3.67 \(\int \cos ^4(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=31 \[ \frac{\cos ^7(a+b x)}{7 b}-\frac{\cos ^5(a+b x)}{5 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.034393, antiderivative size = 31, 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, 14} \[ \frac{\cos ^7(a+b x)}{7 b}-\frac{\cos ^5(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4*Sin[a + b*x]^3,x]

[Out]

-Cos[a + b*x]^5/(5*b) + Cos[a + b*x]^7/(7*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cos ^4(a+b x) \sin ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos ^5(a+b x)}{5 b}+\frac{\cos ^7(a+b x)}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0906027, size = 27, normalized size = 0.87 \[ \frac{\cos ^5(a+b x) (5 \cos (2 (a+b x))-9)}{70 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4*Sin[a + b*x]^3,x]

[Out]

(Cos[a + b*x]^5*(-9 + 5*Cos[2*(a + b*x)]))/(70*b)

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 34, 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 ) ^{2}}{7}}-{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{35}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4*sin(b*x+a)^3,x)

[Out]

1/b*(-1/7*cos(b*x+a)^5*sin(b*x+a)^2-2/35*cos(b*x+a)^5)

________________________________________________________________________________________

Maxima [A]  time = 0.980987, size = 35, normalized size = 1.13 \begin{align*} \frac{5 \, \cos \left (b x + a\right )^{7} - 7 \, \cos \left (b x + a\right )^{5}}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/35*(5*cos(b*x + a)^7 - 7*cos(b*x + a)^5)/b

________________________________________________________________________________________

Fricas [A]  time = 1.6658, size = 62, normalized size = 2. \begin{align*} \frac{5 \, \cos \left (b x + a\right )^{7} - 7 \, \cos \left (b x + a\right )^{5}}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/35*(5*cos(b*x + a)^7 - 7*cos(b*x + a)^5)/b

________________________________________________________________________________________

Sympy [A]  time = 7.05412, size = 46, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{\sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac{2 \cos ^{7}{\left (a + b x \right )}}{35 b} & \text{for}\: b \neq 0 \\x \sin ^{3}{\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)**3,x)

[Out]

Piecewise((-sin(a + b*x)**2*cos(a + b*x)**5/(5*b) - 2*cos(a + b*x)**7/(35*b), Ne(b, 0)), (x*sin(a)**3*cos(a)**
4, True))

________________________________________________________________________________________

Giac [A]  time = 1.16608, size = 36, normalized size = 1.16 \begin{align*} \frac{\cos \left (b x + a\right )^{7}}{7 \, b} - \frac{\cos \left (b x + a\right )^{5}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/7*cos(b*x + a)^7/b - 1/5*cos(b*x + a)^5/b