Integrand size = 25, antiderivative size = 124 \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {\arctan \left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {406} \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {\arctan \left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]
[In]
[Out]
Rule 406
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {\arctan \left (\frac {b x^2-2 \sqrt {a} \sqrt {a-b x^2}}{2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a-b x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a-b x^2}}{b x^2+2 \sqrt {a} \sqrt {a-b x^2}}\right )}{4 a^{3/4} \sqrt {b}} \]
[In]
[Out]
\[\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (-b \,x^{2}+2 a \right )}d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 25.30 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.70 \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {-2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=- \int \frac {1}{- 2 a \sqrt [4]{a - b x^{2}} + b x^{2} \sqrt [4]{a - b x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{1/4}\,\left (2\,a-b\,x^2\right )} \,d x \]
[In]
[Out]