Optimal. Leaf size=34 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0142081, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1593, 325, 205} \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1593
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{b x^2+c x^4} \, dx &=\int \frac{1}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac{1}{b x}-\frac{c \int \frac{1}{b+c x^2} \, dx}{b}\\ &=-\frac{1}{b x}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0123175, size = 34, normalized size = 1. \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 30, normalized size = 0.9 \begin{align*} -{\frac{1}{bx}}-{\frac{c}{b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \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 [A] time = 1.50907, size = 173, normalized size = 5.09 \begin{align*} \left [\frac{x \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) + 1}{b x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.376627, size = 65, normalized size = 1.91 \begin{align*} \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (- \frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (\frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21555, size = 39, normalized size = 1.15 \begin{align*} -\frac{c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]