Optimal. Leaf size=58 \[ \frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0568747, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {514, 446, 80, 63, 208} \[ \frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 514
Rule 446
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{-1+x^2}{\sqrt{a-b+\frac{b}{x^2}} x^3} \, dx &=\int \frac{1-\frac{1}{x^2}}{\sqrt{a-b+\frac{b}{x^2}} x} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1-x}{x \sqrt{a-b+b x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-b+b x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a-b}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \left (-1+\frac{1}{x^2}\right )}\right )}{b}\\ &=\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{\sqrt{a-b}}\right )}{\sqrt{a-b}}\\ \end{align*}
Mathematica [A] time = 0.0609258, size = 100, normalized size = 1.72 \[ \frac{\sqrt{a-b} \left (a x^2-b x^2+b\right )+b x \sqrt{a x^2-b x^2+b} \tanh ^{-1}\left (\frac{x \sqrt{a-b}}{\sqrt{x^2 (a-b)+b}}\right )}{b x^2 \sqrt{a-b} \sqrt{a+b \left (\frac{1}{x^2}-1\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.016, size = 102, normalized size = 1.8 \begin{align*}{\frac{1}{b{x}^{2}}\sqrt{a{x}^{2}-b{x}^{2}+b} \left ( \ln \left ( \sqrt{-b+a}x+\sqrt{a{x}^{2}-b{x}^{2}+b} \right ) bx+\sqrt{a{x}^{2}-b{x}^{2}+b}\sqrt{-b+a} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}-b{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{-b+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.70727, size = 390, normalized size = 6.72 \begin{align*} \left [\frac{\sqrt{a - b} b \log \left (-2 \,{\left (a - b\right )} x^{2} - 2 \, \sqrt{a - b} x^{2} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}} - b\right ) + 2 \,{\left (a - b\right )} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}}}{2 \,{\left (a b - b^{2}\right )}}, \frac{\sqrt{-a + b} b \arctan \left (-\frac{\sqrt{-a + b} x^{2} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}}}{{\left (a - b\right )} x^{2} + b}\right ) +{\left (a - b\right )} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}}}{a b - b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.44019, size = 70, normalized size = 1.21 \begin{align*} - \frac{\begin{cases} - \frac{1}{\sqrt{a} x^{2}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a - b + \frac{b}{x^{2}}}}{b} & \text{otherwise} \end{cases}}{2} - \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a - b}} \sqrt{a - b + \frac{b}{x^{2}}}} \right )}}{\sqrt{- \frac{1}{a - b}} \left (a - b\right )} \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{x^{2} - 1}{\sqrt{a - b + \frac{b}{x^{2}}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]