Optimal. Leaf size=165 \[ \frac{i \text{PolyLog}\left (2,-1+\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}+\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}-\frac{2 \log \left (2-\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.191676, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {205, 2526, 12, 4924, 4868, 2447} \[ \frac{i \text{PolyLog}\left (2,-1+\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}+\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}-\frac{2 \log \left (2-\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2526
Rule 12
Rule 4924
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{c x^2}{a+b x^2}\right )}{a+b x^2} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}-\int \frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} x \left (a+b x^2\right )} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}-\frac{\left (2 \sqrt{a}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{x \left (a+b x^2\right )} \, dx}{\sqrt{b}}\\ &=\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}-\frac{(2 i) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{x \left (i+\frac{\sqrt{b} x}{\sqrt{a}}\right )} \, dx}{\sqrt{a} \sqrt{b}}\\ &=\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (2-\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}+\frac{2 \int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{1+\frac{b x^2}{a}} \, dx}{a}\\ &=\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (2-\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}+\frac{i \text{Li}_2\left (-1+\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [B] time = 0.194038, size = 373, normalized size = 2.26 \[ \frac{4 \text{PolyLog}\left (2,\frac{\sqrt{b} x}{\sqrt{-a}}+1\right )-2 \text{PolyLog}\left (2,\frac{a-\sqrt{-a} \sqrt{b} x}{2 a}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-a} \sqrt{b} x+a}{2 a}\right )-4 \text{PolyLog}\left (2,\frac{a \sqrt{b} x}{(-a)^{3/2}}+1\right )+2 \log \left (\sqrt{-a}-\sqrt{b} x\right ) \log \left (\frac{c x^2}{a+b x^2}\right )-2 \log \left (\sqrt{-a}+\sqrt{b} x\right ) \log \left (\frac{c x^2}{a+b x^2}\right )+\log ^2\left (\sqrt{-a}-\sqrt{b} x\right )-\log ^2\left (\sqrt{-a}+\sqrt{b} x\right )-4 \log \left (\frac{\sqrt{b} x}{\sqrt{-a}}\right ) \log \left (\sqrt{-a}-\sqrt{b} x\right )+2 \log \left (\frac{a-\sqrt{-a} \sqrt{b} x}{2 a}\right ) \log \left (\sqrt{-a}-\sqrt{b} x\right )+4 \log \left (\frac{a \sqrt{b} x}{(-a)^{3/2}}\right ) \log \left (\sqrt{-a}+\sqrt{b} x\right )-2 \log \left (\sqrt{-a}+\sqrt{b} x\right ) \log \left (\frac{\sqrt{-a} \sqrt{b} x+a}{2 a}\right )}{4 \sqrt{-a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}+a}\ln \left ({\frac{c{x}^{2}}{b{x}^{2}+a}} \right ) }\, dx \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\frac{c x^{2}}{a + b x^{2}} \right )}}{a + b x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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