Integrand size = 14, antiderivative size = 127 \[ \int (c+d x)^m \sin (a+b x) \, dx=-\frac {e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3389, 2212} \[ \int (c+d x)^m \sin (a+b x) \, dx=-\frac {e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )}{2 b} \]
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Rule 2212
Rule 3389
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} i \int e^{-i (a+b x)} (c+d x)^m \, dx-\frac {1}{2} i \int e^{i (a+b x)} (c+d x)^m \, dx \\ & = -\frac {e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95 \[ \int (c+d x)^m \sin (a+b x) \, dx=\frac {e^{-\frac {i (b c+a d)}{d}} (c+d x)^m \left (-e^{2 i a} \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )\right )}{2 b} \]
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\[\int \left (d x +c \right )^{m} \sin \left (b x +a \right )d x\]
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none
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.74 \[ \int (c+d x)^m \sin (a+b x) \, dx=-\frac {e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) + e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right )}{2 \, b} \]
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\[ \int (c+d x)^m \sin (a+b x) \, dx=\int \left (c + d x\right )^{m} \sin {\left (a + b x \right )}\, dx \]
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\[ \int (c+d x)^m \sin (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sin \left (b x + a\right ) \,d x } \]
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\[ \int (c+d x)^m \sin (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sin \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x)^m \sin (a+b x) \, dx=\int \sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^m \,d x \]
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