3.187 \(\int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx\)

Optimal. Leaf size=66 \[ -\frac{i \cos ^{-n}(c+d x) \text{Hypergeometric2F1}\left (1,n,n+1,\frac{1}{2} (1+i \tan (c+d x))\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]

[Out]

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

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Rubi [A]  time = 0.0611622, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3084} \[ -\frac{i \cos ^{-n}(c+d x) \, _2F_1\left (1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3084

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> -Simp[(b*(a*Cos[c + d*x] + b*Sin[c + d*x])^n*Hypergeometric2F1[1, n, n + 1, (a + b*Tan[c + d*x])/(2*a)]
)/(2*a*d*n*Cos[c + d*x]^n), x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] &&  !Integer
Q[n]

Rubi steps

\begin{align*} \int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx &=-\frac{i \cos ^{-n}(c+d x) \, _2F_1\left (1,n;1+n;\frac{1}{2} (1+i \tan (c+d x))\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}\\ \end{align*}

Mathematica [A]  time = 2.07308, size = 90, normalized size = 1.36 \[ \frac{\cos ^{-n}(c+d x) \left (n (\tan (c+d x)-i) \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{1}{2} (1+i \tan (c+d x))\right )-2 i (n+1)\right ) (a (\cos (c+d x)+i \sin (c+d x)))^n}{4 d n (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.586, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \right ) \right ) ^{n}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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