Integrand size = 16, antiderivative size = 237 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {3^{-1-m} e^{3 a-\frac {3 b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{-3 a+\frac {3 b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )}{8 b} \]
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Time = 0.24 (sec) , antiderivative size = 237, 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} \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {3^{-m-1} e^{3 a-\frac {3 b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {b (c+d x)}{d}\right )}{8 b}+\frac {3^{-m-1} e^{\frac {3 b c}{d}-3 a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 b (c+d x)}{d}\right )}{8 b} \]
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Rule 2212
Rule 3389
Rule 3393
Rubi steps \begin{align*} \text {integral}& = i \int \left (\frac {3}{4} i (c+d x)^m \sinh (a+b x)-\frac {1}{4} i (c+d x)^m \sinh (3 a+3 b x)\right ) \, dx \\ & = \frac {1}{4} \int (c+d x)^m \sinh (3 a+3 b x) \, dx-\frac {3}{4} \int (c+d x)^m \sinh (a+b x) \, dx \\ & = \frac {1}{8} \int e^{-i (3 i a+3 i b x)} (c+d x)^m \, dx-\frac {1}{8} \int e^{i (3 i a+3 i b x)} (c+d x)^m \, dx-\frac {3}{8} \int e^{-i (i a+i b x)} (c+d x)^m \, dx+\frac {3}{8} \int e^{i (i a+i b x)} (c+d x)^m \, dx \\ & = \frac {3^{-1-m} e^{3 a-\frac {3 b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{-3 a+\frac {3 b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )}{8 b} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.87 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {3^{-1-m} e^{-3 \left (a+\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{6 a} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )-3^{2+m} e^{4 a+\frac {2 b c}{d}} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \left (-\frac {b (c+d x)}{d}\right )^m \left (-3^{2+m} e^{2 a} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )\right )\right )}{8 b} \]
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\[\int \left (d x +c \right )^{m} \sinh \left (b x +a \right )^{3}d x\]
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Time = 0.09 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.43 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {\cosh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) + \cosh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) + 9 \, \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) + 9 \, \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) - \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right )}{24 \, b} \]
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\[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\int \left (c + d x\right )^{m} \sinh ^{3}{\left (a + b x \right )}\, dx \]
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Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-a + \frac {b c}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (a - \frac {b c}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \]
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\[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sinh \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\int {\mathrm {sinh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]
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