Optimal. Leaf size=105 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \]
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Rubi [A] time = 0.03, antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {398} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin {align*} \int \frac {1}{\left (-2 b+a x^2\right ) \sqrt [4]{-b+a x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 163, normalized size = 1.55 \begin {gather*} -\frac {6 b x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {a x^2}{b},\frac {a x^2}{2 b}\right )}{\left (2 b-a x^2\right ) \sqrt [4]{a x^2-b} \left (a x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};\frac {a x^2}{b},\frac {a x^2}{2 b}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};\frac {a x^2}{b},\frac {a x^2}{2 b}\right )\right )+6 b F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {a x^2}{b},\frac {a x^2}{2 b}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.19, size = 105, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{2 \sqrt {2} \sqrt {a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 116.86, size = 338, normalized size = 3.22 \begin {gather*} -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b^{3} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {a x^{2} - b} b \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a b \sqrt {\frac {1}{a^{2} b^{3}}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{2} - b} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{2} \sqrt {\frac {1}{a^{2} b^{3}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{a x^{2} - 2 \, b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{2} - b} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {3}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{2} \sqrt {\frac {1}{a^{2} b^{3}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{2} b^{3}}\right )^{\frac {1}{4}} - {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{a x^{2} - 2 \, 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 (a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - 2 \, b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{2}-2 b \right ) \left (a \,x^{2}-b \right )^{\frac {1}{4}}}\, dx\]
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 \begin {gather*} \int \frac {1}{{\left (a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - 2 \, b\right )}}\,{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 (a\,x^2-b\right )}^{1/4}\,\left (2\,b-a\,x^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}{\left (a x^{2} - 2 b\right ) \sqrt [4]{a x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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