Integrand size = 10, antiderivative size = 152 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}}+\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4947, 327, 303, 1176, 631, 210, 1179, 642} \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{5 \sqrt {2} a^{5/2}}-\frac {\log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{10 \sqrt {2} a^{5/2}}+\frac {\log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{10 \sqrt {2} a^{5/2}}+\frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right ) \]
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Rule 210
Rule 303
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 4947
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac {1}{5} (2 a) \int \frac {x^6}{1+a^2 x^4} \, dx \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )-\frac {2 \int \frac {x^2}{1+a^2 x^4} \, dx}{5 a} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac {\int \frac {1-a x^2}{1+a^2 x^4} \, dx}{5 a^2}-\frac {\int \frac {1+a x^2}{1+a^2 x^4} \, dx}{5 a^2} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )-\frac {\int \frac {1}{\frac {1}{a}-\frac {\sqrt {2} x}{\sqrt {a}}+x^2} \, dx}{10 a^3}-\frac {\int \frac {1}{\frac {1}{a}+\frac {\sqrt {2} x}{\sqrt {a}}+x^2} \, dx}{10 a^3}-\frac {\int \frac {\frac {\sqrt {2}}{\sqrt {a}}+2 x}{-\frac {1}{a}-\frac {\sqrt {2} x}{\sqrt {a}}-x^2} \, dx}{10 \sqrt {2} a^{5/2}}-\frac {\int \frac {\frac {\sqrt {2}}{\sqrt {a}}-2 x}{-\frac {1}{a}+\frac {\sqrt {2} x}{\sqrt {a}}-x^2} \, dx}{10 \sqrt {2} a^{5/2}} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )-\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}}+\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}} \\ & = \frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}}+\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {8 a^{3/2} x^3+12 a^{5/2} x^5 \cot ^{-1}\left (a x^2\right )+6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {a} x\right )-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {a} x\right )-3 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )+3 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{60 a^{5/2}} \]
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Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {x^{5} \operatorname {arccot}\left (a \,x^{2}\right )}{5}+\frac {2 a \left (\frac {x^{3}}{3 a^{2}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{4} \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\right )}{5}\) | \(112\) |
parts | \(\frac {x^{5} \operatorname {arccot}\left (a \,x^{2}\right )}{5}+\frac {2 a \left (\frac {x^{3}}{3 a^{2}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{4} \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\right )}{5}\) | \(112\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {6 \, a x^{5} \operatorname {arccot}\left (a x^{2}\right ) + 4 \, x^{3} - 3 \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right ) + 3 i \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (i \, a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right ) - 3 i \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (-i \, a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right ) + 3 \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (-a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right )}{30 \, a} \]
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Time = 9.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\begin {cases} \frac {x^{5} \operatorname {acot}{\left (a x^{2} \right )}}{5} + \frac {2 x^{3}}{15 a} - \frac {\sqrt [4]{- \frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )}}{5 a^{2}} - \frac {\log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{5 a^{3} \sqrt [4]{- \frac {1}{a^{2}}}} + \frac {\log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{10 a^{3} \sqrt [4]{- \frac {1}{a^{2}}}} - \frac {\operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{5 a^{3} \sqrt [4]{- \frac {1}{a^{2}}}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{5} \, x^{5} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{60} \, a {\left (\frac {8 \, x^{3}}{a^{2}} - \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )}}{a^{2}}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{5} \, x^{5} \arctan \left (\frac {1}{a x^{2}}\right ) + \frac {1}{60} \, a {\left (\frac {8 \, x^{3}}{a^{2}} - \frac {6 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} {\left | a \right |}^{\frac {3}{2}}} - \frac {6 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} {\left | a \right |}^{\frac {3}{2}}} + \frac {3 \, \sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{4}} - \frac {3 \, \sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2} {\left | a \right |}^{\frac {3}{2}}}\right )} \]
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Time = 0.48 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.36 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^5\,\mathrm {acot}\left (a\,x^2\right )}{5}+\frac {2\,x^3}{15\,a}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{5\,a^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5\,a^{5/2}} \]
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