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]

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

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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]]

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])

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*}

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Mathematica [A]  time = 0.09, 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)

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fricas [A]  time = 0.41, size = 26, normalized size = 0.84 \[ \frac {5 \, \cos \left (b x + a\right )^{7} - 7 \, \cos \left (b x + a\right )^{5}}{35 \, b} \]

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

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giac [A]  time = 0.19, size = 27, normalized size = 0.87 \[ \frac {\cos \left (b x + a\right )^{7}}{7 \, b} - \frac {\cos \left (b x + a\right )^{5}}{5 \, b} \]

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

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maple [A]  time = 0.02, size = 34, normalized size = 1.10 \[ \frac {-\frac {\left (\cos ^{5}\left (b x +a \right )\right ) \left (\sin ^{2}\left (b x +a \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (b x +a \right )\right )}{35}}{b} \]

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)

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maxima [A]  time = 0.31, size = 26, normalized size = 0.84 \[ \frac {5 \, \cos \left (b x + a\right )^{7} - 7 \, \cos \left (b x + a\right )^{5}}{35 \, b} \]

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

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mupad [B]  time = 0.38, size = 26, normalized size = 0.84 \[ -\frac {7\,{\cos \left (a+b\,x\right )}^5-5\,{\cos \left (a+b\,x\right )}^7}{35\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^4*sin(a + b*x)^3,x)

[Out]

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

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sympy [A]  time = 8.87, size = 46, normalized size = 1.48 \[ \begin {cases} - \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {2 \cos ^{7}{\left (a + b x \right )}}{35 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\relax (a )} \cos ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]

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))

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