\(\int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx\) [468]

Optimal result

Integrand size = 24, antiderivative size = 56 \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\frac {i a \operatorname {Hypergeometric2F1}\left (2,-1+n,n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-1+n}}{4 d (1-n)} \]

[Out]

1/4*I*a*hypergeom([2, -1+n],[n],1/2+1/2*I*tan(d*x+c))*(a+I*a*tan(d*x+c))^(-1+n)/d/(1-n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 70} \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\frac {i a (a+i a \tan (c+d x))^{n-1} \operatorname {Hypergeometric2F1}\left (2,n-1,n,\frac {1}{2} (i \tan (c+d x)+1)\right )}{4 d (1-n)} \]

[In]

Int[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^n,x]

[Out]

((I/4)*a*Hypergeometric2F1[2, -1 + n, n, (1 + I*Tan[c + d*x])/2]*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(1 - n))

Rule 70

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {(a+x)^{-2+n}}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = \frac {i a \operatorname {Hypergeometric2F1}\left (2,-1+n,n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-1+n}}{4 d (1-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=-\frac {i a \operatorname {Hypergeometric2F1}\left (2,-1+n,n,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-1+n}}{4 d (-1+n)} \]

[In]

Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^n,x]

[Out]

((-1/4*I)*a*Hypergeometric2F1[2, -1 + n, n, (1 + I*Tan[c + d*x])/2]*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(-1 +
n))

Maple [F]

\[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n}d x\]

[In]

int(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^n,x)

[Out]

int(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^n,x)

Fricas [F]

\[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral(1/4*(2*a*e^(2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^n*(e^(4*I*d*x + 4*I*c) + 2*e^(2*I*d*x + 2*I*c
) + 1)*e^(-2*I*d*x - 2*I*c), x)

Sympy [F]

\[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \cos ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(cos(d*x+c)**2*(a+I*a*tan(d*x+c))**n,x)

[Out]

Integral((I*a*(tan(c + d*x) - I))**n*cos(c + d*x)**2, x)

Maxima [F]

\[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)^n*cos(d*x + c)^2, x)

Giac [F]

\[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^n*cos(d*x + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]

[In]

int(cos(c + d*x)^2*(a + a*tan(c + d*x)*1i)^n,x)

[Out]

int(cos(c + d*x)^2*(a + a*tan(c + d*x)*1i)^n, x)