Optimal. Leaf size=36 \[ -\frac{b \text{PolyLog}\left (2,-c x^n\right )}{2 n}+\frac{b \text{PolyLog}\left (2,c x^n\right )}{2 n}+a \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0350596, antiderivative size = 36, 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} \[ -\frac{b \text{PolyLog}\left (2,-c x^n\right )}{2 n}+\frac{b \text{PolyLog}\left (2,c x^n\right )}{2 n}+a \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6095
Rule 5912
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n}\\ &=a \log (x)-\frac{b \text{Li}_2\left (-c x^n\right )}{2 n}+\frac{b \text{Li}_2\left (c x^n\right )}{2 n}\\ \end{align*}
Mathematica [C] time = 0.0750161, size = 39, normalized size = 1.08 \[ \frac{b c x^n \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},c^2 x^{2 n}\right )}{n}+a \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.036, size = 76, normalized size = 2.1 \begin{align*}{\frac{a\ln \left ( c{x}^{n} \right ) }{n}}+{\frac{b\ln \left ( c{x}^{n} \right ){\it Artanh} \left ( c{x}^{n} \right ) }{n}}-{\frac{b{\it dilog} \left ( c{x}^{n} \right ) }{2\,n}}-{\frac{b{\it dilog} \left ( c{x}^{n}+1 \right ) }{2\,n}}-{\frac{b\ln \left ( c{x}^{n} \right ) \ln \left ( c{x}^{n}+1 \right ) }{2\,n}} \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} \,{\left (n \int \frac{\log \left (x\right )}{c x x^{n} + x}\,{d x} + n \int \frac{\log \left (x\right )}{c x x^{n} - x}\,{d x} + \log \left (c x^{n} + 1\right ) \log \left (x\right ) - \log \left (-c x^{n} + 1\right ) \log \left (x\right )\right )} b + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.81852, size = 456, normalized size = 12.67 \begin{align*} -\frac{b n \log \left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - b n \log \left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - b n \log \left (x\right ) \log \left (-\frac{c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1}{c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - 2 \, a n \log \left (x\right ) - b{\rm Li}_2\left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right )\right ) + b{\rm Li}_2\left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \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 (c x^{n} \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 (c x^{n}\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]