Optimal. Leaf size=41 \[ -\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {\text {ArcTan}(a x)}{4 a^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 308, 209}
\begin {gather*} \frac {\text {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 308
Rule 4947
Rubi steps
\begin {align*} \int x^3 \cot ^{-1}(a x) \, dx &=\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 {\tan ^{-1}(a x)}{4 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 41, normalized size = 1.00 \begin {gather*} -\frac {x}{4 a^3}+\frac {x^3}{12 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {\text {ArcTan}(a x)}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 36, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \mathrm {arccot}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}-\frac {a x}{4}+\frac {\arctan \left (a x \right )}{4}}{a^{4}}\) | \(36\) |
default | \(\frac {\frac {a^{4} x^{4} \mathrm {arccot}\left (a x \right )}{4}+\frac {a^{3} x^{3}}{12}-\frac {a x}{4}+\frac {\arctan \left (a x \right )}{4}}{a^{4}}\) | \(36\) |
risch | \(\frac {i x^{4} \ln \left (i a x +1\right )}{8}-\frac {i x^{4} \ln \left (-i a x +1\right )}{8}+\frac {\pi \,x^{4}}{8}+\frac {x^{3}}{12 a}-\frac {x}{4 a^{3}}+\frac {\arctan \left (a x \right )}{4 a^{4}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 38, normalized size = 0.93 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.94, size = 32, normalized size = 0.78 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 39, normalized size = 0.95 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 51, normalized size = 1.24 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 46, normalized size = 1.12 \begin {gather*} \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 . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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