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

Optimal. Leaf size=42 \[ -\frac {\cos (a+b x)}{b}+\frac {2 \cos ^3(a+b x)}{3 b}-\frac {\cos ^5(a+b x)}{5 b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 42, 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 {\cos ^5(a+b x)}{5 b}+\frac {2 \cos ^3(a+b x)}{3 b}-\frac {\cos (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^5,x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 44, normalized size = 1.05 \begin {gather*} -\frac {5 \cos (a+b x)}{8 b}+\frac {5 \cos (3 (a+b x))}{48 b}-\frac {\cos (5 (a+b x))}{80 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^5,x]

[Out]

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

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Maple [A]
time = 0.10, size = 32, normalized size = 0.76

method	result	size
derivativedivides	\(-\frac {\left (\frac {8}{3}+\sin ^{4}\left (b x +a \right )+\frac {4 \left (\sin ^{2}\left (b x +a \right )\right )}{3}\right ) \cos \left (b x +a \right )}{5 b}\)	\(32\)
default	\(-\frac {\left (\frac {8}{3}+\sin ^{4}\left (b x +a \right )+\frac {4 \left (\sin ^{2}\left (b x +a \right )\right )}{3}\right ) \cos \left (b x +a \right )}{5 b}\)	\(32\)
risch	\(-\frac {5 \cos \left (b x +a \right )}{8 b}-\frac {\cos \left (5 b x +5 a \right )}{80 b}+\frac {5 \cos \left (3 b x +3 a \right )}{48 b}\)	\(41\)
norman	\(\frac {-\frac {16}{15 b}-\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {32 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{5}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 34, normalized size = 0.81 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 34, normalized size = 0.81 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.25, size = 60, normalized size = 1.43 \begin {gather*} \begin {cases} - \frac {\sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {8 \cos ^{5}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 5.25, size = 38, normalized size = 0.90 \begin {gather*} -\frac {\cos \left (b x + a\right )^{5}}{5 \, b} + \frac {2 \, \cos \left (b x + a\right )^{3}}{3 \, b} - \frac {\cos \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.36, size = 32, normalized size = 0.76 \begin {gather*} -\frac {\frac {{\cos \left (a+b\,x\right )}^5}{5}-\frac {2\,{\cos \left (a+b\,x\right )}^3}{3}+\cos \left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^5,x)

[Out]

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

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