Optimal. Leaf size=34 \[ \frac{f^{a+b x} \left (e^{i (c+d x)}\right )^n}{b \log (f)+i d n} \]
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Rubi [A] time = 0.0970483, antiderivative size = 34, 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 e^{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 = 0.092996, size = 43, normalized size = 1.26 \[ -\frac{i f^{a+b x} (\cos (c+d x)+i \sin (c+d x))^n}{d n-i b \log (f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.125, size = 86, normalized size = 2.5 \begin{align*}{\frac{{{\rm e}^{ \left ( bx+a \right ) \ln \left ( f \right ) }}}{idn+b\ln \left ( f \right ) }{{\rm e}^{n\ln \left ({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}}+{ \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}} \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988954, size = 68, normalized size = 2. \begin{align*} \frac{-i \, f^{b x} f^{a} \cos \left (d n x + c n\right ) + f^{b x} f^{a} \sin \left (d n x + c n\right )}{d n - i \, b \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3028, size = 70, normalized size = 2.06 \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 = 1.82261, size = 42, normalized size = 1.24 \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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