Integrand size = 18, antiderivative size = 131 \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=\frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f} \]
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Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3398, 3388, 2212} \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=\frac {a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {a e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)} \]
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Rule 2212
Rule 3388
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^m+a (c+d x)^m \cosh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^{1+m}}{d (1+m)}+a \int (c+d x)^m \cosh (e+f x) \, dx \\ & = \frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {1}{2} a \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac {1}{2} a \int e^{i (i e+i f x)} (c+d x)^m \, dx \\ & = \frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.44 \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=-\frac {a e^{-e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} (1+\cosh (e+f x)) \left (-2 e^{e+\frac {c f}{d}} f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m-d e^{2 e} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )+d e^{\frac {2 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )\right ) \text {sech}^2\left (\frac {1}{2} (e+f x)\right )}{4 d f (1+m)} \]
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\[\int \left (d x +c \right )^{m} \left (a +a \cosh \left (f x +e \right )\right )d x\]
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none
Time = 0.08 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.90 \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=-\frac {{\left (a d m + a d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - {\left (a d m + a d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - {\left (a d m + a d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) + {\left (a d m + a d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) - 2 \, {\left (a d f x + a c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 2 \, {\left (a d f x + a c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{2 \, {\left (d f m + d f\right )}} \]
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Exception generated. \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=\text {Exception raised: TypeError} \]
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none
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=-\frac {1}{2} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]
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\[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=\int { {\left (a \cosh \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \]
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Timed out. \[ \int (c+d x)^m (a+a \cosh (e+f x)) \, dx=\int \left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^m \,d x \]
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