\(\int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx\) [941]

Optimal result

Integrand size = 27, antiderivative size = 36 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=-\frac {\left (e^{-i (c+d x)}\right )^n f^{a+b x}}{i d n-b \log (f)} \]

[Out]

-exp(-I*(d*x+c))^n*f^(b*x+a)/(I*d*n-b*ln(f))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4710, 2319, 2325, 2225} \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=-\frac {f^{a+b x} \left (e^{-i (c+d x)}\right )^n}{-b \log (f)+i d n} \]

[In]

Int[f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n,x]

[Out]

-(((E^((-I)*(c + d*x)))^n*f^(a + b*x))/(I*d*n - b*Log[f]))

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2319

Int[(u_.)*((a_.)*(F_)^(v_))^(n_), x_Symbol] :> Dist[(a*F^v)^n/F^(n*v), Int[u*F^(n*v), x], x] /; FreeQ[{F, a, n
}, x] &&  !IntegerQ[n]

Rule 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 4710

Int[(u_.)*(Cos[v_]*(a_.) + (b_.)*Sin[v_])^(n_.), x_Symbol] :> Int[u*(a/E^((a/b)*v))^n, x] /; FreeQ[{a, b, n},
x] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=\frac {i f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n}{d n+i b \log (f)} \]

[In]

Integrate[f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n,x]

[Out]

(I*f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n)/(d*n + I*b*Log[f])

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {f^{x b +a} \left ({\mathrm e}^{i \left (d x +c \right )}\right )^{-n}}{-i d n +b \ln \left (f \right )}\)	\(34\)
norman	\(-\frac {{\mathrm e}^{\left (x b +a \right ) \ln \left (f \right )} {\mathrm e}^{n \ln \left (-\frac {2 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\right )}}{i d n -b \ln \left (f \right )}\)	\(88\)

[In]

int(f^(b*x+a)*(-I*sin(d*x+c)+cos(d*x+c))^n,x,method=_RETURNVERBOSE)

[Out]

1/(-I*d*n+b*ln(f))*f^(b*x+a)*exp(I*(d*x+c))^(-n)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=\frac {f^{b x + a} e^{\left (-i \, d n x - i \, c n\right )}}{-i \, d n + b \log \left (f\right )} \]

[In]

integrate(f^(b*x+a)*(cos(d*x+c)-I*sin(d*x+c))^n,x, algorithm="fricas")

[Out]

f^(b*x + a)*e^(-I*d*n*x - I*c*n)/(-I*d*n + b*log(f))

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).

Time = 0.51 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.97 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=\begin {cases} \frac {f^{a + b x} \left (- i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )^{n}}{b \log {\left (f \right )} - i d n} & \text {for}\: b \neq \frac {i d n}{\log {\left (f \right )}} \\f^{a + \frac {i d n x}{\log {\left (f \right )}}} x \left (- i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )^{n} & \text {otherwise} \end {cases} \]

[In]

integrate(f**(b*x+a)*(cos(d*x+c)-I*sin(d*x+c))**n,x)

[Out]

Piecewise((f**(a + b*x)*(-I*sin(c + d*x) + cos(c + d*x))**n/(b*log(f) - I*d*n), Ne(b, I*d*n/log(f))), (f**(a +
I*d*n*x/log(f))*x*(-I*sin(c + d*x) + cos(c + d*x))**n, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=\frac {f^{b x} f^{a} \cos \left (d n x\right ) - i \, f^{b x} f^{a} \sin \left (d n x\right )}{{\left (-i \, d n + b \log \left (f\right )\right )} \cos \left (c n\right ) + {\left (d n + i \, b \log \left (f\right )\right )} \sin \left (c n\right )} \]

[In]

integrate(f^(b*x+a)*(cos(d*x+c)-I*sin(d*x+c))^n,x, algorithm="maxima")

[Out]

(f^(b*x)*f^a*cos(d*n*x) - I*f^(b*x)*f^a*sin(d*n*x))/((-I*d*n + b*log(f))*cos(c*n) + (d*n + I*b*log(f))*sin(c*n
))

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=\frac {f^{a} e^{\left (-i \, d n x + b x \log \left (f\right ) - i \, c n\right )}}{-i \, d n + b \log \left (f\right )} \]

[In]

integrate(f^(b*x+a)*(cos(d*x+c)-I*sin(d*x+c))^n,x, algorithm="giac")

[Out]

f^a*e^(-I*d*n*x + b*x*log(f) - I*c*n)/(-I*d*n + b*log(f))

Mupad [B] (verification not implemented)

Time = 27.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx=-\frac {f^{a+b\,x}\,{\left ({\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\right )}^n}{-b\,\ln \left (f\right )+d\,n\,1{}\mathrm {i}} \]

[In]

int(f^(a + b*x)*(cos(c + d*x) - sin(c + d*x)*1i)^n,x)

[Out]

-(f^(a + b*x)*exp(- c*1i - d*x*1i)^n)/(d*n*1i - b*log(f))