Optimal. Leaf size=71 \[ \frac{1}{2} e^{i a} b x^m (-i b x)^{-m} \text{Gamma}(m-1,-i b x)+\frac{1}{2} e^{-i a} b x^m (i b x)^{-m} \text{Gamma}(m-1,i b x) \]
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Rubi [A] time = 0.0706639, antiderivative size = 71, 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 = {3308, 2181} \[ \frac{1}{2} e^{i a} b x^m (-i b x)^{-m} \text{Gamma}(m-1,-i b x)+\frac{1}{2} e^{-i a} b x^m (i b x)^{-m} \text{Gamma}(m-1,i b x) \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^{-2+m} \sin (a+b x) \, dx &=\frac{1}{2} i \int e^{-i (a+b x)} x^{-2+m} \, dx-\frac{1}{2} i \int e^{i (a+b x)} x^{-2+m} \, dx\\ &=\frac{1}{2} b e^{i a} x^m (-i b x)^{-m} \Gamma (-1+m,-i b x)+\frac{1}{2} b e^{-i a} x^m (i b x)^{-m} \Gamma (-1+m,i b x)\\ \end{align*}
Mathematica [A] time = 0.0186229, size = 65, normalized size = 0.92 \[ \frac{1}{2} e^{-i a} b x^m \left (e^{2 i a} (-i b x)^{-m} \text{Gamma}(m-1,-i b x)+(i b x)^{-m} \text{Gamma}(m-1,i b x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.072, size = 529, normalized size = 7.5 \begin{align*}{2}^{m-2} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}-{\frac{m}{2}}}{b}^{2}\sqrt{\pi } \left ( 3\,{\frac{{2}^{1-m}{x}^{m-2} \left ({b}^{2} \right ) ^{-1/2+m/2} \left ( 2\,{x}^{2}{b}^{2}+2\,m+2 \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( -1+m \right ) \left ( 3+3\,m \right ) b}}-{\frac{{2}^{2-m}{x}^{m-2} \left ({x}^{2}{b}^{2}-{m}^{2}-m \right ) \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( -1+m \right ) b \left ( 1+m \right ) m} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}+{\frac{m}{2}}}}-3\,{\frac{{2}^{2-m}{x}^{2+m} \left ({b}^{2} \right ) ^{-1/2+m/2}{b}^{3} \left ( bx \right ) ^{-3/2-m}{\it LommelS1} \left ( m+1/2,3/2,bx \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( -1+m \right ) \left ( 3+3\,m \right ) }}+{\frac{{2}^{2-m}{x}^{2+m}{b}^{3} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( -1+m \right ) \left ( 1+m \right ) m} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}+{\frac{m}{2}}} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{1}{2}},bx \right ) } \right ) \sin \left ( a \right ) +{2}^{m-2}{b}^{1-m}\sqrt{\pi } \left ({\frac{{2}^{1-m}{x}^{-1+m}{b}^{-1+m} \left ( -2\,{x}^{2}{b}^{2}+2\,{m}^{2}+2\,m-4 \right ) \sin \left ( bx \right ) }{\sqrt{\pi }m \left ( 2+m \right ) \left ( -1+m \right ) }}-3\,{\frac{{2}^{2-m}{x}^{-1+m}{b}^{-1+m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }m \left ( -3+3\,m \right ) }}+{\frac{{2}^{2-m}{x}^{2+m}{b}^{2+m}\sin \left ( bx \right ) }{\sqrt{\pi }m \left ( 2+m \right ) \left ( -1+m \right ) } \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{3}{2}},bx \right ) }+3\,{\frac{{2}^{2-m}{x}^{2+m}{b}^{2+m} \left ( bx \right ) ^{-5/2-m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ){\it LommelS1} \left ( m+1/2,1/2,bx \right ) }{\sqrt{\pi }m \left ( -3+3\,m \right ) }} \right ) \cos \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 - 2} \sin \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 = 1.82031, size = 149, normalized size = 2.1 \begin{align*} -\frac{e^{\left (-{\left (m - 2\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m - 1, i \, b x\right ) + e^{\left (-{\left (m - 2\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m - 1, -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 - 2} \sin{\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 - 2} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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