3.89 \(\int \cos ^6(a+b x) \sin ^4(a+b x) \, dx\)

Optimal. Leaf size=111 \[ -\frac{\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}-\frac{3 \sin (a+b x) \cos ^7(a+b x)}{80 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{160 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{128 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{256 b}+\frac{3 x}{256} \]

[Out]

(3*x)/256 + (3*Cos[a + b*x]*Sin[a + b*x])/(256*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^5*Si
n[a + b*x])/(160*b) - (3*Cos[a + b*x]^7*Sin[a + b*x])/(80*b) - (Cos[a + b*x]^7*Sin[a + b*x]^3)/(10*b)

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Rubi [A]  time = 0.096975, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}-\frac{3 \sin (a+b x) \cos ^7(a+b x)}{80 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{160 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{128 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{256 b}+\frac{3 x}{256} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^6*Sin[a + b*x]^4,x]

[Out]

(3*x)/256 + (3*Cos[a + b*x]*Sin[a + b*x])/(256*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^5*Si
n[a + b*x])/(160*b) - (3*Cos[a + b*x]^7*Sin[a + b*x])/(80*b) - (Cos[a + b*x]^7*Sin[a + b*x]^3)/(10*b)

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx &=-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3}{10} \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3}{80} \int \cos ^6(a+b x) \, dx\\ &=\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{1}{32} \int \cos ^4(a+b x) \, dx\\ &=\frac{\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3}{128} \int \cos ^2(a+b x) \, dx\\ &=\frac{3 \cos (a+b x) \sin (a+b x)}{256 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac{3 \int 1 \, dx}{256}\\ &=\frac{3 x}{256}+\frac{3 \cos (a+b x) \sin (a+b x)}{256 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac{3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac{\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}\\ \end{align*}

Mathematica [A]  time = 0.195666, size = 62, normalized size = 0.56 \[ \frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{10240 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^6*Sin[a + b*x]^4,x]

[Out]

(120*b*x + 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] - 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] + 2*Sin[10*(a
+ b*x)])/(10240*b)

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Maple [A]  time = 0.011, size = 82, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{10}}-{\frac{3\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( bx+a \right ) }{160} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( bx+a \right ) }{8}} \right ) }+{\frac{3\,bx}{256}}+{\frac{3\,a}{256}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^6*sin(b*x+a)^4,x)

[Out]

1/b*(-1/10*sin(b*x+a)^3*cos(b*x+a)^7-3/80*sin(b*x+a)*cos(b*x+a)^7+1/160*(cos(b*x+a)^5+5/4*cos(b*x+a)^3+15/8*co
s(b*x+a))*sin(b*x+a)+3/256*b*x+3/256*a)

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Maxima [A]  time = 1.00769, size = 65, normalized size = 0.59 \begin{align*} \frac{32 \, \sin \left (2 \, b x + 2 \, a\right )^{5} + 120 \, b x + 120 \, a + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right )}{10240 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

1/10240*(32*sin(2*b*x + 2*a)^5 + 120*b*x + 120*a + 5*sin(8*b*x + 8*a) - 40*sin(4*b*x + 4*a))/b

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Fricas [A]  time = 1.71907, size = 180, normalized size = 1.62 \begin{align*} \frac{15 \, b x +{\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{1280 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

1/1280*(15*b*x + (128*cos(b*x + a)^9 - 176*cos(b*x + a)^7 + 8*cos(b*x + a)^5 + 10*cos(b*x + a)^3 + 15*cos(b*x
+ a))*sin(b*x + a))/b

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Sympy [A]  time = 32.5377, size = 231, normalized size = 2.08 \begin{align*} \begin{cases} \frac{3 x \sin ^{10}{\left (a + b x \right )}}{256} + \frac{15 x \sin ^{8}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{256} + \frac{15 x \sin ^{6}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{128} + \frac{15 x \sin ^{4}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{128} + \frac{15 x \sin ^{2}{\left (a + b x \right )} \cos ^{8}{\left (a + b x \right )}}{256} + \frac{3 x \cos ^{10}{\left (a + b x \right )}}{256} + \frac{3 \sin ^{9}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{256 b} + \frac{7 \sin ^{7}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} + \frac{\sin ^{5}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{10 b} - \frac{7 \sin ^{3}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} - \frac{3 \sin{\left (a + b x \right )} \cos ^{9}{\left (a + b x \right )}}{256 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{6}{\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)**6*sin(b*x+a)**4,x)

[Out]

Piecewise((3*x*sin(a + b*x)**10/256 + 15*x*sin(a + b*x)**8*cos(a + b*x)**2/256 + 15*x*sin(a + b*x)**6*cos(a +
b*x)**4/128 + 15*x*sin(a + b*x)**4*cos(a + b*x)**6/128 + 15*x*sin(a + b*x)**2*cos(a + b*x)**8/256 + 3*x*cos(a
+ b*x)**10/256 + 3*sin(a + b*x)**9*cos(a + b*x)/(256*b) + 7*sin(a + b*x)**7*cos(a + b*x)**3/(128*b) + sin(a +
b*x)**5*cos(a + b*x)**5/(10*b) - 7*sin(a + b*x)**3*cos(a + b*x)**7/(128*b) - 3*sin(a + b*x)*cos(a + b*x)**9/(2
56*b), Ne(b, 0)), (x*sin(a)**4*cos(a)**6, True))

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Giac [A]  time = 1.14561, size = 100, normalized size = 0.9 \begin{align*} \frac{3}{256} \, x + \frac{\sin \left (10 \, b x + 10 \, a\right )}{5120 \, b} + \frac{\sin \left (8 \, b x + 8 \, a\right )}{2048 \, b} - \frac{\sin \left (6 \, b x + 6 \, a\right )}{1024 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{256 \, b} + \frac{\sin \left (2 \, b x + 2 \, a\right )}{512 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="giac")

[Out]

3/256*x + 1/5120*sin(10*b*x + 10*a)/b + 1/2048*sin(8*b*x + 8*a)/b - 1/1024*sin(6*b*x + 6*a)/b - 1/256*sin(4*b*
x + 4*a)/b + 1/512*sin(2*b*x + 2*a)/b