Optimal. Leaf size=402 \[ \frac {a^3 3^{-m-1} e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3 a^3 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{8 f}-\frac {15 a^3 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3 a^3 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {a^3 3^{-m-1} e^{\frac {3 c f}{d}-3 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {5 a^3 (c+d x)^{m+1}}{2 d (m+1)} \]
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Rubi [A] time = 0.55, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2181
Rule 3307
Rule 3312
Rule 3318
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*}
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Mathematica [A] time = 2.42, size = 429, normalized size = 1.07 \[ -\frac {a^3 2^{-m-6} 3^{-m-1} e^{-3 \left (\frac {c f}{d}+e\right )} (c+d x)^m (\cosh (e+f x)+1)^3 \text {sech}^6\left (\frac {1}{2} (e+f x)\right ) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (2^m e^{\frac {3 c f}{d}} \left (d (m+1) e^{\frac {3 c f}{d}} \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {3 f (c+d x)}{d}\right )-20 e^{3 e} f 3^{m+1} (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m\right )+d \left (-3^{m+2}\right ) (m+1) e^{\frac {c f}{d}+5 e} \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )-d e^{6 e} 2^m (m+1) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,-\frac {3 f (c+d x)}{d}\right )-5 d 2^m 3^{m+2} (m+1) e^{\frac {2 c f}{d}+4 e} \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )+5 d 2^m 3^{m+2} (m+1) e^{\frac {4 c f}{d}+2 e} \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )+d 3^{m+2} (m+1) e^{\frac {5 c f}{d}+e} \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )\right )}{d f (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 713, normalized size = 1.77 \[ -\frac {{\left (a^{3} d m + a^{3} d\right )} \cosh \left (\frac {d m \log \left (\frac {3 \, f}{d}\right ) + 3 \, d e - 3 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left (a^{3} d m + a^{3} 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 ) + 45 \, {\left (a^{3} d m + a^{3} 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 ) - 45 \, {\left (a^{3} d m + a^{3} 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 ) - 9 \, {\left (a^{3} d m + a^{3} 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 (a^{3} d m + a^{3} d\right )} \cosh \left (\frac {d m \log \left (-\frac {3 \, f}{d}\right ) - 3 \, d e + 3 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (a^{3} d m + a^{3} d\right )} \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, f}{d}\right ) + 3 \, d e - 3 \, c f}{d}\right ) - 9 \, {\left (a^{3} d m + a^{3} 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 ) - 45 \, {\left (a^{3} d m + a^{3} 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 ) + 45 \, {\left (a^{3} d m + a^{3} 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 ) + 9 \, {\left (a^{3} d m + a^{3} 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 ) + {\left (a^{3} d m + a^{3} d\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, f}{d}\right ) - 3 \, d e + 3 \, c f}{d}\right ) - 60 \, {\left (a^{3} d f x + a^{3} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 60 \, {\left (a^{3} d f x + a^{3} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{24 \, {\left (d f m + d f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cosh \left (f x + e\right ) + a\right )}^{3} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +a \cosh \left (f x +e \right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 373, normalized size = 0.93 \[ -\frac {1}{8} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, e + \frac {3 \, c f}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, e - \frac {3 \, c f}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{3} - \frac {3}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} a^{3} - \frac {3}{2} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{3} + \frac {{\left (d x + c\right )}^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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