Optimal. Leaf size=124 \[ \frac{\tan ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a-b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a-b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.020752, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {397} \[ \frac{\tan ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a-b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a-b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 397
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx &=\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*}
Mathematica [C] time = 0.153459, size = 162, normalized size = 1.31 \[ \frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right ) \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )+6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{2}+2\,a}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{2} - 2 \, a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- 2 a \sqrt [4]{a - b x^{2}} + b x^{2} \sqrt [4]{a - b x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{2} - 2 \, a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]