Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1107, 618, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1107
Rubi steps
\begin {align*} \int \frac {x}{a-b+2 a x^2+a x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a-b+2 a x+a x^2} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{a-b+2 a x^2+a x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.11, size = 91, normalized size = 2.94 \begin {gather*} \left [\frac {\sqrt {a b} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, \sqrt {a b} {\left (x^{2} + 1\right )} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right )}{4 \, a b}, \frac {\sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{a x^{2} + a}\right )}{2 \, a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 23, normalized size = 0.74 \begin {gather*} \frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.84 \begin {gather*} -\frac {\arctanh \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 37, normalized size = 1.19 \begin {gather*} \frac {\log \left (\frac {a x^{2} + a - \sqrt {a b}}{a x^{2} + a + \sqrt {a b}}\right )}{4 \, \sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.34, size = 31, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {a}\,\sqrt {b}\,x^2}{a\,x^2+a-b}\right )}{2\,\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 53, normalized size = 1.71 \begin {gather*} \frac {\sqrt {\frac {1}{a b}} \log {\left (- b \sqrt {\frac {1}{a b}} + x^{2} + 1 \right )}}{4} - \frac {\sqrt {\frac {1}{a b}} \log {\left (b \sqrt {\frac {1}{a b}} + x^{2} + 1 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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