Optimal. Leaf size=103 \[ -\frac {i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \Gamma (m+1,-2 i b x)}{b}+\frac {i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \Gamma (m+1,2 i b x)}{b}+\frac {x^{m+1}}{2 (m+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2181
Rule 3307
Rule 3312
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*}
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Mathematica [A] time = 0.12, size = 90, normalized size = 0.87 \[ \frac {1}{8} x^m \left (e^{2 i a} \left (-2^{-m}\right ) x (-i b x)^{-m-1} \Gamma (m+1,-2 i b x)-e^{-2 i a} 2^{-m} x (i b x)^{-m-1} \Gamma (m+1,2 i b x)+\frac {4 x}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 69, normalized size = 0.67 \[ \frac {4 \, b x x^{m} + {\left (i \, m + i\right )} e^{\left (-m \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 1, 2 i \, b x\right ) + {\left (-i \, m - i\right )} e^{\left (-m \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 1, -2 i \, b x\right )}{8 \, {\left (b m + b\right )}} \]
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 \[ \int x^{m} \cos \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cos ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (m + 1\right )} \int x^{m} \cos \left (2 \, b x + 2 \, a\right )\,{d x} + e^{\left (m \log \relax (x) + \log \relax (x)\right )}}{2 \, {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\cos \left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cos ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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