Optimal. Leaf size=282 \[ \frac{a b e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{f}-\frac{a b e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{f}+\frac{b^2 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}-\frac{b^2 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)}+\frac{b^2 (c+d x)^{m+1}}{2 d (m+1)} \]
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Rubi [A] time = 0.362635, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3317, 3307, 2181, 3312} \[ \frac{a b e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{f}-\frac{a b e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{f}+\frac{b^2 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}-\frac{b^2 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)}+\frac{b^2 (c+d x)^{m+1}}{2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3307
Rule 2181
Rule 3312
Rubi steps
\begin{align*} \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^m+2 a b (c+d x)^m \cosh (e+f x)+b^2 (c+d x)^m \cosh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+(2 a b) \int (c+d x)^m \cosh (e+f x) \, dx+b^2 \int (c+d x)^m \cosh ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+(a b) \int e^{-i (i e+i f x)} (c+d x)^m \, dx+(a b) \int e^{i (i e+i f x)} (c+d x)^m \, dx+b^2 \int \left (\frac{1}{2} (c+d x)^m+\frac{1}{2} (c+d x)^m \cosh (2 e+2 f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+\frac{b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac{a b 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 )}{f}-\frac{a b 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 )}{f}+\frac{1}{2} b^2 \int (c+d x)^m \cosh (2 e+2 f x) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+\frac{b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac{a b 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 )}{f}-\frac{a b 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 )}{f}+\frac{1}{4} b^2 \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac{1}{4} b^2 \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+\frac{b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac{2^{-3-m} b^2 e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{a b 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 )}{f}-\frac{a b 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 )}{f}-\frac{2^{-3-m} b^2 e^{-2 e+\frac{2 c f}{d}} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{2 f (c+d x)}{d}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.720988, size = 254, normalized size = 0.9 \[ \frac{(c+d x)^m \left (8 a b d (m+1) e^{e-\frac{c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )-8 a b d (m+1) e^{\frac{c f}{d}-e} \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )+b^2 d 2^{-m} (m+1) e^{2 e-\frac{2 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )-b^2 d 2^{-m} (m+1) e^{\frac{2 c f}{d}-2 e} \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )+8 a^2 f (c+d x)+4 b^2 f (c+d x)\right )}{8 d f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+b\cosh \left ( fx+e \right ) \right ) ^{2}\, 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 = 2.30203, size = 1191, normalized size = 4.22 \begin{align*} -\frac{{\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac{d m \log \left (\frac{2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac{2 \,{\left (d f x + c f\right )}}{d}\right ) + 8 \,{\left (a b d m + a b 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 ) - 8 \,{\left (a b d m + a b 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 (b^{2} d m + b^{2} d\right )} \cosh \left (\frac{d m \log \left (-\frac{2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) -{\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, \frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (\frac{2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 8 \,{\left (a b d m + a b 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 ) + 8 \,{\left (a b d m + a b 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 (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, -\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (-\frac{2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) - 4 \,{\left ({\left (2 \, a^{2} + b^{2}\right )} d f x +{\left (2 \, a^{2} + b^{2}\right )} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 4 \,{\left ({\left (2 \, a^{2} + b^{2}\right )} d f x +{\left (2 \, a^{2} + b^{2}\right )} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{8 \,{\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 (b \cosh \left (f x + e\right ) + a\right )}^{2}{\left (d x + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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