\(\int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx\) [187]

Optimal result

Integrand size = 33, antiderivative size = 66 \[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=-\frac {i \cos ^{-n}(c+d x) \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \] Output:

-1/2*I*hypergeom([1, n],[1+n],1/2+1/2*I*tan(d*x+c))*(a*cos(d*x+c)+I*a*sin( 
d*x+c))^n/d/n/(cos(d*x+c)^n)
 

Mathematica [A] (verified)

Time = 2.88 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36 \[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\frac {\cos ^{-n}(c+d x) (a (\cos (c+d x)+i \sin (c+d x)))^n \left (-2 i (1+n)+n \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (-i+\tan (c+d x))\right )}{4 d n (1+n)} \] Input:

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

Output:

((a*(Cos[c + d*x] + I*Sin[c + d*x]))^n*((-2*I)*(1 + n) + n*Hypergeometric2 
F1[1, 1 + n, 2 + n, (1 + I*Tan[c + d*x])/2]*(-I + Tan[c + d*x])))/(4*d*n*( 
1 + n)*Cos[c + d*x]^n)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {3042, 3563}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^{-n} (a \cos (c+d x)+i a \sin (c+d x))^ndx\)

\(\Big \downarrow \) 3563

\(\displaystyle -\frac {i \cos ^{-n}(c+d x) \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {1}{2} (i \tan (c+d x)+1)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}\)

Input:

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

Output:

((-1/2*I)*Hypergeometric2F1[1, n, 1 + n, (1 + I*Tan[c + d*x])/2]*(a*Cos[c 
+ d*x] + I*a*Sin[c + d*x])^n)/(d*n*Cos[c + d*x]^n)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3563

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si 
n[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*((a*Cos[c + d*x] + b*Si 
n[c + d*x])^n/(2*a*d*n*Cos[c + d*x]^n))*Hypergeometric2F1[1, n, n + 1, (a + 
 b*Tan[c + d*x])/(2*a)], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] & 
& EqQ[a^2 + b^2, 0] &&  !IntegerQ[n]
 

Maple [F]

\[\int \left (a \cos \left (d x +c \right )+i a \sin \left (d x +c \right )\right )^{n} \cos \left (d x +c \right )^{-n}d x\]

Input:

int((a*cos(d*x+c)+I*a*sin(d*x+c))^n/(cos(d*x+c)^n),x)
 

Output:

int((a*cos(d*x+c)+I*a*sin(d*x+c))^n/(cos(d*x+c)^n),x)
 

Fricas [F]

\[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n}}{\cos \left (d x + c\right )^{n}} \,d x } \] Input:

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^n/(cos(d*x+c)^n),x, algorithm="fri 
cas")
 

Output:

integral(e^(I*d*n*x + I*c*n + n*log(a))/(1/2*(e^(2*I*d*x + 2*I*c) + 1)*e^( 
-I*d*x - I*c))^n, x)
 

Sympy [F]

\[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\int \left (a \left (i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )\right )^{n} \cos ^{- n}{\left (c + d x \right )}\, dx \] Input:

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))**n/(cos(d*x+c)**n),x)
 

Output:

Integral((a*(I*sin(c + d*x) + cos(c + d*x)))**n/cos(c + d*x)**n, x)
 

Maxima [F]

\[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n}}{\cos \left (d x + c\right )^{n}} \,d x } \] Input:

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^n/(cos(d*x+c)^n),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

\[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n}}{\cos \left (d x + c\right )^{n}} \,d x } \] Input:

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^n/(cos(d*x+c)^n),x, algorithm="gia 
c")
 

Output:

integrate((a*cos(d*x + c) + I*a*sin(d*x + c))^n/cos(d*x + c)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\int \frac {{\left (a\,\cos \left (c+d\,x\right )+a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\cos \left (c+d\,x\right )}^n} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx=\int \frac {\left (\cos \left (d x +c \right ) a +\sin \left (d x +c \right ) a i \right )^{n}}{\cos \left (d x +c \right )^{n}}d x \] Input:

int((a*cos(d*x+c)+I*a*sin(d*x+c))^n/(cos(d*x+c)^n),x)
 

Output:

int((cos(c + d*x)*a + sin(c + d*x)*a*i)**n/cos(c + d*x)**n,x)