Optimal. Leaf size=104 \[ \frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {4 x^2}{45 a^4}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^4}{60 a^2}+\frac {23 \log \left (a^2 x^2+1\right )}{90 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {x^5 \cot ^{-1}(a x)}{15 a} \]
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Rubi [A] time = 0.22, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4853, 4917, 266, 43, 4847, 260, 4885} \[ \frac {x^4}{60 a^2}-\frac {4 x^2}{45 a^4}+\frac {23 \log \left (a^2 x^2+1\right )}{90 a^6}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x \cot ^{-1}(a x)}{3 a^5}+\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {x^5 \cot ^{-1}(a x)}{15 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4847
Rule 4853
Rule 4885
Rule 4917
Rubi steps
\begin {align*} \int x^5 \cot ^{-1}(a x)^2 \, dx &=\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} \operatorname {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} \operatorname {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 {\operatorname {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 {\operatorname {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*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 0.76 \[ \frac {30 \left (a^6 x^6+1\right ) \cot ^{-1}(a x)^2+3 a^4 x^4-16 a^2 x^2+46 \log \left (a^2 x^2+1\right )+4 a x \left (3 a^4 x^4-5 a^2 x^2+15\right ) \cot ^{-1}(a x)}{180 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 78, normalized size = 0.75 \[ \frac {3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} + 30 \, {\left (a^{6} x^{6} + 1\right )} \operatorname {arccot}\left (a x\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 15 \, a x\right )} \operatorname {arccot}\left (a x\right ) + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {arccot}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 102, normalized size = 0.98 \[ \frac {x^{6} \mathrm {arccot}\left (a x \right )^{2}}{6}+\frac {x^{5} \mathrm {arccot}\left (a x \right )}{15 a}-\frac {x^{3} \mathrm {arccot}\left (a x \right )}{9 a^{3}}+\frac {x \,\mathrm {arccot}\left (a x \right )}{3 a^{5}}-\frac {\mathrm {arccot}\left (a x \right ) \arctan \left (a x \right )}{3 a^{6}}+\frac {x^{4}}{60 a^{2}}-\frac {4 x^{2}}{45 a^{4}}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90 a^{6}}-\frac {\arctan \left (a x \right )^{2}}{6 a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 95, normalized size = 0.91 \[ \frac {1}{6} \, x^{6} \operatorname {arccot}\left (a x\right )^{2} + \frac {1}{45} \, a {\left (\frac {3 \, a^{4} x^{5} - 5 \, a^{2} x^{3} + 15 \, x}{a^{6}} - \frac {15 \, \arctan \left (a x\right )}{a^{7}}\right )} \operatorname {arccot}\left (a x\right ) + \frac {3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} - 30 \, \arctan \left (a x\right )^{2} + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 85, normalized size = 0.82 \[ \frac {x^6\,{\mathrm {acot}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2+1\right )}{90}-\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}+\frac {{\mathrm {acot}\left (a\,x\right )}^2}{6}-\frac {a^3\,x^3\,\mathrm {acot}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acot}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acot}\left (a\,x\right )}{3}}{a^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.76, size = 104, normalized size = 1.00 \[ \begin {cases} \frac {x^{6} \operatorname {acot}^{2}{\left (a x \right )}}{6} + \frac {x^{5} \operatorname {acot}{\left (a x \right )}}{15 a} + \frac {x^{4}}{60 a^{2}} - \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{9 a^{3}} - \frac {4 x^{2}}{45 a^{4}} + \frac {x \operatorname {acot}{\left (a x \right )}}{3 a^{5}} + \frac {23 \log {\left (a^{2} x^{2} + 1 \right )}}{90 a^{6}} + \frac {\operatorname {acot}^{2}{\left (a x \right )}}{6 a^{6}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{6}}{24} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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