3.1.5 \(\int \cos ^5(a+b x) \, dx\) [5]

Optimal. Leaf size=41 \[ \frac {\sin (a+b x)}{b}-\frac {2 \sin ^3(a+b x)}{3 b}+\frac {\sin ^5(a+b x)}{5 b} \]

[Out]

sin(b*x+a)/b-2/3*sin(b*x+a)^3/b+1/5*sin(b*x+a)^5/b

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \begin {gather*} \frac {\sin ^5(a+b x)}{5 b}-\frac {2 \sin ^3(a+b x)}{3 b}+\frac {\sin (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^5,x]

[Out]

Sin[a + b*x]/b - (2*Sin[a + b*x]^3)/(3*b) + Sin[a + b*x]^5/(5*b)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \cos ^5(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac {\sin (a+b x)}{b}-\frac {2 \sin ^3(a+b x)}{3 b}+\frac {\sin ^5(a+b x)}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 44, normalized size = 1.07 \begin {gather*} \frac {5 \sin (a+b x)}{8 b}+\frac {5 \sin (3 (a+b x))}{48 b}+\frac {\sin (5 (a+b x))}{80 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^5,x]

[Out]

(5*Sin[a + b*x])/(8*b) + (5*Sin[3*(a + b*x)])/(48*b) + Sin[5*(a + b*x)]/(80*b)

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Maple [A]
time = 0.07, size = 32, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\left (\frac {8}{3}+\cos ^{4}\left (b x +a \right )+\frac {4 \left (\cos ^{2}\left (b x +a \right )\right )}{3}\right ) \sin \left (b x +a \right )}{5 b}\)	\(32\)
default	\(\frac {\left (\frac {8}{3}+\cos ^{4}\left (b x +a \right )+\frac {4 \left (\cos ^{2}\left (b x +a \right )\right )}{3}\right ) \sin \left (b x +a \right )}{5 b}\)	\(32\)
risch	\(\frac {5 \sin \left (b x +a \right )}{8 b}+\frac {\sin \left (5 b x +5 a \right )}{80 b}+\frac {5 \sin \left (3 b x +3 a \right )}{48 b}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/5/b*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)

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Maxima [A]
time = 0.28, size = 34, normalized size = 0.83 \begin {gather*} \frac {3 \, \sin \left (b x + a\right )^{5} - 10 \, \sin \left (b x + a\right )^{3} + 15 \, \sin \left (b x + a\right )}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^5,x, algorithm="maxima")

[Out]

1/15*(3*sin(b*x + a)^5 - 10*sin(b*x + a)^3 + 15*sin(b*x + a))/b

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Fricas [A]
time = 0.35, size = 33, normalized size = 0.80 \begin {gather*} \frac {{\left (3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^5,x, algorithm="fricas")

[Out]

1/15*(3*cos(b*x + a)^4 + 4*cos(b*x + a)^2 + 8)*sin(b*x + a)/b

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Sympy [A]
time = 0.23, size = 58, normalized size = 1.41 \begin {gather*} \begin {cases} \frac {8 \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac {4 \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac {\sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cos ^{5}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**5,x)

[Out]

Piecewise((8*sin(a + b*x)**5/(15*b) + 4*sin(a + b*x)**3*cos(a + b*x)**2/(3*b) + sin(a + b*x)*cos(a + b*x)**4/b
, Ne(b, 0)), (x*cos(a)**5, True))

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Giac [A]
time = 0.46, size = 34, normalized size = 0.83 \begin {gather*} \frac {3 \, \sin \left (b x + a\right )^{5} - 10 \, \sin \left (b x + a\right )^{3} + 15 \, \sin \left (b x + a\right )}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^5,x, algorithm="giac")

[Out]

1/15*(3*sin(b*x + a)^5 - 10*sin(b*x + a)^3 + 15*sin(b*x + a))/b

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Mupad [B]
time = 0.10, size = 31, normalized size = 0.76 \begin {gather*} \frac {\frac {{\sin \left (a+b\,x\right )}^5}{5}-\frac {2\,{\sin \left (a+b\,x\right )}^3}{3}+\sin \left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^5,x)

[Out]

(sin(a + b*x) - (2*sin(a + b*x)^3)/3 + sin(a + b*x)^5/5)/b

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