Integrand size = 24, antiderivative size = 60 \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\frac {i a^2 \operatorname {Hypergeometric2F1}\left (3,-2+n,-1+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-2+n}}{8 d (2-n)} \] Output:
1/8*I*a^2*hypergeom([3, -2+n],[-1+n],1/2+1/2*I*tan(d*x+c))*(a+I*a*tan(d*x+ c))^(-2+n)/d/(2-n)
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=-\frac {i a^2 \operatorname {Hypergeometric2F1}\left (3,-2+n,-1+n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-2+n}}{8 d (-2+n)} \] Input:
Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^n,x]
Output:
((-1/8*I)*a^2*Hypergeometric2F1[3, -2 + n, -1 + n, (1 + I*Tan[c + d*x])/2] *(a + I*a*Tan[c + d*x])^(-2 + n))/(d*(-2 + n))
Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3042, 3968, 78}
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.
\(\displaystyle \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^n}{\sec (c+d x)^4}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i a^5 \int \frac {(i \tan (c+d x) a+a)^{n-3}}{(a-i a \tan (c+d x))^3}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {i a^2 (a+i a \tan (c+d x))^{n-2} \operatorname {Hypergeometric2F1}\left (3,n-2,n-1,\frac {i \tan (c+d x) a+a}{2 a}\right )}{8 d (2-n)}\) |
Input:
Int[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^n,x]
Output:
((I/8)*a^2*Hypergeometric2F1[3, -2 + n, -1 + n, (a + I*a*Tan[c + d*x])/(2* a)]*(a + I*a*Tan[c + d*x])^(-2 + n))/(d*(2 - n))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
\[\int \cos \left (d x +c \right )^{4} \left (a +i a \tan \left (d x +c \right )\right )^{n}d x\]
Input:
int(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^n,x)
Output:
int(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^n,x)
\[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")
Output:
integral(1/16*(2*a*e^(2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^n*(e^(8* I*d*x + 8*I*c) + 4*e^(6*I*d*x + 6*I*c) + 6*e^(4*I*d*x + 4*I*c) + 4*e^(2*I* d*x + 2*I*c) + 1)*e^(-4*I*d*x - 4*I*c), x)
\[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \cos ^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(cos(d*x+c)**4*(a+I*a*tan(d*x+c))**n,x)
Output:
Integral((I*a*(tan(c + d*x) - I))**n*cos(c + d*x)**4, x)
\[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cos(d*x+c)^4*(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)^4, x)
\[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cos(d*x+c)^4*(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)^4, x)
Timed out. \[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \] Input:
int(cos(c + d*x)^4*(a + a*tan(c + d*x)*1i)^n,x)
Output:
int(cos(c + d*x)^4*(a + a*tan(c + d*x)*1i)^n, x)
\[ \int \cos ^4(c+d x) (a+i a \tan (c+d x))^n \, dx=\int \left (\tan \left (d x +c \right ) a i +a \right )^{n} \cos \left (d x +c \right )^{4}d x \] Input:
int(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^n,x)
Output:
int((tan(c + d*x)*a*i + a)**n*cos(c + d*x)**4,x)