Integrand size = 8, antiderivative size = 31 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=-\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {arctanh}(a x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6038, 331, 212} \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=\frac {1}{2} a^2 \text {arctanh}(a x)-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \]
[In]
[Out]
Rule 212
Rule 331
Rule 6038
Rubi steps \begin{align*} \text {integral}& = -\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx \\ & = -\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^3 \int \frac {1}{1-a^2 x^2} \, dx \\ & = -\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {arctanh}(a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=-\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {1}{4} a^2 \log (1-a x)+\frac {1}{4} a^2 \log (1+a x) \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(-\frac {-\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}+a x +\operatorname {arccoth}\left (a x \right )}{2 x^{2}}\) | \(26\) |
parts | \(-\frac {\operatorname {arccoth}\left (a x \right )}{2 x^{2}}-\frac {a \left (-\frac {a \ln \left (a x +1\right )}{2}+\frac {1}{x}+\frac {a \ln \left (a x -1\right )}{2}\right )}{2}\) | \(36\) |
derivativedivides | \(a^{2} \left (-\frac {\operatorname {arccoth}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\ln \left (a x +1\right )}{4}-\frac {\ln \left (a x -1\right )}{4}-\frac {1}{2 a x}\right )\) | \(42\) |
default | \(a^{2} \left (-\frac {\operatorname {arccoth}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\ln \left (a x +1\right )}{4}-\frac {\ln \left (a x -1\right )}{4}-\frac {1}{2 a x}\right )\) | \(42\) |
risch | \(-\frac {\ln \left (a x +1\right )}{4 x^{2}}-\frac {\ln \left (-a x +1\right ) a^{2} x^{2}-\ln \left (-a x -1\right ) a^{2} x^{2}+2 a x -\ln \left (a x -1\right )}{4 x^{2}}\) | \(60\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=-\frac {2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{4 \, x^{2}} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=\frac {a^{2} \operatorname {acoth}{\left (a x \right )}}{2} - \frac {a}{2 x} - \frac {\operatorname {acoth}{\left (a x \right )}}{2 x^{2}} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=\frac {1}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a - \frac {\operatorname {arcoth}\left (a x\right )}{2 \, x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.52 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=a {\left (\frac {a}{\frac {a x + 1}{a x - 1} + 1} + \frac {{\left (a x + 1\right )} a \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}}\right )} \]
[In]
[Out]
Time = 4.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {\coth ^{-1}(a x)}{x^3} \, dx=\frac {a\,\mathrm {atan}\left (\frac {a^2\,x}{\sqrt {-a^2}}\right )\,\sqrt {-a^2}}{2}-\frac {\frac {\mathrm {acoth}\left (a\,x\right )}{2}+\frac {a\,x}{2}}{x^2} \]
[In]
[Out]