Optimal. Leaf size=80 \[ \frac {\log (x) \left (a+b x^2\right )}{a \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1112, 266, 36, 29, 31} \[ \frac {\log (x) \left (a+b x^2\right )}{a \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 1112
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (a b+b^2 x^2\right ) \int \frac {1}{x \left (a b+b^2 x^2\right )} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (a b+b^2 x^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (a b+b^2 x^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 a b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (b \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,x^2\right )}{2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (a+b x^2\right ) \log (x)}{a \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.52 \[ \frac {\left (a+b x^2\right ) \left (2 \log (x)-\log \left (a+b x^2\right )\right )}{2 a \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 18, normalized size = 0.22 \[ -\frac {\log \left (b x^{2} + a\right ) - 2 \, \log \relax (x)}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 33, normalized size = 0.41 \[ \frac {1}{2} \, {\left (\frac {\log \left (x^{2}\right )}{a} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{a}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.49 \[ \frac {\left (b \,x^{2}+a \right ) \left (2 \ln \relax (x )-\ln \left (b \,x^{2}+a \right )\right )}{2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 23, normalized size = 0.29 \[ -\frac {\log \left (b x^{2} + a\right )}{2 \, a} + \frac {\log \left (x^{2}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.45, size = 40, normalized size = 0.50 \[ -\frac {\ln \left (\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {a^2}+a^2+a\,b\,x^2\right )+\ln \left (\frac {1}{x^2}\right )}{2\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 15, normalized size = 0.19 \[ \frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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