3.38 \(\int (c \cos (a+b x))^n \, dx\)

Optimal. Leaf size=69 \[ -\frac {\sin (a+b x) (c \cos (a+b x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b c (n+1) \sqrt {\sin ^2(a+b x)}} \]

[Out]

-(c*cos(b*x+a))^(1+n)*hypergeom([1/2, 1/2+1/2*n],[3/2+1/2*n],cos(b*x+a)^2)*sin(b*x+a)/b/c/(1+n)/(sin(b*x+a)^2)
^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2643} \[ -\frac {\sin (a+b x) (c \cos (a+b x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b c (n+1) \sqrt {\sin ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^n,x]

[Out]

-(((c*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*c*(1
+ n)*Sqrt[Sin[a + b*x]^2]))

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 (c \cos (a+b x))^n \, dx &=-\frac {(c \cos (a+b x))^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sin (a+b x)}{b c (1+n) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 64, normalized size = 0.93 \[ -\frac {\sqrt {\sin ^2(a+b x)} \cot (a+b x) (c \cos (a+b x))^n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^n,x]

[Out]

-(((c*Cos[a + b*x])^n*Cot[a + b*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[Sin[a + b
*x]^2])/(b*(1 + n)))

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c \cos \left (b x + a\right )\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*cos(b*x + a))^n, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (b x + a\right )\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^n, x)

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (b x +a \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (b x + a\right )\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^n, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,\cos \left (a+b\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(a + b*x))^n,x)

[Out]

int((c*cos(a + b*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos {\left (a + b x \right )}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*cos(a + b*x))**n, x)

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