Optimal. Leaf size=77 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {398} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin {align*} \int \frac {1}{\left (-2+b x^2\right ) \sqrt [4]{-1+b x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1+b x^2}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1+b x^2}}\right )}{2 \sqrt {2} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 132, normalized size = 1.71 \[ \frac {6 x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};b x^2,\frac {b x^2}{2}\right )}{\left (b x^2-2\right ) \sqrt [4]{b x^2-1} \left (b x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};b x^2,\frac {b x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};b x^2,\frac {b x^2}{2}\right )\right )+6 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};b x^2,\frac {b x^2}{2}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 13.27, size = 274, normalized size = 3.56 \[ \left [\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {b} x}\right ) + \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} - 2 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} b^{\frac {3}{2}} x^{3} + 4 \, \sqrt {b x^{2} - 1} b x^{2} + 4 \, b x^{2} - 4 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{8 \, b}, \frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b}}{b x}\right ) - \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} + 2 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b} b x^{3} - 4 \, \sqrt {b x^{2} - 1} b x^{2} + 4 \, b x^{2} - 4 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {-b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{8 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} - 1\right )}^{\frac {1}{4}} {\left (b x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}-2\right ) \left (b \,x^{2}-1\right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} - 1\right )}^{\frac {1}{4}} {\left (b x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,x^2-1\right )}^{1/4}\,\left (b\,x^2-2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x^{2} - 2\right ) \sqrt [4]{b x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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