3.80 \(\int \cos ^7(a+b x) \sin ^4(a+b x) \, dx\)

Optimal. Leaf size=61 \[ -\frac{\sin ^{11}(a+b x)}{11 b}+\frac{\sin ^9(a+b x)}{3 b}-\frac{3 \sin ^7(a+b x)}{7 b}+\frac{\sin ^5(a+b x)}{5 b} \]

[Out]

Sin[a + b*x]^5/(5*b) - (3*Sin[a + b*x]^7)/(7*b) + Sin[a + b*x]^9/(3*b) - Sin[a + b*x]^11/(11*b)

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Rubi [A]  time = 0.039662, antiderivative size = 61, 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, 270} \[ -\frac{\sin ^{11}(a+b x)}{11 b}+\frac{\sin ^9(a+b x)}{3 b}-\frac{3 \sin ^7(a+b x)}{7 b}+\frac{\sin ^5(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^7*Sin[a + b*x]^4,x]

[Out]

Sin[a + b*x]^5/(5*b) - (3*Sin[a + b*x]^7)/(7*b) + Sin[a + b*x]^9/(3*b) - Sin[a + b*x]^11/(11*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 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 ^7(a+b x) \sin ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\sin ^5(a+b x)}{5 b}-\frac{3 \sin ^7(a+b x)}{7 b}+\frac{\sin ^9(a+b x)}{3 b}-\frac{\sin ^{11}(a+b x)}{11 b}\\ \end{align*}

Mathematica [A]  time = 0.200549, size = 47, normalized size = 0.77 \[ \frac{\sin ^5(a+b x) (3335 \cos (2 (a+b x))+910 \cos (4 (a+b x))+105 \cos (6 (a+b x))+3042)}{36960 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^7*Sin[a + b*x]^4,x]

[Out]

((3042 + 3335*Cos[2*(a + b*x)] + 910*Cos[4*(a + b*x)] + 105*Cos[6*(a + b*x)])*Sin[a + b*x]^5)/(36960*b)

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Maple [A]  time = 0.011, size = 78, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{8}}{11}}-{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{8}}{33}}+{\frac{\sin \left ( bx+a \right ) }{231} \left ({\frac{16}{5}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^7*sin(b*x+a)^4,x)

[Out]

1/b*(-1/11*sin(b*x+a)^3*cos(b*x+a)^8-1/33*sin(b*x+a)*cos(b*x+a)^8+1/231*(16/5+cos(b*x+a)^6+6/5*cos(b*x+a)^4+8/
5*cos(b*x+a)^2)*sin(b*x+a))

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Maxima [A]  time = 0.982321, size = 62, normalized size = 1.02 \begin{align*} -\frac{105 \, \sin \left (b x + a\right )^{11} - 385 \, \sin \left (b x + a\right )^{9} + 495 \, \sin \left (b x + a\right )^{7} - 231 \, \sin \left (b x + a\right )^{5}}{1155 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/1155*(105*sin(b*x + a)^11 - 385*sin(b*x + a)^9 + 495*sin(b*x + a)^7 - 231*sin(b*x + a)^5)/b

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Fricas [A]  time = 1.71904, size = 173, normalized size = 2.84 \begin{align*} \frac{{\left (105 \, \cos \left (b x + a\right )^{10} - 140 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{1155 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

1/1155*(105*cos(b*x + a)^10 - 140*cos(b*x + a)^8 + 5*cos(b*x + a)^6 + 6*cos(b*x + a)^4 + 8*cos(b*x + a)^2 + 16
)*sin(b*x + a)/b

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Sympy [A]  time = 59.6811, size = 88, normalized size = 1.44 \begin{align*} \begin{cases} \frac{16 \sin ^{11}{\left (a + b x \right )}}{1155 b} + \frac{8 \sin ^{9}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{105 b} + \frac{6 \sin ^{7}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{35 b} + \frac{\sin ^{5}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{5 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{7}{\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)**7*sin(b*x+a)**4,x)

[Out]

Piecewise((16*sin(a + b*x)**11/(1155*b) + 8*sin(a + b*x)**9*cos(a + b*x)**2/(105*b) + 6*sin(a + b*x)**7*cos(a
+ b*x)**4/(35*b) + sin(a + b*x)**5*cos(a + b*x)**6/(5*b), Ne(b, 0)), (x*sin(a)**4*cos(a)**7, True))

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Giac [A]  time = 1.16777, size = 111, normalized size = 1.82 \begin{align*} \frac{\sin \left (11 \, b x + 11 \, a\right )}{11264 \, b} + \frac{\sin \left (9 \, b x + 9 \, a\right )}{3072 \, b} - \frac{\sin \left (7 \, b x + 7 \, a\right )}{7168 \, b} - \frac{11 \, \sin \left (5 \, b x + 5 \, a\right )}{5120 \, b} - \frac{\sin \left (3 \, b x + 3 \, a\right )}{512 \, b} + \frac{7 \, \sin \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)^7*sin(b*x+a)^4,x, algorithm="giac")

[Out]

1/11264*sin(11*b*x + 11*a)/b + 1/3072*sin(9*b*x + 9*a)/b - 1/7168*sin(7*b*x + 7*a)/b - 11/5120*sin(5*b*x + 5*a
)/b - 1/512*sin(3*b*x + 3*a)/b + 7/512*sin(b*x + a)/b