Integrand size = 8, antiderivative size = 30 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=-\frac {\cot ^{-1}(a x)}{x}-a \log (x)+\frac {1}{2} a \log \left (1+a^2 x^2\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4947, 272, 36, 29, 31} \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=\frac {1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac {\cot ^{-1}(a x)}{x} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 272
Rule 4947
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^{-1}(a x)}{x}-a \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a x)}{x}-\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {\cot ^{-1}(a x)}{x}-\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {\cot ^{-1}(a x)}{x}-a \log (x)+\frac {1}{2} a \log \left (1+a^2 x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=-\frac {\cot ^{-1}(a x)}{x}-a \log (x)+\frac {1}{2} a \log \left (1+a^2 x^2\right ) \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
method | result | size |
parts | \(-\frac {\operatorname {arccot}\left (a x \right )}{x}-a \left (\ln \left (x \right )-\frac {\ln \left (a^{2} x^{2}+1\right )}{2}\right )\) | \(29\) |
parallelrisch | \(-\frac {2 a \ln \left (x \right ) x -a \ln \left (a^{2} x^{2}+1\right ) x +2 \,\operatorname {arccot}\left (a x \right )}{2 x}\) | \(33\) |
derivativedivides | \(a \left (-\frac {\operatorname {arccot}\left (a x \right )}{a x}+\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\ln \left (a x \right )\right )\) | \(34\) |
default | \(a \left (-\frac {\operatorname {arccot}\left (a x \right )}{a x}+\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\ln \left (a x \right )\right )\) | \(34\) |
risch | \(-\frac {i \ln \left (i a x +1\right )}{2 x}-\frac {2 a \ln \left (x \right ) x -a \ln \left (a^{2} x^{2}+1\right ) x -i \ln \left (-i a x +1\right )+\pi }{2 x}\) | \(54\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=\frac {a x \log \left (a^{2} x^{2} + 1\right ) - 2 \, a x \log \left (x\right ) - 2 \, \operatorname {arccot}\left (a x\right )}{2 \, x} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=- a \log {\left (x \right )} + \frac {a \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {\operatorname {acot}{\left (a x \right )}}{x} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=\frac {1}{2} \, a {\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac {\operatorname {arccot}\left (a x\right )}{x} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=-\frac {1}{2} \, a {\left (\frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x} - \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^{-1}(a x)}{x^2} \, dx=\frac {a\,\left (\ln \left (a^2\,x^2+1\right )-2\,\ln \left (x\right )\right )}{2}-\frac {\mathrm {acot}\left (a\,x\right )}{x} \]
[In]
[Out]