Integrand size = 15, antiderivative size = 55 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 65, 304, 209, 212} \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \]
[In]
[Out]
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^4\right ) \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^4}\right )}{a} \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^4}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \]
[In]
[Out]
Time = 0.81 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{\left (a \,x^{4}+b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}\right )}{4 b^{\frac {1}{4}}}\) | \(57\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.51 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=-\frac {\log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {i \, \log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} + i \, b^{\frac {1}{4}}\right )}{4 \, b^{\frac {1}{4}}} - \frac {i \, \log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} - i \, b^{\frac {1}{4}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {\log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}\right )}{4 \, b^{\frac {1}{4}}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt [4]{a} x \Gamma \left (\frac {5}{4}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=\frac {\arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{2 \, b^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (39) = 78\).
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.38 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=-\frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{4 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{4 \, b} + \frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right )}{8 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right )}{8 \, b} \]
[In]
[Out]
Time = 5.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx=\frac {\mathrm {atan}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )-\mathrm {atanh}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )}{2\,b^{1/4}} \]
[In]
[Out]