Optimal. Leaf size=165 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}+\frac {i \text {Li}_2\left (\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}-1\right )}{\sqrt {a} \sqrt {b}}+\frac {i \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {b}}-\frac {2 \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, 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.
[In]
[Out]
Rule 12
Rule 205
Rule 2447
Rule 2526
Rule 4868
Rule 4924
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 = 0.20, size = 373, normalized size = 2.26 \[ \frac {2 \log \left (\sqrt {-a}-\sqrt {b} x\right ) \log \left (\frac {c x^2}{a+b x^2}\right )-2 \log \left (\sqrt {-a}+\sqrt {b} x\right ) \log \left (\frac {c x^2}{a+b x^2}\right )+4 \text {Li}_2\left (\frac {\sqrt {b} x}{\sqrt {-a}}+1\right )-2 \text {Li}_2\left (\frac {a-\sqrt {-a} \sqrt {b} x}{2 a}\right )+2 \text {Li}_2\left (\frac {a+\sqrt {-a} \sqrt {b} x}{2 a}\right )-4 \text {Li}_2\left (\frac {a \sqrt {b} x}{(-a)^{3/2}}+1\right )+\log ^2\left (\sqrt {-a}-\sqrt {b} x\right )-\log ^2\left (\sqrt {-a}+\sqrt {b} x\right )-4 \log \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ) \log \left (\sqrt {-a}-\sqrt {b} x\right )+2 \log \left (\frac {a-\sqrt {-a} \sqrt {b} x}{2 a}\right ) \log \left (\sqrt {-a}-\sqrt {b} x\right )+4 \log \left (\frac {a \sqrt {b} x}{(-a)^{3/2}}\right ) \log \left (\sqrt {-a}+\sqrt {b} x\right )-2 \log \left (\sqrt {-a}+\sqrt {b} x\right ) \log \left (\frac {\sqrt {-a} \sqrt {b} x+a}{2 a}\right )}{4 \sqrt {-a} \sqrt {b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\frac {c \,x^{2}}{b \,x^{2}+a}\right )}{b \,x^{2}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (\frac {c\,x^2}{b\,x^2+a}\right )}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\frac {c x^{2}}{a + b x^{2}} \right )}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________