\(\int \cos (a+b x) \sin ^5(a+b x) \, dx\) [104]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin ^6(a+b x)}{6 b} \]

[Out]

1/6*sin(b*x+a)^6/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2644, 30} \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin ^6(a+b x)}{6 b} \]

[In]

Int[Cos[a + b*x]*Sin[a + b*x]^5,x]

[Out]

Sin[a + b*x]^6/(6*b)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^5 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\sin ^6(a+b x)}{6 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin ^6(a+b x)}{6 b} \]

[In]

Integrate[Cos[a + b*x]*Sin[a + b*x]^5,x]

[Out]

Sin[a + b*x]^6/(6*b)

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\sin ^{6}\left (b x +a \right )}{6 b}\)	\(14\)
default	\(\frac {\sin ^{6}\left (b x +a \right )}{6 b}\)	\(14\)
norman	\(\frac {32 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{6}}\)	\(32\)
parallelrisch	\(\frac {6 \cos \left (4 b x +4 a \right )+10-15 \cos \left (2 b x +2 a \right )-\cos \left (6 b x +6 a \right )}{192 b}\)	\(41\)
risch	\(-\frac {\cos \left (6 b x +6 a \right )}{192 b}+\frac {\cos \left (4 b x +4 a \right )}{32 b}-\frac {5 \cos \left (2 b x +2 a \right )}{64 b}\)	\(44\)

[In]

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

[Out]

1/6*sin(b*x+a)^6/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2}}{6 \, b} \]

[In]

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

[Out]

-1/6*(cos(b*x + a)^6 - 3*cos(b*x + a)^4 + 3*cos(b*x + a)^2)/b

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\begin {cases} \frac {\sin ^{6}{\left (a + b x \right )}}{6 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((sin(a + b*x)**6/(6*b), Ne(b, 0)), (x*sin(a)**5*cos(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin \left (b x + a\right )^{6}}{6 \, b} \]

[In]

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

[Out]

1/6*sin(b*x + a)^6/b

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin \left (b x + a\right )^{6}}{6 \, b} \]

[In]

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

[Out]

1/6*sin(b*x + a)^6/b

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {{\sin \left (a+b\,x\right )}^6}{6\,b} \]

[In]

int(cos(a + b*x)*sin(a + b*x)^5,x)

[Out]

sin(a + b*x)^6/(6*b)