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 = 0.0973396, antiderivative size = 36, normalized size of antiderivative = 1., 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 4614
Rule 2281
Rule 2287
Rule 2194
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*}
Mathematica [A] time = 0.075935, 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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Maple [B] time = 0.122, size = 86, normalized size = 2.4 \begin{align*}{\frac{{{\rm e}^{ \left ( bx+a \right ) \ln \left ( f \right ) }}}{-idn+b\ln \left ( f \right ) }{{\rm e}^{n\ln \left ({ \left ( 1- \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-1}}-{2\,i\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995875, size = 84, normalized size = 2.33 \begin{align*} \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 \left (f\right )\right )} \cos \left (c n\right ) +{\left (d n + i \, b \log \left (f\right )\right )} \sin \left (c n\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06092, size = 73, normalized size = 2.03 \begin{align*} \frac{f^{b x + a} \left (e^{\left (-i \, d x - i \, c\right )}\right )^{n}}{-i \, d n + b \log \left (f\right )} \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 [A] time = 2.13264, size = 42, normalized size = 1.17 \begin{align*} \frac{f^{a} e^{\left (-i \, d n x + b x \log \left (f\right ) - i \, c n\right )}}{-i \, d n + b \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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