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