Integrand size = 14, antiderivative size = 97 \[ \int x^{1+m} \cos ^2(a+b x) \, dx=\frac {x^{2+m}}{2 (2+m)}+\frac {2^{-4-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (2+m,-2 i b x)}{b^2}+\frac {2^{-4-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (2+m,2 i b x)}{b^2} \]
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Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3393, 3388, 2212} \[ \int x^{1+m} \cos ^2(a+b x) \, dx=\frac {e^{2 i a} 2^{-m-4} x^m (-i b x)^{-m} \Gamma (m+2,-2 i b x)}{b^2}+\frac {e^{-2 i a} 2^{-m-4} x^m (i b x)^{-m} \Gamma (m+2,2 i b x)}{b^2}+\frac {x^{m+2}}{2 (m+2)} \]
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Rule 2212
Rule 3388
Rule 3393
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^{1+m}}{2}+\frac {1}{2} x^{1+m} \cos (2 a+2 b x)\right ) \, dx \\ & = \frac {x^{2+m}}{2 (2+m)}+\frac {1}{2} \int x^{1+m} \cos (2 a+2 b x) \, dx \\ & = \frac {x^{2+m}}{2 (2+m)}+\frac {1}{4} \int e^{-i (2 a+2 b x)} x^{1+m} \, dx+\frac {1}{4} \int e^{i (2 a+2 b x)} x^{1+m} \, dx \\ & = \frac {x^{2+m}}{2 (2+m)}+\frac {2^{-4-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (2+m,-2 i b x)}{b^2}+\frac {2^{-4-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (2+m,2 i b x)}{b^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int x^{1+m} \cos ^2(a+b x) \, dx=\frac {1}{16} x^m \left (\frac {8 x^2}{2+m}+\frac {2^{-m} e^{2 i a} (-i b x)^{-m} \Gamma (2+m,-2 i b x)}{b^2}+\frac {2^{-m} e^{-2 i a} (i b x)^{-m} \Gamma (2+m,2 i b x)}{b^2}\right ) \]
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\[\int x^{1+m} \left (\cos ^{2}\left (b x +a \right )\right )d x\]
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none
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int x^{1+m} \cos ^2(a+b x) \, dx=\frac {4 \, b x x^{m + 1} + {\left (i \, m + 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 2, 2 i \, b x\right ) + {\left (-i \, m - 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 2, -2 i \, b x\right )}{8 \, {\left (b m + 2 \, b\right )}} \]
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\[ \int x^{1+m} \cos ^2(a+b x) \, dx=\int x^{m + 1} \cos ^{2}{\left (a + b x \right )}\, dx \]
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\[ \int x^{1+m} \cos ^2(a+b x) \, dx=\int { x^{m + 1} \cos \left (b x + a\right )^{2} \,d x } \]
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\[ \int x^{1+m} \cos ^2(a+b x) \, dx=\int { x^{m + 1} \cos \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^{1+m} \cos ^2(a+b x) \, dx=\int x^{m+1}\,{\cos \left (a+b\,x\right )}^2 \,d x \]
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