Optimal. Leaf size=23 \[ \frac {1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac {b x^3}{6} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5173, 30} \[ \frac {1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac {b x^3}{6} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5173
Rubi steps
\begin {align*} \int x \tan ^{-1}(\cot (a+b x)) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac {1}{2} b \int x^2 \, dx\\ &=\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(\cot (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.87 \[ \frac {1}{6} x^2 \left (3 \tan ^{-1}(\cot (a+b x))+b x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 17, normalized size = 0.74 \[ -\frac {1}{3} \, b x^{3} + \frac {1}{4} \, {\left (\pi - 2 \, a\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 19, normalized size = 0.83 \[ -\frac {1}{3} \, b x^{3} + \frac {1}{4} \, \pi x^{2} - \frac {1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 54, normalized size = 2.35 \[ \frac {\pi \,x^{2}}{4}-\frac {x^{2} \mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{2}-\frac {-\frac {\left (b x +a \right )^{3}}{3}+\left (b x +a \right )^{2} a -a^{2} \left (b x +a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 17, normalized size = 0.74 \[ -\frac {1}{3} \, b x^{3} + \frac {1}{4} \, {\left (\pi - 2 \, a\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 25, normalized size = 1.09 \[ \frac {\Pi \,x^2}{4}+\frac {b\,x^3}{6}-\frac {x^2\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 49, normalized size = 2.13 \[ \begin {cases} \frac {\pi x^{2}}{4} - \frac {x \operatorname {acot}^{2}{\left (\cot {\left (a + b x \right )} \right )}}{2 b} + \frac {\operatorname {acot}^{3}{\left (\cot {\left (a + b x \right )} \right )}}{6 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \left (- \operatorname {acot}{\left (\cot {\relax (a )} \right )} + \frac {\pi }{2}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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