Integrand size = 17, antiderivative size = 128 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}{-\sqrt {b}+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{b}} \]
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Time = 0.11 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.57, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {272, 65, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}+\sqrt {a x^4-b}+\sqrt {b}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}+\sqrt {a x^4-b}+\sqrt {b}\right )}{4 \sqrt {2} \sqrt [4]{b}} \]
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Rule 65
Rule 210
Rule 272
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
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} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{2 a} \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^4}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^4}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{4 \sqrt {2} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^4}\right )}{4 \sqrt {2} \sqrt [4]{b}} \\ & = \frac {\log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} \sqrt [4]{b}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}+\sqrt {-b+a x^4}\right )}{4 \sqrt {2} \sqrt [4]{b}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}{\sqrt {b}+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{b}} \]
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Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {a \,x^{4}-b}-b^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{4}-b}+b^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{4}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{4}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )\right )}{8 b^{\frac {1}{4}}}\) | \(134\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {1}{4} \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} i \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (i \, b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} i \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (-i \, b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (-b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{4}}} \right )}}{4 \sqrt [4]{a} x \Gamma \left (\frac {5}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} \]
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Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} \]
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Time = 5.81 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.36 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\mathrm {atan}\left (\frac {{\left (a\,x^4-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )-\mathrm {atanh}\left (\frac {{\left (a\,x^4-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{2\,{\left (-b\right )}^{1/4}} \]
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