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

Optimal. Leaf size=56 \[ \frac{i a (a+i a \tan (c+d x))^{n-1} \text{Hypergeometric2F1}\left (2,n-1,n,\frac{1}{2} (1+i \tan (c+d x))\right )}{4 d (1-n)} \]

[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))

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Rubi [A]  time = 0.0572094, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 68} \[ \frac{i a (a+i a \tan (c+d x))^{n-1} \text{Hypergeometric2F1}\left (2,n-1,n,\frac{1}{2} (1+i \tan (c+d x))\right )}{4 d (1-n)} \]

Antiderivative was successfully verified.

[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 3487

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]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-2+n}}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{i a \, _2F_1\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 [B]  time = 13.1871, size = 141, normalized size = 2.52 \[ -\frac{i 2^{n-3} e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^3 \left (e^{i d x}\right )^n \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} \text{Hypergeometric2F1}\left (1,2,n,-e^{2 i (c+d x)}\right ) (a+i a \tan (c+d x))^n}{d (n-1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*2^(-3 + n)*(E^(I*d*x))^n*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(1 + E^((2*I)*(c + d*x)))^3*Hyper
geometric2F1[1, 2, n, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^n)/(d*E^((2*I)*(c + d*x))*(-1 + n)*Sec[c +
d*x]^n*(Cos[d*x] + I*Sin[d*x])^n)

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Maple [F]  time = 1.096, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{4} \, \left (\frac{2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}{\left (e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)