Optimal. Leaf size=131 \[ \frac{a e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{2 f}-\frac{a e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.147174, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3307, 2181} \[ \frac{a e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{2 f}-\frac{a e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int (c+d x)^m (a+a \cosh (e+f x)) \, dx &=\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*}
Mathematica [A] time = 0.31201, size = 189, normalized size = 1.44 \[ -\frac{a e^{-\frac{c f}{d}-e} (c+d x)^m (\cosh (e+f x)+1) \text{sech}^2\left (\frac{1}{2} (e+f x)\right ) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-d e^{2 e} (m+1) \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )+d (m+1) e^{\frac{2 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )-2 f (c+d x) e^{\frac{c f}{d}+e} \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^m\right )}{4 d f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+a\cosh \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21396, size = 586, normalized size = 4.47 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cosh \left (f x + e\right ) + a\right )}{\left (d x + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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