3.101 \(\int \cos ^4(a+b x) \sin ^5(a+b x) \, dx\)

Optimal. Leaf size=46 \[ -\frac{\cos ^9(a+b x)}{9 b}+\frac{2 \cos ^7(a+b x)}{7 b}-\frac{\cos ^5(a+b x)}{5 b} \]

[Out]

-Cos[a + b*x]^5/(5*b) + (2*Cos[a + b*x]^7)/(7*b) - Cos[a + b*x]^9/(9*b)

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Rubi [A]  time = 0.0364733, 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} \[ -\frac{\cos ^9(a+b x)}{9 b}+\frac{2 \cos ^7(a+b x)}{7 b}-\frac{\cos ^5(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4*Sin[a + b*x]^5,x]

[Out]

-Cos[a + b*x]^5/(5*b) + (2*Cos[a + b*x]^7)/(7*b) - Cos[a + b*x]^9/(9*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 ^4(a+b x) \sin ^5(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos ^5(a+b x)}{5 b}+\frac{2 \cos ^7(a+b x)}{7 b}-\frac{\cos ^9(a+b x)}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.143163, size = 37, normalized size = 0.8 \[ \frac{\cos ^5(a+b x) (220 \cos (2 (a+b x))-35 \cos (4 (a+b x))-249)}{2520 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4*Sin[a + b*x]^5,x]

[Out]

(Cos[a + b*x]^5*(-249 + 220*Cos[2*(a + b*x)] - 35*Cos[4*(a + b*x)]))/(2520*b)

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Maple [A]  time = 0.015, size = 52, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5} \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{9}}-{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{5} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{63}}-{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{315}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4*sin(b*x+a)^5,x)

[Out]

1/b*(-1/9*cos(b*x+a)^5*sin(b*x+a)^4-4/63*cos(b*x+a)^5*sin(b*x+a)^2-8/315*cos(b*x+a)^5)

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Maxima [A]  time = 0.996456, size = 49, normalized size = 1.07 \begin{align*} -\frac{35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/315*(35*cos(b*x + a)^9 - 90*cos(b*x + a)^7 + 63*cos(b*x + a)^5)/b

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Fricas [A]  time = 1.71003, size = 95, normalized size = 2.07 \begin{align*} -\frac{35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/315*(35*cos(b*x + a)^9 - 90*cos(b*x + a)^7 + 63*cos(b*x + a)^5)/b

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Sympy [A]  time = 19.9037, size = 68, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{\sin ^{4}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac{4 \sin ^{2}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{35 b} - \frac{8 \cos ^{9}{\left (a + b x \right )}}{315 b} & \text{for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{4}{\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)**4*sin(b*x+a)**5,x)

[Out]

Piecewise((-sin(a + b*x)**4*cos(a + b*x)**5/(5*b) - 4*sin(a + b*x)**2*cos(a + b*x)**7/(35*b) - 8*cos(a + b*x)*
*9/(315*b), Ne(b, 0)), (x*sin(a)**5*cos(a)**4, True))

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Giac [A]  time = 1.13285, size = 92, normalized size = 2. \begin{align*} -\frac{\cos \left (9 \, b x + 9 \, a\right )}{2304 \, b} + \frac{\cos \left (7 \, b x + 7 \, a\right )}{1792 \, b} + \frac{\cos \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac{\cos \left (3 \, b x + 3 \, a\right )}{192 \, b} - \frac{3 \, \cos \left (b x + a\right )}{128 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4*sin(b*x+a)^5,x, algorithm="giac")

[Out]

-1/2304*cos(9*b*x + 9*a)/b + 1/1792*cos(7*b*x + 7*a)/b + 1/320*cos(5*b*x + 5*a)/b - 1/192*cos(3*b*x + 3*a)/b -
3/128*cos(b*x + a)/b