3.102 \(\int \cos ^3(a+b x) \sin ^5(a+b x) \, dx\)

Optimal. Leaf size=31 \[ \frac{\sin ^6(a+b x)}{6 b}-\frac{\sin ^8(a+b x)}{8 b} \]

[Out]

Sin[a + b*x]^6/(6*b) - Sin[a + b*x]^8/(8*b)

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Rubi [A]  time = 0.0323852, 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 = {2564, 14} \[ \frac{\sin ^6(a+b x)}{6 b}-\frac{\sin ^8(a+b x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3*Sin[a + b*x]^5,x]

[Out]

Sin[a + b*x]^6/(6*b) - Sin[a + b*x]^8/(8*b)

Rule 2564

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])

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 ^3(a+b x) \sin ^5(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\sin ^6(a+b x)}{6 b}-\frac{\sin ^8(a+b x)}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.119553, size = 48, normalized size = 1.55 \[ \frac{-72 \cos (2 (a+b x))+12 \cos (4 (a+b x))+8 \cos (6 (a+b x))-3 \cos (8 (a+b x))}{3072 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^3*Sin[a + b*x]^5,x]

[Out]

(-72*Cos[2*(a + b*x)] + 12*Cos[4*(a + b*x)] + 8*Cos[6*(a + b*x)] - 3*Cos[8*(a + b*x)])/(3072*b)

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Maple [A]  time = 0.011, size = 52, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4} \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{8}}-{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{24}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3*sin(b*x+a)^5,x)

[Out]

1/b*(-1/8*cos(b*x+a)^4*sin(b*x+a)^4-1/12*cos(b*x+a)^4*sin(b*x+a)^2-1/24*cos(b*x+a)^4)

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Maxima [A]  time = 1.00128, size = 35, normalized size = 1.13 \begin{align*} -\frac{3 \, \sin \left (b x + a\right )^{8} - 4 \, \sin \left (b x + a\right )^{6}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/24*(3*sin(b*x + a)^8 - 4*sin(b*x + a)^6)/b

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Fricas [A]  time = 1.69535, size = 89, normalized size = 2.87 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{8} - 8 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/24*(3*cos(b*x + a)^8 - 8*cos(b*x + a)^6 + 6*cos(b*x + a)^4)/b

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Sympy [A]  time = 11.6092, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} \frac{\sin ^{8}{\left (a + b x \right )}}{24 b} + \frac{\sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{6 b} & \text{for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{3}{\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)**3*sin(b*x+a)**5,x)

[Out]

Piecewise((sin(a + b*x)**8/(24*b) + sin(a + b*x)**6*cos(a + b*x)**2/(6*b), Ne(b, 0)), (x*sin(a)**5*cos(a)**3,
True))

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Giac [A]  time = 1.14817, size = 35, normalized size = 1.13 \begin{align*} -\frac{3 \, \sin \left (b x + a\right )^{8} - 4 \, \sin \left (b x + a\right )^{6}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a)^5,x, algorithm="giac")

[Out]

-1/24*(3*sin(b*x + a)^8 - 4*sin(b*x + a)^6)/b