Optimal. Leaf size=79 \[ \frac{i e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+4,-i b x)}{2 b^4}-\frac{i e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+4,i b x)}{2 b^4} \]
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Rubi [A] time = 0.0774144, antiderivative size = 79, 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} \[ \frac{i e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+4,-i b x)}{2 b^4}-\frac{i e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+4,i b x)}{2 b^4} \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^{3+m} \sin (a+b x) \, dx &=\frac{1}{2} i \int e^{-i (a+b x)} x^{3+m} \, dx-\frac{1}{2} i \int e^{i (a+b x)} x^{3+m} \, dx\\ &=\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (4+m,-i b x)}{2 b^4}-\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (4+m,i b x)}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0191448, size = 79, normalized size = 1. \[ \frac{i e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+4,-i b x)}{2 b^4}-\frac{i e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+4,i b x)}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.119, size = 454, normalized size = 5.8 \begin{align*}{\frac{{2}^{3+m}\sqrt{\pi }\sin \left ( a \right ) }{{b}^{4}} \left ({b}^{2} \right ) ^{-{\frac{m}{2}}} \left ( 3\,{\frac{{2}^{-4-m}{x}^{3+m}{b}^{3} \left ({b}^{2} \right ) ^{m/2} \left ( 8/3+2/3\,m \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( 4+m \right ) }}-{\frac{{2}^{-3-m}{x}^{1+m}b \left ( -{m}^{2}-7\,m-12 \right ) \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( 4+m \right ) } \left ({b}^{2} \right ) ^{{\frac{m}{2}}}}+{\frac{{2}^{-3-m}{x}^{2+m}{b}^{2} \left ( -{m}^{3}-8\,{m}^{2}-19\,m-12 \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( 4+m \right ) } \left ({b}^{2} \right ) ^{{\frac{m}{2}}} \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{3}{2}},bx \right ) }-{\frac{{2}^{-3-m}{x}^{2+m}{b}^{2} \left ( 2+m \right ) \left ( 1+m \right ) \left ( 3+m \right ) \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }} \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 ) }+{2}^{3+m}{b}^{-4-m}\sqrt{\pi } \left ({\frac{{2}^{-3-m}{x}^{2+m}{b}^{2+m} \left ({m}^{2}+7\,m+10 \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( 5+m \right ) }}-{\frac{{2}^{-3-m}{x}^{2+m}{b}^{2+m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }}}-{\frac{{2}^{-3-m}{x}^{2+m}{b}^{2+m}m \left ( 3+m \right ) \left ( 2+m \right ) \sin \left ( bx \right ) }{\sqrt{\pi }} \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{1}{2}},{\frac{3}{2}},bx \right ) }+{\frac{{2}^{-3-m}{x}^{2+m}{b}^{2+m} \left ( 3+m \right ) \left ( 2+m \right ) \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{1}{2}},bx \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 + 3} \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 = 1.8114, size = 149, normalized size = 1.89 \begin{align*} -\frac{e^{\left (-{\left (m + 3\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 4, i \, b x\right ) + e^{\left (-{\left (m + 3\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 4, -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 + 3} \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 + 3} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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