Optimal. Leaf size=134 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {6-4 \sqrt {2}} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}+a x^2+b}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {6+4 \sqrt {2}} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}+a x^2+b}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.10, antiderivative size = 50, normalized size of antiderivative = 0.37, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1699, 208} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 1699
Rubi steps
\begin {align*} \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx &=b \operatorname {Subst}\left (\int \frac {1}{-b+2 a b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^2+a^2 x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 95, normalized size = 0.71 \begin {gather*} -\frac {i \sqrt {\frac {a^2 x^4}{b^2}+1} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a}{b}} x\right )\right |-1\right )-2 \Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {\frac {i a}{b}} x\right )\right |-1\right )\right )}{\sqrt {\frac {i a}{b}} \sqrt {a^2 x^4+b^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 50, normalized size = 0.37 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 132, normalized size = 0.99 \begin {gather*} \left [\frac {1}{4} \, \sqrt {2} \sqrt {\frac {1}{a b}} \log \left (\frac {a^{2} x^{4} - 2 \, \sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} a b x \sqrt {\frac {1}{a b}} + 2 \, a b x^{2} + b^{2}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ), \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a b}} \arctan \left (\frac {\sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {-\frac {1}{a b}}}{2 \, x}\right )\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 {a x^{2} + b}{\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 38, normalized size = 0.28
method | result | size |
elliptic | \(-\frac {\arctanh \left (\frac {\sqrt {a^{2} x^{4}+b^{2}}\, \sqrt {2}}{2 x \sqrt {a b}}\right ) \sqrt {2}}{2 \sqrt {a b}}\) | \(38\) |
default | \(\frac {\sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticF \left (x \sqrt {\frac {i a}{b}}, i\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}-\frac {2 \sqrt {1-\frac {i a \,x^{2}}{b}}\, \sqrt {1+\frac {i a \,x^{2}}{b}}\, \EllipticPi \left (x \sqrt {\frac {i a}{b}}, -i, \frac {\sqrt {-\frac {i a}{b}}}{\sqrt {\frac {i a}{b}}}\right )}{\sqrt {\frac {i a}{b}}\, \sqrt {a^{2} x^{4}+b^{2}}}\) | \(152\) |
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 {a x^{2} + b}{\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{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 {a\,x^2+b}{\sqrt {a^2\,x^4+b^2}\,\left (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 {a x^{2} + b}{\left (a x^{2} - b\right ) \sqrt {a^{2} x^{4} + b^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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