(2/n)*b + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{a+b \left (c x^n\right )^{2/n}} \, dx=\int \frac {1}{a+b\,{\left (c\,x^n\right )}^{2/n}} \,d x \]

[In]

int(1/(a + b*(c*x^n)^(2/n)),x)

[Out]

int(1/(a + b*(c*x^n)^(2/n)), x)