Optimal. Leaf size=41 \[ -\frac{a^3}{4 x}+\frac{1}{4} a^4 \tanh ^{-1}(a x)-\frac{a}{12 x^3}-\frac{\coth ^{-1}(a x)}{4 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0218977, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5917, 325, 206} \[ -\frac{a^3}{4 x}+\frac{1}{4} a^4 \tanh ^{-1}(a x)-\frac{a}{12 x^3}-\frac{\coth ^{-1}(a x)}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5917
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a x)}{x^5} \, dx &=-\frac{\coth ^{-1}(a x)}{4 x^4}+\frac{1}{4} a \int \frac{1}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a}{12 x^3}-\frac{\coth ^{-1}(a x)}{4 x^4}+\frac{1}{4} a^3 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a}{12 x^3}-\frac{a^3}{4 x}-\frac{\coth ^{-1}(a x)}{4 x^4}+\frac{1}{4} a^5 \int \frac{1}{1-a^2 x^2} \, dx\\ &=-\frac{a}{12 x^3}-\frac{a^3}{4 x}-\frac{\coth ^{-1}(a x)}{4 x^4}+\frac{1}{4} a^4 \tanh ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0118316, size = 57, normalized size = 1.39 \[ -\frac{a^3}{4 x}-\frac{1}{8} a^4 \log (1-a x)+\frac{1}{8} a^4 \log (a x+1)-\frac{a}{12 x^3}-\frac{\coth ^{-1}(a x)}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 47, normalized size = 1.2 \begin{align*} -{\frac{{\rm arccoth} \left (ax\right )}{4\,{x}^{4}}}-{\frac{{a}^{4}\ln \left ( ax-1 \right ) }{8}}-{\frac{a}{12\,{x}^{3}}}-{\frac{{a}^{3}}{4\,x}}+{\frac{{a}^{4}\ln \left ( ax+1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.94443, size = 69, normalized size = 1.68 \begin{align*} \frac{1}{24} \,{\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac{2 \,{\left (3 \, a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a - \frac{\operatorname{arcoth}\left (a x\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5728, size = 100, normalized size = 2.44 \begin{align*} -\frac{6 \, a^{3} x^{3} + 2 \, a x - 3 \,{\left (a^{4} x^{4} - 1\right )} \log \left (\frac{a x + 1}{a x - 1}\right )}{24 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.9597, size = 32, normalized size = 0.78 \begin{align*} \frac{a^{4} \operatorname{acoth}{\left (a x \right )}}{4} - \frac{a^{3}}{4 x} - \frac{a}{12 x^{3}} - \frac{\operatorname{acoth}{\left (a x \right )}}{4 x^{4}} \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\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]