Optimal. Leaf size=402 \[ \frac{a^3 3^{-m-1} e^{3 e-\frac{3 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{3 a^3 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{15 a^3 e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{8 f}-\frac{15 a^3 e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{8 f}-\frac{3 a^3 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}-\frac{a^3 3^{-m-1} e^{\frac{3 c f}{d}-3 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{5 a^3 (c+d x)^{m+1}}{2 d (m+1)} \]
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Rubi [A] time = 0.552822, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3318, 3312, 3307, 2181} \[ \frac{a^3 3^{-m-1} e^{3 e-\frac{3 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{3 a^3 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{15 a^3 e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{8 f}-\frac{15 a^3 e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{8 f}-\frac{3 a^3 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}-\frac{a^3 3^{-m-1} e^{\frac{3 c f}{d}-3 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{5 a^3 (c+d x)^{m+1}}{2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx &=\left (8 a^3\right ) \int (c+d x)^m \sin ^6\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right ) \, dx\\ &=\left (8 a^3\right ) \int \left (\frac{5}{16} (c+d x)^m+\frac{15}{32} (c+d x)^m \cosh (e+f x)+\frac{3}{16} (c+d x)^m \cosh (2 e+2 f x)+\frac{1}{32} (c+d x)^m \cosh (3 e+3 f x)\right ) \, dx\\ &=\frac{5 a^3 (c+d x)^{1+m}}{2 d (1+m)}+\frac{1}{4} a^3 \int (c+d x)^m \cosh (3 e+3 f x) \, dx+\frac{1}{2} \left (3 a^3\right ) \int (c+d x)^m \cosh (2 e+2 f x) \, dx+\frac{1}{4} \left (15 a^3\right ) \int (c+d x)^m \cosh (e+f x) \, dx\\ &=\frac{5 a^3 (c+d x)^{1+m}}{2 d (1+m)}+\frac{1}{8} a^3 \int e^{-i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac{1}{8} a^3 \int e^{i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac{1}{4} \left (3 a^3\right ) \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac{1}{4} \left (3 a^3\right ) \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac{1}{8} \left (15 a^3\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac{1}{8} \left (15 a^3\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx\\ &=\frac{5 a^3 (c+d x)^{1+m}}{2 d (1+m)}+\frac{3^{-1-m} a^3 e^{3 e-\frac{3 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{3\ 2^{-3-m} a^3 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{15 a^3 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 )}{8 f}-\frac{15 a^3 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 )}{8 f}-\frac{3\ 2^{-3-m} a^3 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{3^{-1-m} a^3 e^{-3 e+\frac{3 c f}{d}} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{3 f (c+d x)}{d}\right )}{8 f}\\ \end{align*}
Mathematica [A] time = 2.28873, size = 429, normalized size = 1.07 \[ -\frac{a^3 2^{-m-6} 3^{-m-1} e^{-3 \left (\frac{c f}{d}+e\right )} (c+d x)^m (\cosh (e+f x)+1)^3 \text{sech}^6\left (\frac{1}{2} (e+f x)\right ) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (2^m e^{\frac{3 c f}{d}} \left (d (m+1) e^{\frac{3 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )-20 e^{3 e} f 3^{m+1} (c+d x) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^m\right )+d \left (-3^{m+2}\right ) (m+1) e^{\frac{c f}{d}+5 e} \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )-d e^{6 e} 2^m (m+1) \left (\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )-5 d 2^m 3^{m+2} (m+1) e^{\frac{2 c f}{d}+4 e} \left (\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )+5 d 2^m 3^{m+2} (m+1) e^{\frac{4 c f}{d}+2 e} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )+d 3^{m+2} (m+1) e^{\frac{5 c f}{d}+e} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )\right )}{d f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.15, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+a\cosh \left ( fx+e \right ) \right ) ^{3}\, 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 = 2.40286, size = 1665, normalized size = 4.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3}{\left (d x + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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