Integrand size = 8, antiderivative size = 46 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=\frac {a}{6 x^2}-\frac {\cot ^{-1}(a x)}{3 x^3}+\frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (1+a^2 x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 272, 46} \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=\frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)}{3 x^3}+\frac {a}{6 x^2} \]
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Rule 46
Rule 272
Rule 4947
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{3} a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{6} a \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{6} a \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right ) \\ & = \frac {a}{6 x^2}-\frac {\cot ^{-1}(a x)}{3 x^3}+\frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (1+a^2 x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=-\frac {\cot ^{-1}(a x)}{3 x^3}-\frac {1}{6} a \left (-\frac {1}{x^2}-2 a^2 \log (x)+a^2 \log \left (1+a^2 x^2\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91
method | result | size |
parts | \(-\frac {\operatorname {arccot}\left (a x \right )}{3 x^{3}}-\frac {a \left (-\frac {1}{2 x^{2}}-a^{2} \ln \left (x \right )+\frac {a^{2} \ln \left (a^{2} x^{2}+1\right )}{2}\right )}{3}\) | \(42\) |
derivativedivides | \(a^{3} \left (-\frac {\operatorname {arccot}\left (a x \right )}{3 a^{3} x^{3}}+\frac {1}{6 a^{2} x^{2}}+\frac {\ln \left (a x \right )}{3}-\frac {\ln \left (a^{2} x^{2}+1\right )}{6}\right )\) | \(44\) |
default | \(a^{3} \left (-\frac {\operatorname {arccot}\left (a x \right )}{3 a^{3} x^{3}}+\frac {1}{6 a^{2} x^{2}}+\frac {\ln \left (a x \right )}{3}-\frac {\ln \left (a^{2} x^{2}+1\right )}{6}\right )\) | \(44\) |
parallelrisch | \(\frac {2 a^{3} \ln \left (x \right ) x^{3}-a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-a^{3} x^{3}+a x -2 \,\operatorname {arccot}\left (a x \right )}{6 x^{3}}\) | \(52\) |
risch | \(-\frac {i \ln \left (i a x +1\right )}{6 x^{3}}-\frac {-2 a^{3} \ln \left (x \right ) x^{3}+a^{3} \ln \left (-a^{2} x^{2}-1\right ) x^{3}-i \ln \left (-i a x +1\right )-a x +\pi }{6 x^{3}}\) | \(66\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=-\frac {a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) - a x + 2 \, \operatorname {arccot}\left (a x\right )}{6 \, x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=\frac {a^{3} \log {\left (x \right )}}{3} - \frac {a^{3} \log {\left (a^{2} x^{2} + 1 \right )}}{6} + \frac {a}{6 x^{2}} - \frac {\operatorname {acot}{\left (a x \right )}}{3 x^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=-\frac {1}{6} \, {\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{3 \, x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=\frac {1}{6} \, {\left (a^{2} {\left (\frac {1}{a^{2} x^{2}} - \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} - \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x^{3}}\right )} a \]
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Time = 0.93 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \frac {\cot ^{-1}(a x)}{x^4} \, dx=\left \{\begin {array}{cl} -\frac {\pi }{6\,x^3} & \text {\ if\ \ }a=0\\ \frac {a^4\,\ln \left (x\right )-\frac {a^4\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {a^2}{2\,x^2}}{3\,a}-\frac {\mathrm {acot}\left (a\,x\right )}{3\,x^3} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
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