Optimal. Leaf size=79 \[ \frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \text{Gamma}(m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \text{Gamma}(m-2,-i b x) \]
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Rubi [A] time = 0.0717161, 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} \[ \frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \text{Gamma}(m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \text{Gamma}(m-2,-i b x) \]
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{1}{2} i b^2 e^{i a} x^m (-i b x)^{-m} \Gamma (-2+m,-i b x)+\frac{1}{2} i b^2 e^{-i a} x^m (i b x)^{-m} \Gamma (-2+m,i b x)\\ \end{align*}
Mathematica [A] time = 0.0156524, size = 79, normalized size = 1. \[ \frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \text{Gamma}(m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \text{Gamma}(m-2,-i b x) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.079, size = 599, normalized size = 7.6 \begin{align*} \text{result too large to display} \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 = 1.7661, 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 - 2, i \, b x\right ) + e^{\left (-{\left (m - 3\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m - 2, -i \, b x\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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