Optimal. Leaf size=65 \[ -\frac{1}{2} e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m,-i b x)-\frac{1}{2} e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m,i b x) \]
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Rubi [A] time = 0.0731398, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3307, 2181} \[ -\frac{1}{2} e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m,-i b x)-\frac{1}{2} e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m,i b x) \]
Antiderivative was successfully verified.
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Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^{-1+m} \cos (a+b x) \, dx &=\frac{1}{2} \int e^{-i (a+b x)} x^{-1+m} \, dx+\frac{1}{2} \int e^{i (a+b x)} x^{-1+m} \, dx\\ &=-\frac{1}{2} e^{i a} x^m (-i b x)^{-m} \Gamma (m,-i b x)-\frac{1}{2} e^{-i a} x^m (i b x)^{-m} \Gamma (m,i b x)\\ \end{align*}
Mathematica [A] time = 0.022577, size = 62, normalized size = 0.95 \[ \frac{1}{2} e^{-i a} x^m \left (-e^{2 i a} (-i b x)^{-m} \text{Gamma}(m,-i b x)-(i b x)^{-m} \text{Gamma}(m,i b x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.078, size = 427, normalized size = 6.6 \begin{align*}{2}^{-1+m} \left ({b}^{2} \right ) ^{-{\frac{m}{2}}}\sqrt{\pi } \left ( 3\,{\frac{{x}^{-1+m}{2}^{-m} \left ({b}^{2} \right ) ^{m/2} \left ( 2\,{x}^{2}{b}^{2}+2\,m+4 \right ) \sin \left ( bx \right ) }{\sqrt{\pi }m \left ( 6+3\,m \right ) b}}+{\frac{{2}^{1-m}{x}^{-1+m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }mb} \left ({b}^{2} \right ) ^{{\frac{m}{2}}}}-3\,{\frac{{x}^{2+m}{2}^{1-m} \left ({b}^{2} \right ) ^{m/2}{b}^{2} \left ( bx \right ) ^{-3/2-m}{\it LommelS1} \left ( m+3/2,3/2,bx \right ) \sin \left ( bx \right ) }{\sqrt{\pi }m \left ( 6+3\,m \right ) }}-{\frac{{x}^{2+m}{2}^{1-m}{b}^{2} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }m} \left ({b}^{2} \right ) ^{{\frac{m}{2}}} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{1}{2}},{\frac{1}{2}},bx \right ) } \right ) \cos \left ( a \right ) -{2}^{-1+m}{b}^{-m}\sqrt{\pi } \left ({\frac{{2}^{1-m}{x}^{m}{b}^{m}\sin \left ( bx \right ) }{\sqrt{\pi } \left ( 1+m \right ) }}-{\frac{{2}^{1-m}{x}^{m}{b}^{m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( 1+m \right ) m}}-{\frac{{x}^{2+m}{b}^{2+m}{2}^{1-m}\sin \left ( bx \right ) }{\sqrt{\pi } \left ( 1+m \right ) } \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{1}{2}},{\frac{3}{2}},bx \right ) }+{\frac{{x}^{2+m}{b}^{2+m}{2}^{1-m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( 1+m \right ) m} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{1}{2}},bx \right ) } \right ) \sin \left ( a \right ) \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*} \int x^{m - 1} \cos \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70119, size = 142, normalized size = 2.18 \begin{align*} \frac{i \, e^{\left (-{\left (m - 1\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m, i \, b x\right ) - i \, e^{\left (-{\left (m - 1\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m, -i \, b x\right )}{2 \, b} \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 - 1} \cos{\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 - 1} \cos \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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