Optimal. Leaf size=103 \[ -\frac{i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \text{Gamma}(m+1,-2 i b x)}{b}+\frac{i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \text{Gamma}(m+1,2 i b x)}{b}+\frac{x^{m+1}}{2 (m+1)} \]
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Rubi [A] time = 0.132942, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3312, 3307, 2181} \[ -\frac{i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \text{Gamma}(m+1,-2 i b x)}{b}+\frac{i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \text{Gamma}(m+1,2 i b x)}{b}+\frac{x^{m+1}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^m \cos ^2(a+b x) \, dx &=\int \left (\frac{x^m}{2}+\frac{1}{2} x^m \cos (2 a+2 b x)\right ) \, dx\\ &=\frac{x^{1+m}}{2 (1+m)}+\frac{1}{2} \int x^m \cos (2 a+2 b x) \, dx\\ &=\frac{x^{1+m}}{2 (1+m)}+\frac{1}{4} \int e^{-i (2 a+2 b x)} x^m \, dx+\frac{1}{4} \int e^{i (2 a+2 b x)} x^m \, dx\\ &=\frac{x^{1+m}}{2 (1+m)}-\frac{i 2^{-3-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (1+m,-2 i b x)}{b}+\frac{i 2^{-3-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (1+m,2 i b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.11927, size = 90, normalized size = 0.87 \[ \frac{1}{8} x^m \left (e^{2 i a} \left (-2^{-m}\right ) x (-i b x)^{-m-1} \text{Gamma}(m+1,-2 i b x)-e^{-2 i a} 2^{-m} x (i b x)^{-m-1} \text{Gamma}(m+1,2 i b x)+\frac{4 x}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cos \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (m + 1\right )} \int x^{m} \cos \left (2 \, b x + 2 \, a\right )\,{d x} + e^{\left (m \log \left (x\right ) + \log \left (x\right )\right )}}{2 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94255, size = 203, normalized size = 1.97 \begin{align*} \frac{4 \, b x x^{m} +{\left (i \, m + i\right )} e^{\left (-m \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 1, 2 i \, b x\right ) +{\left (-i \, m - i\right )} e^{\left (-m \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 1, -2 i \, b x\right )}{8 \,{\left (b m + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cos ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cos \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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