Optimal. Leaf size=23 \[ \frac {1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac {b x^4}{12} \]
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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2168, 30} \[ \frac {1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac {b x^4}{12} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int x^2 \tan ^{-1}(\tan (a+b x)) \, dx &=\frac {1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac {1}{3} b \int x^3 \, dx\\ &=-\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(\tan (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.87 \[ -\frac {1}{12} x^3 \left (b x-4 \tan ^{-1}(\tan (a+b x))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 13, normalized size = 0.57 \[ \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 20, normalized size = 0.87 \[ -\frac {b \,x^{4}}{12}+\frac {x^{3} \arctan \left (\tan \left (b x +a \right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 81, normalized size = 3.52 \[ \frac {\frac {4 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )} \arctan \left (\tan \left (b x + a\right )\right )}{b^{2}} - \frac {{\left (b x + a\right )}^{4} - 4 \, {\left (b x + a\right )}^{3} a + 6 \, {\left (b x + a\right )}^{2} a^{2}}{b^{2}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 19, normalized size = 0.83 \[ \frac {x^3\,\mathrm {atan}\left (\mathrm {tan}\left (a+b\,x\right )\right )}{3}-\frac {b\,x^4}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 32, normalized size = 1.39 \[ - \frac {b x^{4}}{12} + \frac {x^{3} \left (\operatorname {atan}{\left (\tan {\left (a + b x \right )} \right )} + \pi \left \lfloor {\frac {a + b x - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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