Integrand size = 17, antiderivative size = 73 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\frac {x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {260, 205, 211} \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )} \]
[In]
[Out]
Rule 205
Rule 211
Rule 260
Rubi steps \begin{align*} \text {integral}& = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {\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 )}{2 a} \\ & = \frac {x}{2 a \left (a+b \left (c x^n\right )^{2/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 )}{2 a^{3/2} \sqrt {b}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\frac {x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.35 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.18
method | result | size |
risch | \(\frac {x}{2 a \left (a +b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}\right )}+\frac {\arctan \left (\frac {b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\right )}{2 a \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\) | \(305\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.99 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\left [\frac {2 \, a b c^{\frac {2}{n}} x - {\left (b c^{\frac {2}{n}} x^{2} + a\right )} \sqrt {-a b c^{\frac {2}{n}}} \log \left (\frac {b c^{\frac {2}{n}} x^{2} - 2 \, \sqrt {-a b c^{\frac {2}{n}}} x - a}{b c^{\frac {2}{n}} x^{2} + a}\right )}{4 \, {\left (a^{2} b^{2} c^{\frac {4}{n}} x^{2} + a^{3} b c^{\frac {2}{n}}\right )}}, \frac {a b c^{\frac {2}{n}} x + {\left (b c^{\frac {2}{n}} x^{2} + a\right )} \sqrt {a b c^{\frac {2}{n}}} \arctan \left (\frac {\sqrt {a b c^{\frac {2}{n}}} x}{a}\right )}{2 \, {\left (a^{2} b^{2} c^{\frac {4}{n}} x^{2} + a^{3} b c^{\frac {2}{n}}\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {2}{n}}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx=\int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{2/n}\right )}^2} \,d x \]
[In]
[Out]