Optimal. Leaf size=36 \[ \frac{x^{m+1} \tan ^{-1}(\cot (a+b x))}{m+1}+\frac{b x^{m+2}}{m^2+3 m+2} \]
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Rubi [A] time = 0.0210559, antiderivative size = 36, normalized size of antiderivative = 1., 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} \[ \frac{x^{m+1} \tan ^{-1}(\cot (a+b x))}{m+1}+\frac{b x^{m+2}}{m^2+3 m+2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^m \tan ^{-1}(\cot (a+b x)) \, dx &=\frac{x^{1+m} \tan ^{-1}(\cot (a+b x))}{1+m}+\frac{b \int x^{1+m} \, dx}{1+m}\\ &=\frac{b x^{2+m}}{2+3 m+m^2}+\frac{x^{1+m} \tan ^{-1}(\cot (a+b x))}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0479835, size = 31, normalized size = 0.86 \[ \frac{x^{m+1} \left ((m+2) \tan ^{-1}(\cot (a+b x))+b x\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 56, normalized size = 1.6 \begin{align*}{\frac{\pi \,{x}^{1+m}}{2+2\,m}}-{\frac{b{x}^{2}{{\rm e}^{m\ln \left ( x \right ) }}}{2+m}}-{\frac{ \left ({\rm arccot} \left (\cot \left ( bx+a \right ) \right )-bx \right ) x{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07195, size = 101, normalized size = 2.81 \begin{align*} -\frac{{\left (2 \,{\left (b m + b\right )} x^{2} -{\left (\pi{\left (m + 2\right )} - 2 \, a m - 4 \, a\right )} x\right )} x^{m}}{2 \,{\left (m^{2} + 3 \, m + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09397, size = 84, normalized size = 2.33 \begin{align*} -\frac{2 \, b m x^{2} x^{m} - \pi m x x^{m} + 2 \, a m x x^{m} + 2 \, b x^{2} x^{m} - 2 \, \pi x x^{m} + 4 \, a x x^{m}}{2 \,{\left (m^{2} + 3 \, m + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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