Optimal. Leaf size=263 \[ \frac {a^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a^2 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{f}-\frac {a^2 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{f}-\frac {a^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a^2 (c+d x)^{m+1}}{2 d (m+1)} \]
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Rubi [A] time = 0.34, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3318, 3312, 3307, 2181} \[ \frac {a^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a^2 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {f (c+d x)}{d}\right )}{f}-\frac {a^2 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {f (c+d x)}{d}\right )}{f}-\frac {a^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a^2 (c+d x)^{m+1}}{2 d (m+1)} \]
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))^2 \, dx &=\left (4 a^2\right ) \int (c+d x)^m \sin ^4\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx\\ &=\left (4 a^2\right ) \int \left (\frac {3}{8} (c+d x)^m+\frac {1}{2} (c+d x)^m \cosh (e+f x)+\frac {1}{8} (c+d x)^m \cosh (2 e+2 f x)\right ) \, dx\\ &=\frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {1}{2} a^2 \int (c+d x)^m \cosh (2 e+2 f x) \, dx+\left (2 a^2\right ) \int (c+d x)^m \cosh (e+f x) \, dx\\ &=\frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {1}{4} a^2 \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac {1}{4} a^2 \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx+a^2 \int e^{-i (i e+i f x)} (c+d x)^m \, dx+a^2 \int e^{i (i e+i f x)} (c+d x)^m \, dx\\ &=\frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {2^{-3-m} a^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^2 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^2 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} a^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*}
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Mathematica [A] time = 1.09, size = 302, normalized size = 1.15 \[ -\frac {a^2 2^{-m-5} e^{-2 \left (\frac {c f}{d}+e\right )} (c+d x)^m (\cosh (e+f x)+1)^2 \text {sech}^4\left (\frac {1}{2} (e+f x)\right ) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-3 f 2^{m+2} (c+d x) e^{2 \left (\frac {c f}{d}+e\right )} \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m-d e^{4 e} (m+1) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )-d 2^{m+3} (m+1) e^{\frac {c f}{d}+3 e} \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )+d 2^{m+3} (m+1) e^{\frac {3 c f}{d}+e} \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )+d (m+1) e^{\frac {4 c f}{d}} \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )\right )}{d f (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 493, normalized size = 1.87 \[ -\frac {{\left (a^{2} d m + a^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 8 \, {\left (a^{2} d m + a^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - 8 \, {\left (a^{2} d m + a^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - {\left (a^{2} d m + a^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (a^{2} d m + a^{2} d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 8 \, {\left (a^{2} d m + a^{2} d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) + 8 \, {\left (a^{2} d m + a^{2} d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) + {\left (a^{2} d m + a^{2} d\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) - 12 \, {\left (a^{2} d f x + a^{2} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 12 \, {\left (a^{2} d f x + a^{2} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{8 \, {\left (d f m + d f\right )}} \]
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 \[ \int {\left (a \cosh \left (f x + e\right ) + a\right )}^{2} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +a \cosh \left (f x +e \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 209, normalized size = 0.79 \[ -\frac {1}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} a^{2} - {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{2} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]
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 \[ \int {\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]
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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