3.99 \(\int \cos ^6(a+b x) \sin ^5(a+b x) \, dx\)

Optimal. Leaf size=46 \[ -\frac{\cos ^{11}(a+b x)}{11 b}+\frac{2 \cos ^9(a+b x)}{9 b}-\frac{\cos ^7(a+b x)}{7 b} \]

[Out]

-Cos[a + b*x]^7/(7*b) + (2*Cos[a + b*x]^9)/(9*b) - Cos[a + b*x]^11/(11*b)

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Rubi [A]  time = 0.0356549, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2565, 270} \[ -\frac{\cos ^{11}(a+b x)}{11 b}+\frac{2 \cos ^9(a+b x)}{9 b}-\frac{\cos ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^6*Sin[a + b*x]^5,x]

[Out]

-Cos[a + b*x]^7/(7*b) + (2*Cos[a + b*x]^9)/(9*b) - Cos[a + b*x]^11/(11*b)

Rule 2565

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^6(a+b x) \sin ^5(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos ^7(a+b x)}{7 b}+\frac{2 \cos ^9(a+b x)}{9 b}-\frac{\cos ^{11}(a+b x)}{11 b}\\ \end{align*}

Mathematica [A]  time = 0.272674, size = 37, normalized size = 0.8 \[ \frac{\cos ^7(a+b x) (364 \cos (2 (a+b x))-63 \cos (4 (a+b x))-365)}{5544 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^6*Sin[a + b*x]^5,x]

[Out]

(Cos[a + b*x]^7*(-365 + 364*Cos[2*(a + b*x)] - 63*Cos[4*(a + b*x)]))/(5544*b)

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Maple [A]  time = 0.013, size = 52, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{7} \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{11}}-{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{7} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{99}}-{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{693}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^6*sin(b*x+a)^5,x)

[Out]

1/b*(-1/11*cos(b*x+a)^7*sin(b*x+a)^4-4/99*cos(b*x+a)^7*sin(b*x+a)^2-8/693*cos(b*x+a)^7)

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Maxima [A]  time = 0.988885, size = 49, normalized size = 1.07 \begin{align*} -\frac{63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}}{693 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/693*(63*cos(b*x + a)^11 - 154*cos(b*x + a)^9 + 99*cos(b*x + a)^7)/b

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Fricas [A]  time = 1.74097, size = 97, normalized size = 2.11 \begin{align*} -\frac{63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}}{693 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/693*(63*cos(b*x + a)^11 - 154*cos(b*x + a)^9 + 99*cos(b*x + a)^7)/b

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Sympy [A]  time = 54.0469, size = 68, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{\sin ^{4}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{7 b} - \frac{4 \sin ^{2}{\left (a + b x \right )} \cos ^{9}{\left (a + b x \right )}}{63 b} - \frac{8 \cos ^{11}{\left (a + b x \right )}}{693 b} & \text{for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{6}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**6*sin(b*x+a)**5,x)

[Out]

Piecewise((-sin(a + b*x)**4*cos(a + b*x)**7/(7*b) - 4*sin(a + b*x)**2*cos(a + b*x)**9/(63*b) - 8*cos(a + b*x)*
*11/(693*b), Ne(b, 0)), (x*sin(a)**5*cos(a)**6, True))

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Giac [B]  time = 1.13953, size = 111, normalized size = 2.41 \begin{align*} -\frac{\cos \left (11 \, b x + 11 \, a\right )}{11264 \, b} - \frac{\cos \left (9 \, b x + 9 \, a\right )}{9216 \, b} + \frac{5 \, \cos \left (7 \, b x + 7 \, a\right )}{7168 \, b} + \frac{\cos \left (5 \, b x + 5 \, a\right )}{1024 \, b} - \frac{5 \, \cos \left (3 \, b x + 3 \, a\right )}{1536 \, b} - \frac{5 \, \cos \left (b x + a\right )}{512 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^5,x, algorithm="giac")

[Out]

-1/11264*cos(11*b*x + 11*a)/b - 1/9216*cos(9*b*x + 9*a)/b + 5/7168*cos(7*b*x + 7*a)/b + 1/1024*cos(5*b*x + 5*a
)/b - 5/1536*cos(3*b*x + 3*a)/b - 5/512*cos(b*x + a)/b