Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.013011, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {240, 212, 206, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 240
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\\ \end{align*}
Mathematica [A] time = 0.0293327, size = 76, normalized size = 1.33 \[ \frac{-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4 \sqrt [4]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}\, dx \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 = 1.6457, size = 281, normalized size = 4.93 \begin{align*} \frac{\arctan \left (\frac{\frac{x \sqrt{\frac{\sqrt{b} x^{2} + \sqrt{b x^{4} + a}}{x^{2}}}}{b^{\frac{1}{4}}} - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{b^{\frac{1}{4}}}}{x}\right )}{b^{\frac{1}{4}}} + \frac{\log \left (\frac{b^{\frac{1}{4}} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}{4 \, b^{\frac{1}{4}}} - \frac{\log \left (-\frac{b^{\frac{1}{4}} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}{4 \, b^{\frac{1}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.27058, size = 36, normalized size = 0.63 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.14823, size = 278, normalized size = 4.88 \begin{align*} \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{4 \, b} + \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{4 \, b} - \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}} \log \left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{8 \, b} + \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}} \log \left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]