Optimal. Leaf size=28 \[ \frac{1}{10} \text{PolyLog}\left (2,-\frac{1}{a x^5}\right )-\frac{1}{10} \text{PolyLog}\left (2,\frac{1}{a x^5}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0217677, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6096, 5913} \[ \frac{1}{10} \text{PolyLog}\left (2,-\frac{1}{a x^5}\right )-\frac{1}{10} \text{PolyLog}\left (2,\frac{1}{a x^5}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6096
Rule 5913
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\coth ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=\frac{1}{10} \text{Li}_2\left (-\frac{1}{a x^5}\right )-\frac{1}{10} \text{Li}_2\left (\frac{1}{a x^5}\right )\\ \end{align*}
Mathematica [A] time = 0.0134413, size = 26, normalized size = 0.93 \[ \frac{1}{10} \left (\text{PolyLog}\left (2,-\frac{1}{a x^5}\right )-\text{PolyLog}\left (2,\frac{1}{a x^5}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.146, size = 85, normalized size = 3. \begin{align*} \ln \left ( x \right ){\rm arccoth} \left (a{x}^{5}\right )+{\frac{1}{2}\sum _{{\it \_R1}={\it RootOf} \left ( a{{\it \_Z}}^{5}-1 \right ) }\ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) }-{\frac{1}{2}\sum _{{\it \_R1}={\it RootOf} \left ( a{{\it \_Z}}^{5}+1 \right ) }\ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.990651, size = 140, normalized size = 5. \begin{align*} -\frac{1}{2} \, a{\left (\frac{\log \left (a x^{5} + 1\right )}{a} - \frac{\log \left (a x^{5} - 1\right )}{a}\right )} \log \left (x\right ) - \frac{1}{10} \, a{\left (\frac{\log \left (a x^{5} - 1\right ) \log \left (a x^{5}\right ) +{\rm Li}_2\left (-a x^{5} + 1\right )}{a} - \frac{\log \left (a x^{5} + 1\right ) \log \left (-a x^{5}\right ) +{\rm Li}_2\left (a x^{5} + 1\right )}{a}\right )} + \operatorname{arcoth}\left (a x^{5}\right ) \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{\operatorname{arcoth}\left (a x^{5}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \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{\operatorname{arcoth}\left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]