Integrand size = 16, antiderivative size = 267 \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=-\frac {3 e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )}{8 b} \]
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Time = 0.21 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3393, 3389, 2212} \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=-\frac {3 e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-m-1} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 i b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-m-1} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 i b (c+d x)}{d}\right )}{8 b} \]
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Rule 2212
Rule 3389
Rule 3393
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{4} (c+d x)^m \sin (a+b x)-\frac {1}{4} (c+d x)^m \sin (3 a+3 b x)\right ) \, dx \\ & = -\left (\frac {1}{4} \int (c+d x)^m \sin (3 a+3 b x) \, dx\right )+\frac {3}{4} \int (c+d x)^m \sin (a+b x) \, dx \\ & = -\left (\frac {1}{8} i \int e^{-i (3 a+3 b x)} (c+d x)^m \, dx\right )+\frac {1}{8} i \int e^{i (3 a+3 b x)} (c+d x)^m \, dx+\frac {3}{8} i \int e^{-i (a+b x)} (c+d x)^m \, dx-\frac {3}{8} i \int e^{i (a+b x)} (c+d x)^m \, dx \\ & = -\frac {3 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 )}{8 b}-\frac {3 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 )}{8 b}+\frac {3^{-1-m} e^{3 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 {3 i b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{-3 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 {3 i b (c+d x)}{d}\right )}{8 b} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.94 \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=\frac {3^{-1-m} e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-3^{2+m} e^{2 i \left (2 a+\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )-3^{2+m} e^{2 i a+\frac {4 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )+e^{6 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )\right )}{8 b} \]
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\[\int \left (d x +c \right )^{m} \left (\sin ^{3}\left (b x +a \right )\right )d x\]
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none
Time = 0.11 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70 \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=-\frac {9 \, e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) - e^{\left (-\frac {d m \log \left (-\frac {3 i \, b}{d}\right ) + 3 i \, b c - 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) + 9 \, e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right ) - e^{\left (-\frac {d m \log \left (\frac {3 i \, b}{d}\right ) - 3 i \, b c + 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )}{24 \, b} \]
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\[ \int (c+d x)^m \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{m} \sin ^{3}{\left (a + b x \right )}\, dx \]
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\[ \int (c+d x)^m \sin ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sin \left (b x + a\right )^{3} \,d x } \]
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\[ \int (c+d x)^m \sin ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sin \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int (c+d x)^m \sin ^3(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]
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