3.246 \(\int \cos (c+d x) (b \cos (c+d x))^n \, dx\)

Optimal. Leaf size=69 \[ -\frac{\sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0491827, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {16, 2643} \[ -\frac{\sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*(b*Cos[c + d*x])^n,x]

[Out]

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

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \cos (c+d x) (b \cos (c+d x))^n \, dx &=\frac{\int (b \cos (c+d x))^{1+n} \, dx}{b}\\ &=-\frac{(b \cos (c+d x))^{2+n} \, _2F_1\left (\frac{1}{2},\frac{2+n}{2};\frac{4+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0540431, size = 70, normalized size = 1.01 \[ -\frac{\sqrt{\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{d (n+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^n,x]

[Out]

-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Cot[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*S
qrt[Sin[c + d*x]^2])/(d*(2 + n)))

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Maple [F]  time = 1.167, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( b\cos \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)*(b*cos(d*x+c))^n,x)

[Out]

int(cos(d*x+c)*(b*cos(d*x+c))^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*cos(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c))^n*cos(d*x + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*cos(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c))^n*cos(d*x + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos{\left (c + d x \right )}\right )^{n} \cos{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*cos(d*x+c))**n,x)

[Out]

Integral((b*cos(c + d*x))**n*cos(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(b*cos(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c))^n*cos(d*x + c), x)