Integrand size = 14, antiderivative size = 110 \[ \int (c+d x)^m \cosh (a+b x) \, dx=\frac {e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )}{2 b}-\frac {e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )}{2 b} \]
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Time = 0.07 (sec) , antiderivative size = 110, 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 = {3388, 2212} \[ \int (c+d x)^m \cosh (a+b x) \, dx=\frac {e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {b (c+d x)}{d}\right )}{2 b}-\frac {e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {b (c+d x)}{d}\right )}{2 b} \]
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Rule 2212
Rule 3388
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int e^{-i (i a+i b x)} (c+d x)^m \, dx+\frac {1}{2} \int e^{i (i a+i b x)} (c+d x)^m \, dx \\ & = \frac {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 )}{2 b}-\frac {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 )}{2 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int (c+d x)^m \cosh (a+b x) \, dx=\frac {e^{-a-\frac {b c}{d}} (c+d x)^m \left (e^{2 a} \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )-e^{\frac {2 b c}{d}} \left (b \left (\frac {c}{d}+x\right )\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )\right )}{2 b} \]
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\[\int \left (d x +c \right )^{m} \cosh \left (b x +a \right )d x\]
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none
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.53 \[ \int (c+d x)^m \cosh (a+b x) \, dx=-\frac {\cosh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) - \cosh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) - \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) + \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right )}{2 \, b} \]
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Exception generated. \[ \int (c+d x)^m \cosh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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none
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int (c+d x)^m \cosh (a+b x) \, dx=-\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-a + \frac {b c}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (a - \frac {b c}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} \]
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\[ \int (c+d x)^m \cosh (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cosh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x)^m \cosh (a+b x) \, dx=\int \mathrm {cosh}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^m \,d x \]
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