Optimal. Leaf size=79 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {398} \begin {gather*} -\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}} \end {gather*}
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 = 0.15, size = 137, normalized size = 1.73 \begin {gather*} \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 )}{\sqrt [4]{-b x^2-1} \left (b x^2+2\right ) \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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.14, size = 90, normalized size = 1.14 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-b x^2-1}}{\sqrt {b} x}\right )}{2 \sqrt {2} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x \left (-b x^2-1\right )^{3/4}}{\sqrt {2} \left (b x^2+1\right )}\right )}{2 \sqrt {2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 29.24, size = 273, normalized size = 3.46 \begin {gather*} \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} + 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} + 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {b} - 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} - 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} + 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} - 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {-b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{8 \, b}\right ] \end {gather*}
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 \begin {gather*} \int -\frac {1}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-b \,x^{2}-2\right ) \left (-b \,x^{2}-1\right )^{\frac {1}{4}}}\, dx \end {gather*}
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 \begin {gather*} -\int \frac {1}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
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 \begin {gather*} -\int \frac {1}{{\left (-b\,x^2-1\right )}^{1/4}\,\left (b\,x^2+2\right )} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {1}{b x^{2} \sqrt [4]{- b x^{2} - 1} + 2 \sqrt [4]{- b x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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