Integrand size = 8, antiderivative size = 41 \[ \int x^3 \cot ^{-1}(a x) \, dx=-\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {\arctan (a x)}{4 a^4} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 41, 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, 308, 209} \[ \int x^3 \cot ^{-1}(a x) \, dx=\frac {\arctan (a x)}{4 a^4}-\frac {x}{4 a^3}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {x^3}{12 a} \]
[In]
[Out]
Rule 209
Rule 308
Rule 4947
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {1}{4} a \int \frac {x^4}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {1}{4} a \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{4 a^3} \\ & = -\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {\arctan (a x)}{4 a^4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int x^3 \cot ^{-1}(a x) \, dx=-\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {\arctan (a x)}{4 a^4} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}-\frac {a x}{4}+\frac {\arctan \left (a x \right )}{4}}{a^{4}}\) | \(36\) |
default | \(\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}-\frac {a x}{4}+\frac {\arctan \left (a x \right )}{4}}{a^{4}}\) | \(36\) |
parallelrisch | \(\frac {3 a^{4} x^{4} \operatorname {arccot}\left (a x \right )+a^{3} x^{3}-3 a x -3 \,\operatorname {arccot}\left (a x \right )}{12 a^{4}}\) | \(36\) |
parts | \(\frac {x^{4} \operatorname {arccot}\left (a x \right )}{4}+\frac {a \left (\frac {\frac {1}{3} a^{2} x^{3}-x}{a^{4}}+\frac {\arctan \left (a x \right )}{a^{5}}\right )}{4}\) | \(39\) |
risch | \(\frac {i x^{4} \ln \left (i a x +1\right )}{8}-\frac {i x^{4} \ln \left (-i a x +1\right )}{8}+\frac {\pi \,x^{4}}{8}+\frac {x^{3}}{12 a}-\frac {x}{4 a^{3}}+\frac {\arctan \left (a x \right )}{4 a^{4}}\) | \(59\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int x^3 \cot ^{-1}(a x) \, dx=\frac {a^{3} x^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )}{12 \, a^{4}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int x^3 \cot ^{-1}(a x) \, dx=\begin {cases} \frac {x^{4} \operatorname {acot}{\left (a x \right )}}{4} + \frac {x^{3}}{12 a} - \frac {x}{4 a^{3}} - \frac {\operatorname {acot}{\left (a x \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int x^3 \cot ^{-1}(a x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arccot}\left (a x\right ) + \frac {1}{12} \, a {\left (\frac {a^{2} x^{3} - 3 \, x}{a^{4}} + \frac {3 \, \arctan \left (a x\right )}{a^{5}}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int x^3 \cot ^{-1}(a x) \, dx=\frac {1}{12} \, {\left (\frac {3 \, x^{4} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{3} {\left (\frac {3}{a^{2} x^{2}} - 1\right )}}{a^{2}} - \frac {3 \, \arctan \left (\frac {1}{a x}\right )}{a^{5}}\right )} a \]
[In]
[Out]
Time = 0.86 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int x^3 \cot ^{-1}(a x) \, dx=\left \{\begin {array}{cl} \frac {\pi \,x^4}{8} & \text {\ if\ \ }a=0\\ \frac {3\,\mathrm {atan}\left (a\,x\right )-3\,a\,x+a^3\,x^3}{12\,a^4}+\frac {x^4\,\mathrm {acot}\left (a\,x\right )}{4} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
[In]
[Out]