\(\int \cos (c+d x) (a+i a \tan (c+d x))^n \, dx\) [474]

Optimal result

Integrand size = 22, antiderivative size = 88 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^n \, dx=-\frac {i 2^{-\frac {1}{2}+n} a \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-n,\frac {1}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {3}{2}-n} (a+i a \tan (c+d x))^{-1+n}}{d} \] Output:

-I*2^(-1/2+n)*a*cos(d*x+c)*hypergeom([-1/2, 3/2-n],[1/2],1/2-1/2*I*tan(d*x 
+c))*(1+I*tan(d*x+c))^(3/2-n)*(a+I*a*tan(d*x+c))^(-1+n)/d
 

Mathematica [A] (verified)

Time = 12.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.66 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^n \, dx=-\frac {i 2^{-1+n} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{-1+n} \left (1+e^{2 i (c+d x)}\right )^{-1+n} \operatorname {Hypergeometric2F1}\left (-1+n,-\frac {1}{2}+n,\frac {1}{2}+n,-e^{2 i (c+d x)}\right ) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (-1+2 n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3042, 3986, 3042, 4006, 80, 79}

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.

\(\displaystyle \int \cos (c+d x) (a+i a \tan (c+d x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (c+d x))^n}{\sec (c+d x)}dx\)

\(\Big \downarrow \) 3986

\(\displaystyle \cos (c+d x) \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{n-\frac {1}{2}}}{\sqrt {a-i a \tan (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \cos (c+d x) \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{n-\frac {1}{2}}}{\sqrt {a-i a \tan (c+d x)}}dx\)

\(\Big \downarrow \) 4006

\(\displaystyle \frac {a^2 \cos (c+d x) \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{n-\frac {3}{2}}}{(a-i a \tan (c+d x))^{3/2}}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {a 2^{n-\frac {3}{2}} \cos (c+d x) \sqrt {a-i a \tan (c+d x)} (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^n \int \frac {\left (\frac {1}{2} i \tan (c+d x)+\frac {1}{2}\right )^{n-\frac {3}{2}}}{(a-i a \tan (c+d x))^{3/2}}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {i 2^{n-\frac {1}{2}} \cos (c+d x) (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-n,\frac {1}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d}\)

Input:

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

Output:

((-I)*2^(-1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[-1/2, 3/2 - n, 1/2, (1 - 
 I*Tan[c + d*x])/2]*(1 + I*Tan[c + d*x])^(1/2 - n)*(a + I*a*Tan[c + d*x])^ 
n)/d
 

Defintions of rubi rules used

rule 79

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

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 3042

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

rule 3986

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(n_.), x_Symbol] :> Simp[(d*Sec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/ 
2)*(a - b*Tan[e + f*x])^(m/2))   Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b* 
Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + 
 b^2, 0]
 

rule 4006

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f)   Subst[Int[(a + b*x)^(m - 1)*( 
c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n 
}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(cos(d*x+c)*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

Integral((I*a*(tan(c + d*x) - I))**n*cos(c + d*x), x)
 

Maxima [F]

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

integrate(cos(d*x+c)*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(cos(d*x+c)*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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