Optimal. Leaf size=36 \[ -\frac {f^{a+b x} \left (e^{-i (c+d x)}\right )^n}{-b \log (f)+i d n} \]
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Rubi [A] time = 0.10, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4614, 2281, 2287, 2194} \[ -\frac {f^{a+b x} \left (e^{-i (c+d x)}\right )^n}{-b \log (f)+i d n} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2281
Rule 2287
Rule 4614
Rubi steps
\begin {align*} \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx &=\int \left (e^{-i (c+d x)}\right )^n f^{a+b x} \, dx\\ &=\left (e^{i n (c+d x)} \left (e^{-i (c+d x)}\right )^n\right ) \int e^{-i n (c+d x)} f^{a+b x} \, dx\\ &=\left (e^{i n (c+d x)} \left (e^{-i (c+d x)}\right )^n\right ) \int \exp (-i c n+a \log (f)-x (i d n-b \log (f))) \, dx\\ &=-\frac {\left (e^{-i (c+d x)}\right )^n f^{a+b x}}{i d n-b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 43, normalized size = 1.19 \[ \frac {i f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n}{d n+i b \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 30, normalized size = 0.83 \[ \frac {f^{b x + a} e^{\left (-i \, d n x - i \, c n\right )}}{-i \, d n + b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.83, size = 31, normalized size = 0.86 \[ \frac {f^{a} e^{\left (-i \, d n x + b x \log \relax (f) - i \, c n\right )}}{-i \, d n + b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 86, normalized size = 2.39 \[ \frac {{\mathrm e}^{\left (b x +a \right ) \ln \relax (f )} {\mathrm e}^{n \ln \left (\frac {1-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\right )}}{-i d n +b \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 62, normalized size = 1.72 \[ \frac {f^{b x} f^{a} \cos \left (d n x\right ) - i \, f^{b x} f^{a} \sin \left (d n x\right )}{{\left (-i \, d n + b \log \relax (f)\right )} \cos \left (c n\right ) + {\left (d n + i \, b \log \relax (f)\right )} \sin \left (c n\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 35, normalized size = 0.97 \[ -\frac {f^{a+b\,x}\,{\left ({\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\right )}^n}{-b\,\ln \relax (f)+d\,n\,1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.66, size = 107, normalized size = 2.97 \[ \begin {cases} - \frac {f^{a} f^{b x} \left (- i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )^{n}}{- b \log {\relax (f )} + i d n} & \text {for}\: b \neq \frac {i d n}{\log {\relax (f )}} \\f^{a} x \left (- i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )^{n} e^{i d n x} + \frac {i f^{a} \left (- i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )^{n} e^{i d n x}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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