Optimal. Leaf size=65 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0220998, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {1152, 217, 203} \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1152
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2}}{\sqrt{a^2-b^2 x^4}} \, dx &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a-b x^2}} \, dx}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x}{\sqrt{a-b x^2}}\right )}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0428736, size = 50, normalized size = 0.77 \[ \frac{i \log \left (\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 69, normalized size = 1.1 \begin{align*}{\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{1}{b} \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }}}}} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \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{\sqrt{b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.98855, size = 270, normalized size = 4.15 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-\frac{2 \, b^{2} x^{4} + a b x^{2} - 2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{-b} x - a^{2}}{b x^{2} + a}\right )}{2 \, b}, -\frac{\arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{b}}{b^{2} x^{3} + a b x}\right )}{\sqrt{b}}\right ] \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{\sqrt{a + b x^{2}}}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, 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{\sqrt{b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]