Optimal. Leaf size=148 \[ -\frac {a e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.14, antiderivative size = 148, normalized size of antiderivative = 1.00, 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, 3308, 2181} \[ -\frac {a e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {i f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^m (a+a \sin (e+f x)) \, dx &=\int \left (a (c+d x)^m+a (c+d x)^m \sin (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^{1+m}}{d (1+m)}+a \int (c+d x)^m \sin (e+f x) \, dx\\ &=\frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {1}{2} (i a) \int e^{-i (e+f x)} (c+d x)^m \, dx-\frac {1}{2} (i a) \int e^{i (e+f x)} (c+d x)^m \, dx\\ &=\frac {a (c+d x)^{1+m}}{d (1+m)}-\frac {a e^{i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i f (c+d x)}{d}\right )}{2 f}-\frac {a e^{-i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i f (c+d x)}{d}\right )}{2 f}\\ \end {align*}
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Mathematica [A] time = 2.93, size = 199, normalized size = 1.34 \[ -\frac {a (c+d x)^m (\sin (e+f x)+1) \left (d (m+1) \left (-\frac {i f (c+d x)}{d}\right )^{-m} \left (\cos \left (e-\frac {c f}{d}\right )+i \sin \left (e-\frac {c f}{d}\right )\right ) \Gamma \left (m+1,-\frac {i f (c+d x)}{d}\right )+d (m+1) \left (\frac {i f (c+d x)}{d}\right )^{-m} \left (\cos \left (e-\frac {c f}{d}\right )-i \sin \left (e-\frac {c f}{d}\right )\right ) \Gamma \left (m+1,\frac {i f (c+d x)}{d}\right )-2 c f-2 d (e+f x)+2 d e\right )}{2 d f (m+1) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 136, normalized size = 0.92 \[ -\frac {{\left (a d m + a d\right )} e^{\left (-\frac {d m \log \left (\frac {i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, c f}{d}\right ) + {\left (a d m + a d\right )} e^{\left (-\frac {d m \log \left (-\frac {i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, c f}{d}\right ) - 2 \, {\left (a d f x + a c f\right )} {\left (d x + c\right )}^{m}}{2 \, {\left (d f m + d f\right )}} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +a \sin \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \int {\left (d x + c\right )}^{m} \sin \left (f x + e\right )\,{d x} + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \left (c + d x\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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