Optimal. Leaf size=30 \[ \frac{1}{4} b \text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\frac{1}{4} b \text{PolyLog}\left (2,\frac{c}{x^2}\right )+a \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0327894, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6095, 5912} \[ \frac{1}{4} b \text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\frac{1}{4} b \text{PolyLog}\left (2,\frac{c}{x^2}\right )+a \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6095
Rule 5912
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )}{x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=a \log (x)+\frac{1}{4} b \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{4} b \text{Li}_2\left (\frac{c}{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0139595, size = 28, normalized size = 0.93 \[ \frac{1}{4} b \left (\text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\text{PolyLog}\left (2,\frac{c}{x^2}\right )\right )+a \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.03, size = 154, normalized size = 5.1 \begin{align*} -a\ln \left ({x}^{-1} \right ) -b\ln \left ({x}^{-1} \right ){\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) +{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1+{\frac{1}{x}\sqrt{-c}} \right ) }+{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1-{\frac{1}{x}\sqrt{-c}} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1+{\frac{1}{x}\sqrt{-c}} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1-{\frac{1}{x}\sqrt{-c}} \right ) }-{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1-{\frac{1}{x}\sqrt{c}} \right ) }-{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1+{\frac{1}{x}\sqrt{c}} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1-{\frac{1}{x}\sqrt{c}} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1+{\frac{1}{x}\sqrt{c}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b \int \frac{\log \left (\frac{c}{x^{2}} + 1\right ) - \log \left (-\frac{c}{x^{2}} + 1\right )}{x}\,{d x} + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]