Optimal. Leaf size=77 \[ -\frac{\log \left (a x^4+2 a x^2+a-b\right )}{4 (a-b)}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a-b)}+\frac{\log (x)}{a-b} \]
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Rubi [A] time = 0.0717382, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1114, 705, 29, 634, 618, 206, 628} \[ -\frac{\log \left (a x^4+2 a x^2+a-b\right )}{4 (a-b)}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a-b)}+\frac{\log (x)}{a-b} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 705
Rule 29
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \left (a-b+2 a x+a x^2\right )} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 (a-b)}+\frac{\operatorname{Subst}\left (\int \frac{-2 a-a x}{a-b+2 a x+a x^2} \, dx,x,x^2\right )}{2 (a-b)}\\ &=\frac{\log (x)}{a-b}-\frac{\operatorname{Subst}\left (\int \frac{2 a+2 a x}{a-b+2 a x+a x^2} \, dx,x,x^2\right )}{4 (a-b)}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a-b+2 a x+a x^2} \, dx,x,x^2\right )}{2 (a-b)}\\ &=\frac{\log (x)}{a-b}-\frac{\log \left (a-b+2 a x^2+a x^4\right )}{4 (a-b)}+\frac{a \operatorname{Subst}\left (\int \frac{1}{4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )}{a-b}\\ &=\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \left (1+x^2\right )}{\sqrt{b}}\right )}{2 (a-b) \sqrt{b}}+\frac{\log (x)}{a-b}-\frac{\log \left (a-b+2 a x^2+a x^4\right )}{4 (a-b)}\\ \end{align*}
Mathematica [A] time = 0.0484948, size = 90, normalized size = 1.17 \[ \frac{\left (\sqrt{a}+\sqrt{b}\right ) \log \left (\sqrt{a} \left (x^2+1\right )-\sqrt{b}\right )+\left (\sqrt{b}-\sqrt{a}\right ) \log \left (\sqrt{a} \left (x^2+1\right )+\sqrt{b}\right )-4 \sqrt{b} \log (x)}{4 \sqrt{b} (b-a)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 71, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{a-b}}-{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a-b \right ) }{4\,a-4\,b}}+{\frac{a}{2\,a-2\,b}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53002, size = 350, normalized size = 4.55 \begin{align*} \left [-\frac{\sqrt{\frac{a}{b}} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \,{\left (b x^{2} + b\right )} \sqrt{\frac{a}{b}} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \,{\left (a - b\right )}}, -\frac{2 \, \sqrt{-\frac{a}{b}} \arctan \left (\frac{b \sqrt{-\frac{a}{b}}}{a x^{2} + a}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \,{\left (a - b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.29165, size = 184, normalized size = 2.39 \begin{align*} \left (- \frac{1}{4 \left (a - b\right )} - \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) \log{\left (x^{2} + \frac{4 a b \left (- \frac{1}{4 \left (a - b\right )} - \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a - b\right )} - \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \left (- \frac{1}{4 \left (a - b\right )} + \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) \log{\left (x^{2} + \frac{4 a b \left (- \frac{1}{4 \left (a - b\right )} + \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a - b\right )} + \frac{\sqrt{a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \frac{\log{\left (x \right )}}{a - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.18689, size = 96, normalized size = 1.25 \begin{align*} -\frac{a \arctan \left (\frac{a x^{2} + a}{\sqrt{-a b}}\right )}{2 \, \sqrt{-a b}{\left (a - b\right )}} - \frac{\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \,{\left (a - b\right )}} + \frac{\log \left (x^{2}\right )}{2 \,{\left (a - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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