Integrand size = 10, antiderivative size = 104 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=-\frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {23 \log \left (1+a^2 x^2\right )}{90 a^6} \]
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Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 5037, 272, 45, 4931, 266, 5005} \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {4 x^2}{45 a^4}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^4}{60 a^2}+\frac {23 \log \left (a^2 x^2+1\right )}{90 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {x^5 \cot ^{-1}(a x)}{15 a} \]
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Rule 45
Rule 266
Rule 272
Rule 4931
Rule 4947
Rule 5005
Rule 5037
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {1}{3} a \int \frac {x^6 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {\int x^4 \cot ^{-1}(a x) \, dx}{3 a}-\frac {\int \frac {x^4 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a} \\ & = \frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {1}{15} \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {\int x^2 \cot ^{-1}(a x) \, dx}{3 a^3}+\frac {\int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^3} \\ & = -\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {1}{30} \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac {\int \cot ^{-1}(a x) \, dx}{3 a^5}-\frac {\int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^5}-\frac {\int \frac {x^3}{1+a^2 x^2} \, dx}{9 a^2} \\ & = \frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {1}{30} \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {\int \frac {x}{1+a^2 x^2} \, dx}{3 a^4}-\frac {\text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )}{18 a^2} \\ & = -\frac {x^2}{30 a^4}+\frac {x^4}{60 a^2}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {\log \left (1+a^2 x^2\right )}{5 a^6}-\frac {\text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{18 a^2} \\ & = -\frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {23 \log \left (1+a^2 x^2\right )}{90 a^6} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.76 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {-16 a^2 x^2+3 a^4 x^4+4 a x \left (15-5 a^2 x^2+3 a^4 x^4\right ) \cot ^{-1}(a x)+30 \left (1+a^6 x^6\right ) \cot ^{-1}(a x)^2+46 \log \left (1+a^2 x^2\right )}{180 a^6} \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(\frac {30 a^{6} x^{6} \operatorname {arccot}\left (a x \right )^{2}+12 a^{5} x^{5} \operatorname {arccot}\left (a x \right )+3 a^{4} x^{4}-20 a^{3} x^{3} \operatorname {arccot}\left (a x \right )+16-16 a^{2} x^{2}+60 \,\operatorname {arccot}\left (a x \right ) a x +30 \operatorname {arccot}\left (a x \right )^{2}+46 \ln \left (a^{2} x^{2}+1\right )}{180 a^{6}}\) | \(90\) |
parts | \(\frac {x^{6} \operatorname {arccot}\left (a x \right )^{2}}{6}+\frac {\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{5}-\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{3}+\operatorname {arccot}\left (a x \right ) a x -\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\frac {a^{4} x^{4}}{20}-\frac {4 a^{2} x^{2}}{15}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{30}-\frac {\arctan \left (a x \right )^{2}}{2}}{3 a^{6}}\) | \(96\) |
derivativedivides | \(\frac {\frac {a^{6} x^{6} \operatorname {arccot}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{15}-\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{9}+\frac {\operatorname {arccot}\left (a x \right ) a x}{3}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{3}+\frac {a^{4} x^{4}}{60}-\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90}-\frac {\arctan \left (a x \right )^{2}}{6}}{a^{6}}\) | \(98\) |
default | \(\frac {\frac {a^{6} x^{6} \operatorname {arccot}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{15}-\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{9}+\frac {\operatorname {arccot}\left (a x \right ) a x}{3}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{3}+\frac {a^{4} x^{4}}{60}-\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90}-\frac {\arctan \left (a x \right )^{2}}{6}}{a^{6}}\) | \(98\) |
risch | \(-\frac {\left (a^{6} x^{6}+1\right ) \ln \left (i a x +1\right )^{2}}{24 a^{6}}+\frac {\left (15 i \pi \,a^{6} x^{6}+15 x^{6} \ln \left (-i a x +1\right ) a^{6}+6 i a^{5} x^{5}-10 i a^{3} x^{3}+30 i a x +15 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{180 a^{6}}+\frac {i x^{3} \ln \left (-i a x +1\right )}{18 a^{3}}-\frac {x^{6} \ln \left (-i a x +1\right )^{2}}{24}+\frac {x^{6} \pi ^{2}}{24}-\frac {i x \ln \left (-i a x +1\right )}{6 a^{5}}+\frac {\pi \,x^{5}}{30 a}-\frac {i \pi \,x^{6} \ln \left (-i a x +1\right )}{12}+\frac {x^{4}}{60 a^{2}}-\frac {\pi \,x^{3}}{18 a^{3}}-\frac {i x^{5} \ln \left (-i a x +1\right )}{30 a}-\frac {4 x^{2}}{45 a^{4}}+\frac {\pi x}{6 a^{5}}-\frac {\ln \left (-i a x +1\right )^{2}}{24 a^{6}}-\frac {\pi \arctan \left (a x \right )}{6 a^{6}}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90 a^{6}}\) | \(267\) |
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.75 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} + 30 \, {\left (a^{6} x^{6} + 1\right )} \operatorname {arccot}\left (a x\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 15 \, a x\right )} \operatorname {arccot}\left (a x\right ) + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \]
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Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {acot}^{2}{\left (a x \right )}}{6} + \frac {x^{5} \operatorname {acot}{\left (a x \right )}}{15 a} + \frac {x^{4}}{60 a^{2}} - \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{9 a^{3}} - \frac {4 x^{2}}{45 a^{4}} + \frac {x \operatorname {acot}{\left (a x \right )}}{3 a^{5}} + \frac {23 \log {\left (a^{2} x^{2} + 1 \right )}}{90 a^{6}} + \frac {\operatorname {acot}^{2}{\left (a x \right )}}{6 a^{6}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{6}}{24} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.91 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {1}{6} \, x^{6} \operatorname {arccot}\left (a x\right )^{2} + \frac {1}{45} \, a {\left (\frac {3 \, a^{4} x^{5} - 5 \, a^{2} x^{3} + 15 \, x}{a^{6}} - \frac {15 \, \arctan \left (a x\right )}{a^{7}}\right )} \operatorname {arccot}\left (a x\right ) + \frac {3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} - 30 \, \arctan \left (a x\right )^{2} + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \]
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\[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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Time = 0.95 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.82 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {x^6\,{\mathrm {acot}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2+1\right )}{90}-\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}+\frac {{\mathrm {acot}\left (a\,x\right )}^2}{6}-\frac {a^3\,x^3\,\mathrm {acot}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acot}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acot}\left (a\,x\right )}{3}}{a^6} \]
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