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

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

[Out]

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

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Rubi [A]  time = 0.0631324, 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 = {3083} \[ -\frac{i \sin ^{-n}(c+d x) \, _2F_1\left (1,n;n+1;-\frac{1}{2} i (\cot (c+d x)+i)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3083

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

Rubi steps

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

Mathematica [C]  time = 3.45183, size = 367, normalized size = 5.56 \[ -\frac{4 \sin \left (\frac{1}{2} (c+d x)\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \sin ^{-n}(c+d x) (a (\cos (c+d x)+i \sin (c+d x)))^n \left (\text{Hypergeometric2F1}\left (1-2 n,1-n,2-n,-i \tan \left (\frac{1}{2} (c+d x)\right )\right )+F_1\left (1-n;-2 n,1;2-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )\right )}{d (n-1) \left (2 F_1\left (1-n;-2 n,1;2-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )+\frac{\left (1-i \tan \left (\frac{1}{2} (c+d x)\right )\right ) \left (-2 n (i \sin (c+d x)+\cos (c+d x)-1) F_1\left (2-n;1-2 n,1;3-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )-(i \sin (c+d x)+\cos (c+d x)-1) F_1\left (2-n;-2 n,2;3-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )+(n-2) (\cos (c+d x)+1) \left (1+i \tan \left (\frac{1}{2} (c+d x)\right )\right )^{2 n}\right )}{n-2}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*Cos[(c + d*x)/2]*(AppellF1[1 - n, -2*n, 1, 2 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]] + Hypergeomet
ric2F1[1 - 2*n, 1 - n, 2 - n, (-I)*Tan[(c + d*x)/2]])*Sin[(c + d*x)/2]*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^n)/
(d*(-1 + n)*Sin[c + d*x]^n*(2*AppellF1[1 - n, -2*n, 1, 2 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]] + ((-
2*n*AppellF1[2 - n, 1 - 2*n, 1, 3 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]]*(-1 + Cos[c + d*x] + I*Sin[c
+ d*x]) - AppellF1[2 - n, -2*n, 2, 3 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]]*(-1 + Cos[c + d*x] + I*S
in[c + d*x]) + (-2 + n)*(1 + Cos[c + d*x])*(1 + I*Tan[(c + d*x)/2])^(2*n))*(1 - I*Tan[(c + d*x)/2]))/(-2 + n))
)

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Maple [F]  time = 0.631, 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 ( \sin \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/(sin(d*x+c)^n),x)

[Out]

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

[Out]

integrate((a*cos(d*x + c) + I*a*sin(d*x + c))^n*sin(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 (-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\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/(sin(d*x+c)^n),x, algorithm="fricas")

[Out]

integral((a*e^(I*d*x + I*c))^n/(1/2*(-I*e^(2*I*d*x + 2*I*c) + I)*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/(sin(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}}{\sin \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/(sin(d*x+c)^n),x, algorithm="giac")

[Out]

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