Optimal. Leaf size=75 \[ -\frac{e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+1,-i b x)}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+1,i b x)}{2 b} \]
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Rubi [A] time = 0.0657262, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3308, 2181} \[ -\frac{e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+1,-i b x)}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+1,i b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \sin (a+b x) \, dx &=\frac{1}{2} i \int e^{-i (a+b x)} x^m \, dx-\frac{1}{2} i \int e^{i (a+b x)} x^m \, dx\\ &=-\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (1+m,-i b x)}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (1+m,i b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.013591, size = 75, normalized size = 1. \[ -\frac{e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+1,-i b x)}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+1,i b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.062, size = 378, normalized size = 5. \begin{align*}{2}^{m} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}-{\frac{m}{2}}}\sqrt{\pi } \left ( 3\,{\frac{{2}^{-1-m} \left ({b}^{2} \right ) ^{1/2+m/2}{x}^{m} \left ( 6+2\,m \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( 1+m \right ) \left ( 9+3\,m \right ) b}}+{\frac{{x}^{m}{2}^{-m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( 1+m \right ) b} \left ({b}^{2} \right ) ^{{\frac{1}{2}}+{\frac{m}{2}}}}+{\frac{{2}^{-m}{x}^{2+m}bm\sin \left ( bx \right ) }{\sqrt{\pi } \left ( 1+m \right ) } \left ({b}^{2} \right ) ^{{\frac{1}{2}}+{\frac{m}{2}}} \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{1}{2}},{\frac{3}{2}},bx \right ) }-{\frac{{2}^{-m}{x}^{2+m}b \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( 1+m \right ) } \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}{b}^{-1-m}\sqrt{\pi } \left ({\frac{{x}^{1+m}{b}^{1+m}{2}^{-m}\sin \left ( bx \right ) }{\sqrt{\pi } \left ( 2+m \right ) }}-{\frac{{2}^{-m}{x}^{2+m}{b}^{2+m}\sin \left ( bx \right ) }{\sqrt{\pi } \left ( 2+m \right ) } \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{3}{2}},bx \right ) }-3\,{\frac{{2}^{-1-m}{x}^{2+m}{b}^{2+m} \left ( 4/3+2/3\,m \right ) \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 } \left ( 2+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} \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 = 1.78999, size = 132, normalized size = 1.76 \begin{align*} -\frac{e^{\left (-m \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 1, i \, b x\right ) + e^{\left (-m \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} \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} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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