Optimal. Leaf size=89 \[ -\frac {2}{3} a^4 \log (x)+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {a^3 \cot ^{-1}(a x)}{2 x}-\frac {a^2}{12 x^2}+\frac {1}{3} a^4 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {a \cot ^{-1}(a x)}{6 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4853, 4919, 266, 44, 36, 29, 31, 4885} \[ -\frac {a^2}{12 x^2}+\frac {1}{3} a^4 \log \left (a^2 x^2+1\right )-\frac {2}{3} a^4 \log (x)+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {\cot ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 4853
Rule 4885
Rule 4919
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx &=-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {1}{2} a \int \frac {\cot ^{-1}(a x)}{x^4 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {1}{2} a \int \frac {\cot ^{-1}(a x)}{x^4} \, dx+\frac {1}{2} a^3 \int \frac {\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {1}{6} a^2 \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx+\frac {1}{2} a^3 \int \frac {\cot ^{-1}(a x)}{x^2} \, dx-\frac {1}{2} a^5 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} a^4 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \operatorname {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 {1}{4} a^4 \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {1}{6} a^4 \log (x)+\frac {1}{12} a^4 \log \left (1+a^2 x^2\right )-\frac {1}{4} a^4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} a^6 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)+\frac {1}{3} a^4 \log \left (1+a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 0.91 \[ \frac {\left (a^4 x^4-1\right ) \cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)-\frac {a^2}{12 x^2}-\frac {a \left (3 a^2 x^2-1\right ) \cot ^{-1}(a x)}{6 x^3}+\frac {1}{3} a^4 \log \left (a^2 x^2+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 78, normalized size = 0.88 \[ \frac {4 \, a^{4} x^{4} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{4} x^{4} \log \relax (x) - a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )^{2} - 2 \, {\left (3 \, a^{3} x^{3} - a x\right )} \operatorname {arccot}\left (a x\right )}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 91, normalized size = 1.02 \[ \frac {1}{12} \, {\left ({\left (3 \, \arctan \left (\frac {1}{a x}\right )^{2} - \frac {6 \, \arctan \left (\frac {1}{a x}\right )}{a x} - \frac {1}{a^{2} x^{2}} + \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a^{3} x^{3}} + 4 \, \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} a^{3} - \frac {3 \, \arctan \left (\frac {1}{a x}\right )^{2}}{a x^{4}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 91, normalized size = 1.02 \[ -\frac {\mathrm {arccot}\left (a x \right )^{2}}{4 x^{4}}+\frac {a \,\mathrm {arccot}\left (a x \right )}{6 x^{3}}-\frac {a^{3} \mathrm {arccot}\left (a x \right )}{2 x}-\frac {a^{4} \mathrm {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}-\frac {a^{2}}{12 x^{2}}-\frac {2 a^{4} \ln \left (a x \right )}{3}+\frac {a^{4} \ln \left (a^{2} x^{2}+1\right )}{3}-\frac {a^{4} \arctan \left (a x \right )^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 95, normalized size = 1.07 \[ -\frac {1}{6} \, {\left (3 \, a^{3} \arctan \left (a x\right ) + \frac {3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a \operatorname {arccot}\left (a x\right ) - \frac {{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \relax (x) + 1\right )} a^{2}}{12 \, x^{2}} - \frac {\operatorname {arccot}\left (a x\right )^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 73, normalized size = 0.82 \[ {\mathrm {acot}\left (a\,x\right )}^2\,\left (\frac {a^4}{4}-\frac {1}{4\,x^4}\right )-\frac {2\,a^4\,\ln \relax (x)}{3}+\frac {a^4\,\ln \left (a^2\,x^2+1\right )}{3}-\frac {a^2}{12\,x^2}-\frac {a^2\,\mathrm {acot}\left (a\,x\right )\,\left (\frac {a\,x^2}{2}-\frac {1}{6\,a}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 80, normalized size = 0.90 \[ - \frac {2 a^{4} \log {\relax (x )}}{3} + \frac {a^{4} \log {\left (a^{2} x^{2} + 1 \right )}}{3} + \frac {a^{4} \operatorname {acot}^{2}{\left (a x \right )}}{4} - \frac {a^{3} \operatorname {acot}{\left (a x \right )}}{2 x} - \frac {a^{2}}{12 x^{2}} + \frac {a \operatorname {acot}{\left (a x \right )}}{6 x^{3}} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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